# Continuous Differentiability of the Value Function of Semilinear Parabolic
Infinite Time Horizon Optimal Control Problems on $L^{2}(\Omega)$ under
Control Constraints ††thanks: The authors were supported by the ERC advanced
grant 668998 (OCLOC) under the EU’s H2020 research program.

Karl Kunisch Institute for Mathematics and Scientific Computing, University of
Graz, Heinrichstrasse 36, A-8010 Graz, Austria, and Radon Institute, Austrian
Academy of Science, Linz, Austria. (karl.kunisch@uni-graz.at). Buddhika
Priyasad Institute for Mathematics and Scientific Computing, University of
Graz, Heinrichstrasse 36, A-8010 Graz, Austria. (b.sembukutti-liyanage@uni-
graz.at).

###### Abstract

An abstract framework guaranteeing the local continuous differentiability of
the value function associated with optimal stabilization problems subject to
abstract semilinear parabolic equations subject to a norm constraint on the
controls is established. It guarantees that the value function satisfies the
associated Hamilton-Jacobi-Bellman equation in the classical sense. The
applicability of the developed framework is demonstrated for specific
semilinear parabolic equations.

##  1 Introduction.

Continuous differentiability of the value function with respect to the initial
datum is an important problem in optimal feedback control theory. Indeed, if
the value function is $C^{1}$ then it is the solution of a Hamilton Jacobi
Bellman (HJB) equation and its negative gradient can be used to define on
optimal state feedback law. The subject matter of this paper addresses local
continuous differentiability of the value function $\mathcal{V}$ for infinite
horizon optimal control problems subject to semilinear parabolic control
problems and norm constraints on the control. Such problems are intimately
related to stabilization problems which are often cast as infinite horizon
optimal control problems. Investigating infinite horizon problems constitutes
one of the specificities of this paper. Another one is the fact that we focus
on the differentiability of $\mathcal{V}$ on (subsets of) $L^{2}(\Omega)$.
Thus we need to consider the semilinear equations with initial data $y_{0}\in
L^{2}(\Omega)$. As a consequence the solutions of the semilinear equations
only enjoy low Sobolev-space regularity. This restricts the class of
nonlinearities, compared to those which are admissible if the states are in
$L^{\infty}((0,\infty)\times\Omega)$, which is the situation typically
addressed in the literature on optimal control [Cas] and [Tro2]. The latter
necessitates to take the initial conditions in spaces strictly smaller than
$L^{2}(\Omega)$. Here we consider $L^{2}(\Omega)$, first due to intrinsic
interest, secondly because ultimately the HJB equation should be solved
numerically, which is easier in an $L^{2}(\Omega)$ setting than in other
topologies, like $H^{1}(\Omega)$. Let us also recall that one of the
approaches to solve the HJB equation is given by the policy iteration. It
assumes that the value function is $C^{1}$.

The underlying analysis demands stability and sensitivity analysis of infinite
dimensional optimal control problems subject to nonlinear equations. For this
purpose we utilize the theory of generalized equations as established by [Don]
and [Rob]. It involves first order approximations of the state and adjoint
equations, which lead to restrictions on the class of nonlinearities which can
be admitted. We refer to the section on examples in this respect.

The current investigations are to some degree a continuation of work the first
author’s work on optimal feedback control for infinite dimensional systems. In
[BKP1, BKP2, BKP3] Taylor approximations of the value function for problems
with a concrete structure, namely, bilinear control systems, and the Navier
Stokes equations were investigated and differentiability of the value function
was obtained as a by product. In these investigations norm constraints were
not considered. Here we admit norm constraints and we focus on semilinear
equations. Let us also notice that the systems investigated in [BKP1, BKP2,
BKP3] share the property that the second derivatives with respect to the state
variable of the nonlinearity in the state equation do not depend on the state
itself anymore.

Let us also compare our work to the developments in the field of parametric
sensitivity analysis of semilinear parabolic equations under control
constraints. There are many papers focusing on stability and sensitivity
analysis of finite time horizon problems along with pointwise control
constraints, see e.g. [BM, GHH, Gri, GV, Mal, MT, Tro1, Wac], and the
literature there. First, of these papers, except for [GV], consider the case
with initial data in $L^{2}(\Omega)$. In [GV] again the third derivative of
the nonlinearity is zero. Secondly, all of them consider the finite horizon
case. Since we treat infinite horizon problems we have to guarantee
stabilizability (for small initial data) under control constraints. Then we
use a fixed point argument to obtain well-posedness of the system. Well-
posedness and stability with respect to parameters of the adjoint equation is
significantly more involved for infinite horizon problems than for finite
horizon problems. It requires techniques, differently from those used in the
finite horizon case. Another aspect is the proper characterization of the
adjoint state at $t=\infty$.

In the finite dimensional case, there is, of course a tremendous amount of
work on the treatment of the value function if it is not $C^{1}$. Fewer papers
concentrate on the case where the value function enjoys smoothness properties.
We mention [Goe] and [CF] in this respect.

In order to achieve the goal we desire, we lay out the following setup. In
Section 2, we consider an abstract parametric optimization problem with an
equality constraint and another convex constraint. Existence of an optimal
solution, of a multiplier associate to the equality constraint, and Lipschitz
stability of the component of the state variable which lies in the complement
of the kernel of the linearized constraint will be established. This result is
necessary but not sufficient for the further developments, since stability is
obtained in a norm which is too weak and since the stability estimate does not
involve the component in the kernel of the linearized constraint and the
multiplier, i.e. the adjoint states, yet. At the level of Section 2 this
remains as Assumption (H7). In Section 3 we specify the concrete optimal
stabilization problem and a set of conditions, most importantly on the
nonlinearity of the state equation, under which Assumption (H7) can be
established, for initial data $y_{0}\in L^{2}(\Omega)$. Section 3 also
contains a summary of the main results of this paper. They are stated as
theorems with a little stronger assumptions than eventually necessary, for the
saker of easing the presentation. Section 4 is dedicated to the proof of
verifying the assumptions of the general setup of Section 2 for the concrete
optimal control problem stated in Section 3. As conclusion we obtain the
Lipschitz continuity in the appropriate norms of the all variables appearing
in the optimality system with respect to the parameter of interest, which is
the initial condition $y_{0}$, in our case. Since our analysis is a local one
involving second order optimality conditions, solutions to the optimality
system are related to local solutions to the optimal control problem. As a
corollary to these results we obtain that the local value function is Fréchet
differentiable. In Section 5, we show that in the neighborhood of global
solutions the value function $\mathcal{V}$ satisfies the Hamilton-Jacobi-
Bellman (HJB) equation in the strong sense. Finally, Section 6 is devoted to
demonstrating that the developed framework is applicable for some concrete
examples, namely for linear systems, Fisher’s equations, and parabolic
equations with global Lipschitz nonlinearities. All our results require a
smallness assumption on the initial conditions $y_{0}$. Two aspects need to be
taken into consideration in this respect. First $y_{0}$ has to be sufficiently
small so that the controlled system is stable. Secondly a second order
optimality condition is needed. For this to hold a sufficient condition is
provided by smallness of the adjoint state, which in turn can be implied by
smallness of $y_{0}$. We stress that these two issues are of related, but
independent nature.

##  2 Lipschitz stability for an abstract optimization problem.

Here we present a stability result for an abstract, infinite dimensional
optimization problem which will be the building block for the results below.
This result is geared towards exploiting the specific nature of optimization
problem with differential equations as constraints. First existence of a dual
variable will result from a regular point condition. Subsequently the
Lipschitz stability result is obtained in two steps. In the first one, we rely
on the relationship between the linearized optimality conditions and an
associated linear-quadratic optimal optimization problem, with an extra convex
constraint. This approach is useful since it provides the existence of
solutions to the linearized system on the basis of variational techniques.
However it dictates a certain norms for the involved quantities. These norms
are too weak for our goal of obtaining Lipschitz continuity of the adjoint
variables in such a manner that differentiability of the cost with respect to
the initial conditions can be argued. Therefore, in a second step we exploit
the specific structure of the optimality systems, using the fact the it is
related to a parabolic optimal control problem, to obtain the Lipschitz
continuity in the stronger norms. This two step approach is also present in
some of the earlier work on stability and sensitivity analysis which was
quoted in the introduction. But due to that fact these papers considered
finite horizon problems it came as a byproduct which improved the regularity
of the adjoints. In our work it is essential to reach our goal. This is why we
decided to formalize this two step approach which was not done in earlier
work.

Concretely, we consider the optimization problem

$\begin{cases}\min\ f(x)\\\ e(x,q)=0,\ x\in C.\end{cases}$ ($P_{q}$)

with a parameter dependent equality constraint, and a general constraint
described by $x\in C$, where $C$ is a closed convex subset of a real Hilbert
space $X$. Further, $W$ is a real Hilbert space and $P$ is a normed linear
space. In the application that we have in mind, the parameter $q$ will appear
as the initial condition in the dynamical system. The following Assumption
(H1) is assumed to hold throughout.

######  Assumption H1.

$q_{0}\in P$ is a nominal reference parameter,
$x_{0}$ is a local solution ($P_{q_{0}}$),
$f:X\longrightarrow\mathbb{R}^{+}$ is twice continuously differentiable in a
neighborhood of $x_{0}$,
$e:X\times P\longrightarrow W$ is continuous, and twice continuously
differentiable w.r.t. $x$, with first and second derivative Lipschitz
continuous in a neighborhood of $(x_{0},q_{0})$.

The derivatives with respect to $x$ will be denoted by primes and the
derivatives w.r.t. $y$ and $u$ later on, are denoted by subscripts. They are
all considered in the sense of Lebesgue derivatives.

We introduce the Lagrangian ${\mathcal{L}}:X\times P\times
W^{*}\longrightarrow\mathbb{R}$ associated to ($P_{q}$) by

${\mathcal{L}}(x,q,\lambda)=f(x)+\langle\lambda,e(x,q)\rangle_{W^{*},W}.$
(2.1)

Next further relevant assumptions are introduced:

######  Assumption H2 (regular point condition).

$0\in\text{int }e^{\prime}(x_{0},q_{0})(C-x_{0}),$

where $int$ denotes the interior in the $W$ topology. This regularity
condition implies the existence of a Lagrange multiplier $\lambda_{0}\in
W^{*}$, see e.g. [MZ] such that the following first order condition holds:

$\begin{cases}\langle{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0}),c-x_{0}\rangle_{X^{*},X}\geq
0,\ \forall c\in C,\\\ e(x_{0},q_{0})=0.\end{cases}$ (2.2)

It is equivalent to

$\displaystyle\begin{cases}0\in{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0})+\partial\mathbf{I}_{C}(x_{0}),\quad&\text{in
}X^{*},\\\ e(x_{0},q_{0})=0,\quad&\text{in }W,\end{cases}$ (2.3)

where $\partial\mathbf{I}_{C}(x)$ denotes the subdifferential of the indicator
function of the set $C$ at $x\in X$.

Let $\displaystyle A\in{\mathcal{L}}(X,X^{*})$ denote the operator
representation of
$\displaystyle{\mathcal{L}}^{\prime\prime}(x_{0},q_{0},\lambda_{0})$, i.e.

$\langle
Ax_{1},x_{2}\rangle_{X^{*},X}={\mathcal{L}}^{\prime\prime}(x_{0},q_{0},\lambda_{0})(x_{1},x_{2})$
(2.4)

and define

$E=e^{\prime}(x_{0},q_{0})\in{\mathcal{L}}(X,W).$ (2.5)

We further require

######  Assumption H3 (positive definiteness).

$\exists\kappa>0:\ \langle Ax,x\rangle_{X^{*},X}\geq\kappa\left\lVert
x\right\rVert^{2}_{X},\ \forall x\in\text{ker }E.$

The stability result of $(x_{0},\lambda_{0})$ with respect to perturbation of
$q$ at $q_{0}$ will be based on Robinson’s strong regularity condition which
involves the following linearized form of the optimality condition,

$\displaystyle\begin{cases}0\in{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0})+A(x-x_{0})+E^{*}(\lambda-\lambda_{0})+\partial\mathbf{I}_{C}(x)&\text{in
}X^{*},\\\ 0=e(x_{0},q_{0})+E(x-x_{0})&\text{in }W.\end{cases}$ (2.6)

We define a multivalued operator $\displaystyle{\mathcal{T}}:X\times
W^{*}\longrightarrow X^{*}\times W$ by

${\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}f^{\prime}(x_{0})-Ax_{0}\\\
-Ex_{0}\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix},$ (2.7)

and observe that (2.6) is equivalent to

$\displaystyle 0\in{\mathcal{T}}\begin{pmatrix}x\\\ \lambda\end{pmatrix}.$

Here it is understood that ${\mathcal{T}}$ is evaluated at
$(x_{0},q_{0},\lambda_{0})\in X\times P\times W^{*}$. But ${\mathcal{T}}$ is
not yet the mapping for which we need to verify the Robinson-Dontchev strong
regularity condition in our context. It relates to the fact that we must to
treat the multiplier $\lambda$ in smaller space than $W^{*}$. Before we can
properly specify this condition some additional preparation is necessary. We
first introduce Banach spaces:

$\underline{X}\subset X,\ \underline{W^{*}}\subset W^{*},\
\underline{X^{*}}\subset X^{*},$ (2.8)

with continuous injections. A restriction of ${\mathcal{T}}$ will be defined
as multivalued operator
$\displaystyle\underline{{\mathcal{T}}}:\underline{X}\times\underline{W^{*}}\to\underline{X^{*}}\times
W$. Indeed, in applications to optimal control problems extra regularity of
multipliers can be obtained by investigating the solutions (2.3), see e.g.
Section 3. In the context of optimal stabilization problems this structural
property will become transparent in Proposition 4.1 and Proposition 4.2, see
also [BKP3, Proposition 15]. It will turn out to be essential for our
purposes. But this situation where the multiplier has extra regularity is also
of abstract interest. When studying stability in this setting this means that
the second coordinate of the domain of ${\mathcal{T}}$ needs to be changed
from $W^{*}$ to $\underline{W^{*}}$. This entails that the range space of
${\mathcal{T}}$ has to be modified appropriately, in order to obtain stability
of the $\lambda$ coordinate. For this purpose we introduce
$\underline{X^{*}}\subset X^{*}$. The reason for further restricting $X$ to
$\underline{X}$ will become evident in the proof of Proposition 4.2. It is
related to the fact that we consider infinite horizon problems.

Now we adapt the conditions on $f$ and $e$ to the choice of the spaces in
(2.8).

######  Assumption H4.

There exists a neighborhood
$\displaystyle\widetilde{U}_{1}\times\widetilde{U}_{2}\subset\underline{X}\times
P$ of $(x_{0},q_{0})$ such that

  1. (i)

the restriction of $x\mapsto f^{\prime}(x)$ to $\underline{X}$ defines a
mapping from $\underline{f^{\prime}}(x)$ from
$\widetilde{U}_{1}\subset\underline{X}$ to $\underline{X^{*}}$,

  2. (ii)

the restriction $e^{\prime}(x,q)^{*}\in{\mathcal{L}}(W^{*},X^{*})$ to
$\underline{W^{*}}$ defines operators
$\underline{e^{\prime}(x,q)^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})$
for every $(x,q)\in\widetilde{U}_{1}\times\widetilde{U}_{2}$.

With these assumption holding we define the restricted linearized Lagrangian

$\underline{{\mathcal{L}}^{\prime}}:\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}\subset\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\quad\text{by}\quad\underline{{\mathcal{L}}^{\prime}}(x,q,\lambda)=\underline{f^{\prime}}(x)+\underline{e^{\prime}(x,q)^{*}}\lambda.$
(2.9)

Next we adapt $\partial\mathbf{I}_{C}\subset X^{*}$ to the situation of (2.8)
and define for $x\in\underline{X}$ the set valued mapping

${\underline{\partial\mathbf{I}_{C}}(x)}=\left\\{y\in\underline{X^{*}}:\langle
y,v-x\rangle_{X^{*},X}\leq 0,\ \forall v\in
C\cap\underline{X}\right\\}\subset\underline{X^{*}}.$ (2.10)

We henceforth assume that
$(x_{0},\lambda_{0})\in\underline{X}\times\underline{W^{*}}$, it will also
follow as a special case of (H7) below. The following assumption will
guarantee that the restriction $\underline{{\mathcal{T}}}$ of ${\mathcal{T}}$
is well-defined as operator from $\underline{X}\times\underline{W^{*}}$ to
$\underline{X^{*}}\times W$, and the one beyond is needed for Lipschitz
continuous dependence of local solutions to ($P_{q}$) with respect to $q$.

######  Assumption H5.

$\displaystyle\underline{{\mathcal{L}}^{\prime}}:\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}\subset\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}$ is Fréchet
differentiable with respect to $x$, and
$(\underline{{\mathcal{L}}^{\prime}})^{\prime}$ is continuous at
$(x_{0},q_{0},\lambda_{0})\in\underline{X}\times P\times\underline{W^{*}}$.

######  Assumption H6.

There exists $\nu>0$ such that:

$\displaystyle\left\lVert e(x,q_{1})-e(x,q_{2})\right\rVert_{W}$
$\displaystyle\leq\nu\left\lVert q_{1}-q_{2}\right\rVert_{P},\
\forall(x,q_{1})\text{ and
}(x,q_{2})\in\widetilde{U}_{1}\times\widetilde{U}_{2},$ (2.11a)
$\displaystyle\left\lVert{\underline{e^{\prime}(x,q_{1})^{*}}-\underline{e^{\prime}(x,q_{2})^{*}}}\right\rVert_{{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})}$
$\displaystyle\leq\nu\left\lVert q_{1}-q_{2}\right\rVert_{P},\
\forall(x,q_{1})\text{ and
}(x,q_{2})\in\widetilde{U}_{1}\times\widetilde{U}_{2}.$ (2.11b)

Let us further set

$\underline{E^{*}}=\underline{e^{\prime}(x_{0},q_{0})^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})\text{
and
}\underline{A}=(\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0}))^{\prime}\in{\mathcal{L}}(\underline{X},\underline{X^{*}}).$

With Assumptions (H1)-(H5) holding (2.3) can be expressed as

$\displaystyle\begin{cases}0\in\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0})+\underline{\partial\mathbf{I}_{C}}(x_{0}),\quad&\text{in
}\underline{X^{*}},\\\ e(x_{0},q_{0})=0,\quad&\text{in }W.\end{cases}$ (2.12)

Moreover (2.6) restricted to $\underline{X}\times\underline{X^{*}}$ result in:

$\displaystyle
0\in\begin{cases}\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0})+\underline{A}(x-x_{0})+\underline{E^{*}}(\lambda-\lambda_{0})+\underline{\partial\mathbf{I}_{C}}(x)&\text{
in }\underline{X^{*}},\\\ e(x_{0},q_{0})+E(x-x_{0})&\text{ in }W,\end{cases}$
(2.13)

and the multivalued operator
$\displaystyle\underline{{\mathcal{T}}}:\underline{X}\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\times
W$ related to (2.7) is defined as

$\underline{{\mathcal{T}}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\underline{A}&\underline{E^{*}}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\underline{f^{\prime}}(x_{0})-\underline{A}x_{0}\\\
-Ex_{0}\end{pmatrix}+\begin{pmatrix}\underline{\partial\mathbf{I}_{C}}(x)\\\
0\end{pmatrix}.$ (2.14)

Observe that (2.13) is equivalent to

$\displaystyle 0\in\underline{{\mathcal{T}}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}.$

Existence and Lipschitz continuity of solutions in a neighborhood
$(x_{0},q_{0},\lambda_{0})$ will follow from the strong regularity assumption
which requires us to show that there exist neighborhoods
$\hat{V}\subset\underline{X^{*}}\times W$ of $0$ and
$\hat{U}=\hat{U}_{1}\times\hat{U}_{2}\subset\underline{X}\times\underline{W^{*}}$
of $(x_{0},q_{0})$ such that $\underline{{\mathcal{T}}}^{-1}$ has the
properties that $\underline{{\mathcal{T}}}^{-1}(\hat{V})\cap\hat{U}$ is
single-valued and that it is Lipschitz continuous from $\hat{V}$ to $\hat{U}$,
see [Don], (and also [Rob], [IK, Definition 2.2, p 31], in case
$\underline{X}=X,\ \underline{W^{*}}=W^{*},\ \underline{X^{*}}=X^{*}$). We
approach the strong regularity assumption in two steps. In the first one we
argue invertibility of ${\mathcal{T}}$ and Lipschitz continuity of the
variable $x$ in $X$. For this purpose we exploit the symmetry of
${\mathcal{T}}$ and consider an associated variational problem. In our
specific situation the inverse of ${\mathcal{T}}$ \- and consequently of
$\underline{{\mathcal{T}}}$ \- is single-valued and thus the restriction to
the neighborhood $\hat{U}$ is not needed. Existence and Lipschitz continuity
of $\lambda$ as well as Lipschitz continuity of $x$ in the small space
$\underline{X}\times\underline{W^{*}}$ remains an assumption in the generality
of problem ($P_{q}$). It will be verified in a second step for the optimal
stabilization problems in the following sections.

######  Assumption H7.

For $(\beta_{1},\beta_{2})\in\hat{V}\subset\underline{X^{*}}\times W$ , the
solution
$\displaystyle\left(x_{(\beta_{1},\beta_{2})},\lambda_{(\beta_{1},\beta_{2})}\right)$
to $\displaystyle{\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\beta_{1}\\\ \beta_{2}\end{pmatrix}$ lies
in $\underline{X}\times\underline{W^{*}}$. Moreover there exists a constant
$k>0$ such that

$\left\lVert
x_{(\beta_{1},\beta_{2})}-x_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{\underline{X}}+\left\lVert\lambda_{(\beta_{1},\beta_{2})}-\lambda_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{\underline{W^{*}}}\leq
k\left[\left\lVert(\beta_{1},\beta_{2})-(\hat{\beta}_{1},\hat{\beta}_{2})\right\rVert_{\underline{X^{*}}\times
W}+\left\lVert
x_{(\beta_{1},\beta_{2})}-x_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{X}\right]$

for all
$(\beta_{1},\beta_{2})\in\hat{V},(\hat{\beta}_{1},\hat{\beta}_{2})\in\hat{V}$.

This condition is used after the existence of
$x_{\beta}=x_{(\beta_{1},\beta_{2})}$ was already established. Note that for
$(\beta_{1},\beta_{2})^{T}=0$ we have
$(x_{(0,0)},\lambda_{(0,0)})=(x_{0},\lambda_{0})$ and hence (H7) in particular
implies that $(x_{0},\lambda_{0})\in\underline{X}\times\underline{W^{*}}$. We
arrive at the announced stability result.

######  Theorem 2.1.

Assume that (H1)-(H7) hold at a local solution $x_{0}$ of ($P_{q_{0}}$). Then
there exist a neighborhood
$U=U(x_{0},\lambda_{0})\subset\underline{X}\times\underline{W^{*}}$, a
neighborhood $N=N(q_{0})\subset P$, and a constant $\mu$ such that for all
$q\in N$ there exists a unique $(x(q),\lambda(q))\in U$ satisfying

$\displaystyle
0\in\begin{cases}\underline{{\mathcal{L}}^{\prime}}(x(q),q,\lambda(q))+\underline{\partial\mathbf{I}_{C}}(x(q)),\quad&\text{
in }\underline{X^{*}},\\\ e(x(q),q),\quad&\text{ in }W,\end{cases}$ (2.15)

and

$\left\lVert(x(q_{1}),\lambda(q_{1}))-(x(q_{2}),\lambda(q_{2}))\right\rVert_{\underline{X}\times\underline{W^{*}}}\leq\mu\left\lVert
q_{1}-q_{2}\right\rVert_{P},\ \forall q_{1},q_{2}\in N.$ (2.16)

In addition there exists a nontrivial neighborhood $\widetilde{N}\subset N$ of
$q_{0}$ such that $x(q)$ is a local solution of ($P_{q}$) for
$q\in\widetilde{N}$.

For the proof we shall employ the following lemma in which
$A\in{\mathcal{L}}(X,X^{*})$ and $E\in{\mathcal{L}}(X,W)$ denote generic
operators. For the sake of completeness we also include its proof.

######  Lemma 2.1.

Let $(\tilde{a},\tilde{b})\in X^{*}\times W$, assume that
$A\in{\mathcal{L}}(X,X^{*})$ is self-adjoint and satisfies (H3), and that the
set $\displaystyle S(\tilde{b})=\\{x\in C:\ Ex=\tilde{b}\\}$ is nonempty. Then
the problem

$\begin{cases}\min_{x\in C}\tilde{J}(x)=\min_{x\in C}\frac{1}{2}\langle
Ax,x\rangle_{X^{*},X}+\langle\tilde{a},x\rangle_{X^{*},X},\\\
Ex=\tilde{b},\end{cases}$ (2.17)

admits a unique solution $x=x(\tilde{a},\tilde{b})$ satisfying

$\langle Ax+\tilde{a},v-x\rangle_{X^{*},X}\geq 0,\ \text{for all }v\in
S(\tilde{b}).$ (2.18)

If moreover the regular point condition $0\in\text{int
}E(C-x(\tilde{a},\tilde{b}))$ holds, then there exists
$\lambda=\lambda(\tilde{a},\tilde{b})\in W^{*}$ such that

$0\in\begin{cases}\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\tilde{a}\\\
-\tilde{b}\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix}.\end{cases}$ (2.19)

###### Proof.

Since $C$ is a closed and convex, $S(\tilde{b})$ is closed and convex. By
assumption $S(\tilde{b})$ is nonempty. Hence there exists an $x\in C$ such
that $Ex=\tilde{b}$. Note that $x$ can be uniquely decomposed as $x=w+y$, with
$y\in\text{ker}E,\,w\in\text{ker}E^{\perp}$ and $Ew=\tilde{b}$. By (H3) the
functional $\tilde{J}$ is bounded from below and coercive on $S(\tilde{b})$.
Hence there exists a minimizing sequence $\\{x_{n}\\}$ in $S(\tilde{b})$ such
that $\displaystyle\lim_{n\rightarrow\infty}\tilde{J}(x_{n})=\inf_{x\in
S(\tilde{b})}\tilde{J}(x)$. Each $x_{n}$ can be decomposed as $x_{n}=w+y_{n}$,
with $y_{n}\in\text{ker}E$. By (H3) the sequences
$\displaystyle\\{y_{n}\\}_{n=1}^{\infty}$ and hence
$\displaystyle\\{x_{n}\\}_{n=1}^{\infty}$ are bounded. Thus there exists a
subsequence $\displaystyle\\{x_{n_{k}}\\}$ with weak limit
$x=x(\tilde{a},\tilde{b})$ in $S(\tilde{b})$. Since $\tilde{J}$ weakly lower
semi-continuous, we have that
$\displaystyle\tilde{J}(x)\leq\liminf_{k\rightarrow\infty}\tilde{J}(x_{n_{k}})$
and $x$ minimizes $\tilde{J}$ over $S(\tilde{b})$. This further implies that
$\displaystyle\langle Ax+\tilde{a},v-x\rangle_{X^{*},X}\geq 0$ for all $v\in
S(\tilde{b})$. Uniqueness of $x$ follows from (H3).

The regular point condition implies the existence of a multiplier
$\lambda=\lambda(\tilde{a},\tilde{b})\in W^{*}$ such that (2.19) holds. See
e.g. [IK, Theorem 1.6] ∎

Proof of the Theorem 2.1.

  1. (i)

The proof of the first assertion of the Theorem 2.1 is based on the implicit
function theorem of Dontchev for generalized equations, see [Don, Theorem 2.4,
Remark 2.5]. We introduce the mapping
$\underline{{\mathcal{F}}}:\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\times W$ given by

$\underline{{\mathcal{F}}}(x,q,\lambda)=\begin{pmatrix}\underline{{\mathcal{L}}^{\prime}}(x,q,\lambda)\\\
e(x,q)\end{pmatrix},$

and observe that Assumption (H6) implies that for all $(x,q_{1},\lambda),$ and
$(x,q_{2},\lambda)\in\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}$

$\left\lVert\underline{{\mathcal{F}}}(x,q_{1},\lambda)-\underline{{\mathcal{F}}}(x,q_{2},\lambda)\right\rVert_{\underline{X^{*}}\times
W}\leq\nu\left(1+\left\lVert\lambda\right\rVert_{\underline{W^{*}}}\right)\left\lVert
q_{1}-q_{2}\right\rVert_{P}.$ (2.20)

By (H1) and (H5), and using the integral mean value theorem it can be argued
that

$\begin{pmatrix}x\\\
\lambda\end{pmatrix}\to\begin{pmatrix}\underline{A}&\underline{E^{*}}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\underline{f^{\prime}}(x_{0})-\underline{A}x_{0}\\\
-Ex_{0}\end{pmatrix}$

strongly approximates $\underline{{\mathcal{F}}}$ at
$(x_{0},q_{0},\lambda_{0})$, [Don]. In the next two steps the strong
regularity condition for $\underline{{\mathcal{T}}}$ will be verified.

  2. (ii)

(Existence). Let, at first, $(\beta_{1},\beta_{2})\in X^{*}\times W$ and
consider $\displaystyle{\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\beta_{1}\\\ \beta_{2}\end{pmatrix}$ which
is equivalent to,

$0\in\begin{pmatrix}a\\\ -b\end{pmatrix}+\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix},$ (2.21)

with $a=f^{\prime}(x_{0})-Ax_{0}-\beta_{1},\ b=Ex_{0}+\beta_{2}$, and $A,E$
defined in (2.4), (2.5). To solve (2.21) we consider,

$\begin{cases}\min_{x\in C}\ \frac{1}{2}\langle Ax,x\rangle_{X^{*},X}+\langle
a,x\rangle_{X^{*},X},\\\ Ex=b.\end{cases}$ (2.22)

This corresponds to (2.17) with $\tilde{a}=a$, $\tilde{b}=b$ and feasible set
$\displaystyle S(\beta_{2})=\\{x\in C:Ex=b\\}$. Clearly $\displaystyle
x_{0}\in S(0)=\\{x\in C:Ex=Ex_{0}\\}$. By (H2) and [IK, Theorem I.2.8], there
exists a neighborhood of the origin $\displaystyle\tilde{V}\subset X^{*}\times
W$ such that $S(\beta_{2})$ is not empty for all
$(\beta_{1},\beta_{2})\in\tilde{V}$. Thus by Lemma 2.1 there exists a unique
solution $x=x(\beta_{1},\beta_{2})$ to (2.22) for each
$(\beta_{1},\beta_{2})\in\tilde{V}$. By [IK, Theorem I.2.11, I.2.12, I.2.15],
possibly after reducing $\tilde{V}$, these solutions depend Hölder
continuously on $(\beta_{1},\beta_{2})\in\tilde{V}\subset X^{*}\times W$, with
exponent $\frac{1}{2}$. The regular point condition for the solution
$x(\beta_{1},\beta_{2})$ is

$0\in\text{int }E(C-x(\beta_{1},\beta_{2}))=\text{int }E(C-x_{0})-\beta_{2},$

which is satisfied due to (H2), possibly after again reducing $\tilde{V}$.
Hence there exists a Lagrange multiplier
$\lambda=\lambda(\beta_{1},\beta_{2})$ associated to $Ex=b$, and (2.21) admits
a unique solution $(x(\beta_{1},\beta_{2}),\lambda(\beta_{1},\beta_{2}))$
since it is the first order optimality condition for (2.17).

  3. (iii)

(Uniqueness and Lipschitz continuity) Let $(\beta_{1},\beta_{2})\in\tilde{V}$
and $(\hat{\beta_{1}},\hat{\beta_{2}})\in\tilde{V}$ with corresponding
solutions $(x,\lambda)\in X\times W^{*}$ and $(\hat{x},\hat{\lambda})\in
X\times W^{*}$. This implies that

$\begin{cases}\langle a+Ax+E^{*}\lambda,c-x\rangle_{X^{*},X}\geq 0,\ \forall
c\in C,\\\ Ex=b,\text{ with }a=f^{\prime}(x_{0})-Ax_{0}-\beta_{1},\
b=Ex_{0}+\beta_{2},\end{cases}$ (2.23)

and

$\begin{cases}\langle\hat{a}+A\hat{x}+E^{*}\hat{\lambda},c-\hat{x}\rangle_{X^{*},X}\geq
0,\ \forall c\in C,\\\ E\hat{x}=\hat{b},\text{with
}\hat{a}=f^{\prime}(x_{0})-Ax_{0}-\hat{\beta_{1}},\
\hat{b}=Ex_{0}+\hat{\beta_{2}}.\end{cases}$ (2.24)

By the first equations in (2.23) and (2.24) we obtain that

$\langle a+Ax+E^{*}\lambda,\hat{x}-x\rangle_{X^{*},X}\geq 0,\
\langle\hat{a}+A\hat{x}+E^{*}\hat{\lambda},x-\hat{x}\rangle_{X^{*},X}\geq 0,\
x,\hat{x}\in C.$ (2.25)

Combining these inequalities, we have that

$\langle
a-\hat{a}+A(x-\hat{x})+E^{*}(\lambda-\hat{\lambda}),x-\hat{x}\rangle_{X^{*},X}\leq
0.$ (2.26)

The second equalities in (2.23) and (2.24) imply that

$E(x-\hat{x})=b-\hat{b}.$ (2.27)

Let us set

$\delta x=\hat{x}-x,\ \delta\lambda=\hat{\lambda}-\lambda,\ \delta
a=\hat{a}-a,\ \delta b=\hat{b}-b.$

Then $\delta\beta_{1}=-(\hat{\beta_{1}}-\beta_{1})$ and
$\delta\beta_{2}=\hat{\beta_{2}}-\beta_{2}$, and (2.26), (2.27) result in

$\langle\delta x,A\delta x\rangle_{X,X^{*}}+\langle\delta\lambda,E\delta
x\rangle_{W^{*},W}-\langle\delta\beta_{1},\delta x\rangle_{X^{*},X}\leq 0,$
(2.28)

and

$E\delta x=\delta b.$ (2.29)

By (H2) the operator $E$ is surjective. Hence by the closed range theorem
expressing $\delta x=\delta v+\delta w\in\text{ker }E+\text{range }E^{*}$
implies that $\displaystyle E\delta x=E\delta w=\delta\beta_{2}$. Again by the
closed range theorem there exists $k_{1}>0$:

$\left\lVert\delta w\right\rVert_{X}\leq
k_{1}\left\lVert\delta\beta_{2}\right\rVert_{W}.$ (2.30)

From the first equation in (2.21) we have

$Ax+E^{*}\lambda-
Ax_{0}+f^{\prime}(x_{0})-\beta_{1}\in-\partial{\mathbf{I}}_{C}(x).$

Next we restrict the perturbation parameters to satisfy
$(\beta_{1},\beta_{2})\in(\underline{X^{*}}\times W)\cap\tilde{V}$. Due to
Assumptions (H4) and (H7) we have
$(x,\lambda)\in\underline{X}\times\underline{W^{*}}$,

$\underline{A}x+{\underline{E^{*}}}\lambda-\underline{A}x_{0}+\underline{f^{\prime}}(x_{0})-\beta_{1}\in\underline{X^{*}}$

and hence

$\underline{A}x+{\underline{E^{*}}}\lambda-\underline{A}x_{0}+\underline{f^{\prime}}(x_{0})-\beta_{1}\in-\underline{\partial\mathbf{I}_{C}}(x).$

The analogous equation holds with $(x,\lambda,\beta_{1})$ replaced by
$(\hat{x},\hat{\lambda},\hat{\beta}_{1})$.
By (H3), (2.28) and Assumption (H7) we find

$\displaystyle\kappa\left\lVert\delta v\right\rVert^{2}$
$\displaystyle\leq\langle\delta v,A\delta v\rangle_{X,X^{*}}=\langle\delta
x,A\delta x\rangle_{X,X^{*}}-2\langle\delta v,A\delta
w\rangle_{X,X^{*}}-\langle\delta w,A\delta w\rangle_{X,X^{*}}$ (2.31)
$\displaystyle\leq-\langle\delta\lambda,E\delta
x\rangle_{W^{*},W}+\langle\delta\beta_{1},\delta
x\rangle_{X,X^{*}}-2\langle\delta v,A\delta w\rangle_{X,X^{*}}-\langle\delta
w,A\delta w\rangle_{X,X^{*}}$
$\displaystyle\leq\tilde{k}\left\lVert\delta\lambda\right\rVert_{\underline{W}^{*}}\left\lVert\delta\beta_{2}\right\rVert_{W}+\left\lVert\delta\beta_{1}\right\rVert_{X^{*}}\left\lVert\delta
x\right\rVert_{X}+\left\lVert A\right\rVert\left\lVert\delta
w\right\rVert_{X}\left(2\left\lVert\delta v\right\rVert_{X}+\left\lVert\delta
w\right\rVert_{X}\right)$ $\displaystyle\leq\tilde{k}k(\left\lVert\delta
w\right\rVert_{X}+\left\lVert\delta
v\right\rVert_{X}+\left\lVert\delta\beta\right\rVert_{\underline{X}^{*}\times
W})\left\lVert\delta\beta_{2}\right\rVert_{W}+(\left\lVert\delta\beta_{1}\right\rVert_{X^{*}}+2\left\lVert
A\right\rVert\left\lVert\delta w\right\rVert_{X})(\left\lVert\delta
v\right\rVert_{X}+\left\lVert\delta w\right\rVert_{X}),$

where $\tilde{k}$ denotes the embedding constant of $\underline{W}^{*}$ into
$W^{*}$. Using (2.30) and rearranging terms there exists a constant $k_{2}>0$
such that

$\left\lVert\delta v\right\rVert_{X}\leq
k_{2}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X}^{*}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right).$
(2.32)

Applying (2.30) again this implies the existence of $k_{3}$ such that

$\left\lVert\delta x\right\rVert_{X}\leq
k_{3}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X}^{*}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right)\text{
for all }(\beta_{1},\beta_{2})\in(\underline{X^{*}}\times W)\cap\tilde{V}.$
(2.33)

Another application of (H7) and (2.33) imply the existence of a constant
$k_{4}$ and a neighborhood $\hat{V}$ of the origin in $\underline{X^{*}}\times
W$ such that the desired Lipschitz stability estimate for
$(\underline{{\mathcal{T}}})^{-1}$

$\left\lVert\delta
x\right\rVert_{\underline{X}}+\left\lVert\delta\lambda\right\rVert_{\underline{W^{*}}}\leq
k_{4}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X^{*}}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right)\text{
for all }(\beta_{1},\beta_{2})\in\hat{V}\subset\underline{X^{*}}\times W$
(2.34)

holds.

  4. (iv)

As a consequence of the previous two steps $\underline{{\mathcal{T}}}$ is
strongly regular at $(x_{0},q_{0},\lambda_{0})$. Together with step (i),
Dontchev’s theorem is applicable [Don, Theorem 2.4, Remark 2.5], and (2.15),
and (2.16) follow.

  5. (v)

(Local solution to ($P_{q}$)) Now we show that there exists a neighborhood
$\tilde{N}$ of $q_{0}$ such that for $q\in\tilde{N}$ the second order
sufficient optimality condition is satisfied at $x(q)$, so that $x(q)$ is a
local solution of ($P_{q}$) by eg. [IK, Theorem 2.12, p42]. Due to (H3) and
regularity of $f,e$ we obtain

${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))(h,h)\geq\frac{\kappa}{2}\left\lVert
h\right\rVert^{2},\ \text{for all }h\in\text{ker }E,\ \text{if }q\in
N(q_{0}).$ (2.35)

Let us define $\displaystyle E_{q}=(e_{y}(x(q),q))$ for $q\in N(q_{0})$. By
the surjectivity of $E_{q_{0}}$ and regularity of $e$ there exists a
neighborhood $\tilde{N}\subset N(q_{0})$ such that $E_{q}$ is surjective for
all $q\in\tilde{N}$. There exist $\delta_{0},\gamma>0$ such that

${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))(h+z,h+z)\geq\delta_{0}\left\lVert
h+z\right\rVert^{2},\ \text{for all }h\in\text{ker }E,z\in X$ (2.36)

satisfying $\left\lVert z\right\rVert\leq\gamma\left\lVert h\right\rVert$ by
[IK, Lemma 2.13, p43]. Let us define the orthogonal projection onto ker$\
E_{q}$ given by $\displaystyle P_{\text{ker
}E_{q}}=I-E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}$. We choose $\tilde{N}$ so that

$\left\lVert
E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}-E_{q_{0}}^{*}(E_{q_{0}}E_{q_{0}}^{*})^{-1}E_{q_{0}}\right\rVert\leq\frac{\gamma}{1+\gamma}$

for all $q\in\tilde{N}$. For $x\in\text{ker }E_{q}$, we have $x=h+z$ for
$h\in\text{ker }E,\ z\in(\text{ker }E)^{\perp}$ and $\left\lVert
x\right\rVert^{2}=\left\lVert h\right\rVert^{2}+\left\lVert
z\right\rVert^{2}$. Thus,

$\left\lVert z\right\rVert\leq\left\lVert
E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}x-E_{q_{0}}^{*}(E_{q_{0}}E_{q_{0}}^{*})^{-1}E_{q_{0}}x\right\rVert\leq\frac{\gamma}{1+\gamma}\big{(}\left\lVert
h\right\rVert+\left\lVert z\right\rVert\big{)}$

and hence $\left\lVert z\right\rVert\leq\gamma\left\lVert h\right\rVert$. From
(2.35) this implies

${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))\geq\delta_{0}\left\lVert
x\right\rVert^{2},\ \text{for all }x\in\text{ker }E.$

This concludes the proof. ∎

##  3 Differentiability of value function for optimal stabilization subject to
semi-linear parabolic equations.

Here we describe the optimal control problems which we shall analyze and state
the main results.

###  3.1 Notation

Let $\Omega$ be an open connected bounded subset of $\mathbb{R}^{d}$ with a
Lipschitz continuous boundary $\Gamma$. The associated space-time cylinder is
denoted by $Q=\Omega\times(0,\infty)$ and the associated lateral boundary by
$\Sigma=\Gamma\times(0,\infty)$. We define the Hilbert spaces

$Y=L^{2}(\Omega),\quad V=H^{1}_{0}(\Omega),\text{ and }\quad
U=L^{2}(0,\infty;\,{\mathcal{U}}),$

where ${\mathcal{U}}$ is a Hilbert space which will be identified with its
dual. Observe that the embedding $V\subset Y$ is dense and compact. Further
$V\subset Y\subset V^{*}$, is a Gelfand triple. Here $V^{*}$ denotes the
topological dual of $V$ with respect to the pivot space $Y$. For any
$T\in(0,\infty)$ we define the space

$W(0,T)=\bigg{\\{}y\in L^{2}(0,T;V);\ \frac{dy}{dt}\in
L^{2}(0,T;V^{*})\bigg{\\}},$

and for $T=\infty$, we write $W_{\infty}$ and $I=(0,\infty)$. We further set
$W^{0}_{\infty}=\\{y\in W_{\infty}:y(0)=0\\}$. We also set

$W(T,\infty)=\bigg{\\{}y\in L^{2}(T,\infty;V);\ \frac{dy}{dt}\in
L^{2}(T,\infty;V^{*})\bigg{\\}}.$

We shall frequently use that $W_{\infty}$ embeds continuously into
$C([0,\infty),Y)$, see e.g. [LM, Theorem 4.2] and that
$\displaystyle\lim_{t\to\infty}y(t)=0$, for $y\in W_{\infty}$, see e.g. [CK].
The set of admissible controls $U_{ad}$ is chosen to be

$U_{ad}\subset\\{u\in U:\|u(t)\|_{{\mathcal{U}}}\leq\eta,\text{ for }a.e.\
t>0\\}$ (3.1)

where $\eta$ is a positive constant. We further set
${\mathcal{U}}_{ad}=\\{v\in{\mathcal{U}}:\|v\|_{{\mathcal{U}}}\leq\eta\\}$ and
denote by $\displaystyle\mathbb{P}_{{\mathcal{U}}_{ad}}$ the projection of
${\mathcal{U}}$ on ${\mathcal{U}}_{ad}$. For this choice of admissible
controls the dynamical system can be stabilized for all sufficiently small
initial conditions in $Y$, see Corollary 4.3 and Remark 4.1.

For $\delta>0$ and $\bar{y}\in Y$, we define the open neighborhoods
$B_{Y}(\delta)=\left\\{y\in Y:\left\lVert y\right\rVert_{Y}<\delta\right\\},$
and $\quad B_{Y}(\bar{y},\delta)=\left\\{y\in Y:\left\lVert
y-\bar{y}\right\rVert_{Y}<\delta\right\\}$.

###  3.2 Problem formulation and assumptions.

We focus on the stabilization problem for an abstract semi-linear parabolic
equation formulated as infinite horizon optimal control problem under control
constraints:

$\displaystyle(\mathcal{P})\qquad\mathcal{V}(y_{0})=\min_{(y,u)\in
W_{\infty}\times U_{ad}}\ J(y,u)$ $\displaystyle=\min_{(y,u)\in
W_{\infty}\times U_{ad}}\ \frac{1}{2}\int_{0}^{\infty}\left\lVert
y(t)\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u(t)\right\rVert^{2}_{Y}dt,$ (3.2a) subject to the semilinear parabolic
equation $\displaystyle y_{t}$
$\displaystyle={\mathcal{A}}y+{\mathcal{F}}(y)+Bu\quad\text{ in
}L^{2}(I;V^{*})$ (3.2b) $\displaystyle y(x,0)$ $\displaystyle=y_{0}\quad\text{
in }Y.$ (3.2c)

Throughout ${\mathcal{F}}$ is the substitution operator associated to a
mapping $\mathfrak{f}:\mathbb{R}\to\mathbb{R}$ so that
$({\mathcal{F}}y)(t)=\mathfrak{f}(y(t))$. Sufficient conditions which
guarantee the existence of solutions to (3.2b), (3.2c), as well as solutions
$(\bar{y},\bar{u})$ to ($\mathcal{P}$), for $y_{0}\in Y$ sufficiently small,
will be given below. We shall also make use of the adjoint equation associated
to an optimal state $\bar{y}$, given by

$-p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(\bar{y})^{*}p=-\bar{y}\quad\text{
in }L^{2}(I;V^{*}).$ (3.2d)

Its adjoint state $p$ which will be considered in $L^{2}(I;V)$ or in
$W_{\infty}$. The following assumption will be essential.

####  3.2.1 Assumptions A.

  * A1

The operator ${\mathcal{A}}$ with domain ${\mathcal{D}}({\mathcal{A}})\subset
Y$ and range in $Y$, generates a strongly continuous analytic semigroup
$\displaystyle e^{{\mathcal{A}}t}$ on $Y$ and can be extended to
${\mathcal{A}}\in{\mathcal{L}}(V,V^{*})$.

  * A2

$B\in{\mathcal{L}}({\mathcal{U}},Y)$ and there exists a stabilizing feedback
operator $K\in{\mathcal{L}}(Y,{\mathcal{U}})$ such that the semigroup
$\displaystyle e^{({\mathcal{A}}-BK)t}$ is exponentially stable on $Y$.

  * A3

The nonlinearity $\displaystyle{\mathcal{F}}:W_{\infty}\to L^{2}(I;V^{*})$ is
twice continuously Fréchet differentiable, with second Fréchet derivative
${\mathcal{F}}^{\prime\prime}$ bounded on bounded subsets of $W_{\infty}$, and
${\mathcal{F}}(0)=0$.

  * A4

${\mathcal{F}}:W(0,T)\to L^{1}(0,T;{\mathcal{H}}^{*})$ is weak-to-weak
continuous for every $T>0$, for some Hilbert space ${\mathcal{H}}$ which
embeds densely in $V$.

Note that
$\displaystyle\left(L^{1}(0,T;{\mathcal{H}}^{*})\right)^{*}=L^{\infty}(0,T;{\mathcal{H}})$,
see [Emm, Theorem 7.1.23(iv), p 164]. Moreover,
$L^{\infty}(0,T;{\mathcal{H}})$ is dense in $L^{2}(0,T;V)$, see [MS, Lemma
A.1, p 2231].

  * A5

${\mathcal{F}}^{\prime}(\bar{y})\in{\mathcal{L}}{(L^{2}(I;V),L^{2}(I;V^{*}))}$.

######  Remark 3.1.

The requirement that $\mathcal{F}(0)=0$ in (A2) is consistent with the fact
that we focus on the stabilization problem with $0$ as steady state for
(3.2b). Without loss of generality we further assume that

$\mathcal{F}^{\prime}(0)=0,$ (3.3)

which can always be achieved by making $\mathcal{F}^{\prime}(0)$ to be
perturbation of $\mathcal{A}$.

######  Remark 3.2.

Let us assume that (A3) holds. Then in view of the fact that ${\mathcal{F}}$
is a substitution operator we have
$[{\mathcal{F}}^{\prime}(y)v](t)=\mathfrak{f}^{\prime}(y(t))v(t)$ for $y$ and
$v$ in $W_{\infty}$, and
${\mathcal{F}}^{\prime}(y)\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$. Its
adjoint $[{\mathcal{F}}^{\prime}(y)^{*}v](t)=\mathfrak{f}^{\prime}(y(t))v(t)$,
for $v\in L^{2}(I;V)$, satisfying
${\mathcal{F}}^{\prime}(y)^{*}\in{\mathcal{L}}(L^{2}(I;V),W_{\infty}^{*})$. It
has a natural restriction to an operator
$\underline{{\mathcal{F}}^{\prime}(y)^{*}}\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$.
With (A3) holding it is differentiable and
$[\underline{{\mathcal{F}}^{\prime}(y)}^{*}]^{\prime}$ is a bilinear form on
$W_{\infty}\times W_{\infty}$ with values in $L^{2}(I;V^{*})$.

####  3.2.2 Abstract setup.

Here we relate problem ($\mathcal{P}$) to the abstract problem ($P_{q}$),
which is used with the following spaces:

$\begin{array}[]{l}X=W_{\infty}\times U,\;W=L^{2}(I;V^{*})\times
Y,\;P=Y,\;C=U_{ad},\;X^{*}=W_{\infty}^{*}\times U,\;W^{*}=L^{2}(I;V)\times
Y,\\\\[7.3194pt] \underline{X}=W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}})),\quad\underline{X^{*}}=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})),\quad\underline{W^{*}}=\widetilde{W}_{\infty},\end{array}$
(3.4)

where $\widetilde{\mathcal{W}}=\\{(\varphi,\varphi(0)):\varphi\in
W_{\infty}\\}$, endowed with the norm of $W_{\infty}$. At times we identify
$\widetilde{\mathcal{W}}$ with $W_{\infty}$. We recall that the dual space of
$\displaystyle W_{\infty}=L^{2}(I;V)\cap W^{1,2}(I;V^{*})$ is $\displaystyle
W^{*}_{\infty}=L^{2}(I;V^{*})+(W^{1,2}(I;V^{*}))^{*}$, endowed with the norm
$\displaystyle\left\lVert
z\right\rVert_{W^{*}_{\infty}}=\inf_{z=z_{1}+z_{2}}\left\lVert
z_{1}\right\rVert_{L^{2}(I;V^{*})}+\left\lVert
z_{2}\right\rVert_{W^{1,2}(I;V^{*})^{*}}$, where $\displaystyle z_{1}\in
L^{2}(I;V^{*}),z_{2}\in(W^{1,2}(I;V^{*}))^{*}$.

To express ($P_{q}$) for the present case, we set $x=(y,u)\in W_{\infty}\times
U$, and the parameter $q$ becomes the initial condition $y_{0}\in Y$. Further
$f:W_{\infty}\times U\longrightarrow\mathbb{R}$ is given by

$f(y,u)=\frac{1}{2}\int_{0}^{\infty}\left\lVert
y(t)\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u(t)\right\rVert^{2}_{{\mathcal{U}}}dt,$ (3.5)

and $e(x,q)=e(y,u,y_{0})$ is

$e(y,u,y_{0})=\begin{pmatrix}y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\\\
y(0)-y_{0}\end{pmatrix}:W_{\infty}\times U\times Y\longrightarrow
L^{2}(I;V^{*})\times Y$ (3.6)

By (A3) the mapping $e$ is Fréchet differentiable with respect to
$\displaystyle x=(y,u)\in W_{\infty}\times U$ and thus for $(y,u,y_{0})\in
W_{\infty}\times U\times Y$ we have

$e^{\prime}(y,u,y_{0})(v,w)=\begin{pmatrix}v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v-Bw\\\
v(0,\cdot)\end{pmatrix}:W_{\infty}\times U\longrightarrow L^{2}(I;V^{*})\times
Y.$ (3.7)

The Lagrange functional ${\mathcal{L}}:W_{\infty}\times U\times Y\times
L^{2}(I;V)\times Y\longrightarrow\mathbb{R}$ corresponding to our optimal
control problem is given by

${\mathcal{L}}(y,u,y_{0},p,p_{1})=J(y,u)+\int_{0}^{\infty}\langle
p,y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\rangle_{V,V^{*}}dt+(p_{1},y(0)-y_{0})_{Y},$

where $(p,p_{1})\in L^{2}(I;V)\times Y$ corresponds to the abstract Lagrange
multiplier $\lambda\in W^{*}$.

In the remainder of this subsection we specify the mappings ${\mathcal{T}}$
and $\underline{{\mathcal{T}}}$ for problem ($\mathcal{P}$). This will
facilitate the proofs of the main results further below.

At first we take a closer look to the adjoint
$\displaystyle\widetilde{E}^{*}:=e^{\prime}(y,u,y_{0})^{*}\in{\mathcal{L}}(L^{2}(I;V)\times
Y,W^{*}_{\infty}\times U)$ at a generic element $(y,u,y_{0})\in
W_{\infty}\times U\times Y$. It is characterized by the property that for all
$(v,w)\in W_{\infty}\times U,\ (p,p_{1})\in L^{2}(I;V)\times Y$ we have

$\langle\widetilde{E}(v,w),(p,p_{1})\rangle_{L^{2}(I;V^{*})\times
Y,L^{2}(I;V)\times Y}=\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{W_{\infty},W^{*}_{\infty}}+(w,\widetilde{E}_{2}^{*}(p,p_{1}))_{U}$

where

$\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{W_{\infty},W^{*}_{\infty}}=\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+(v(0),p_{1})_{Y},$

and

$(w,\widetilde{E}_{2}^{*}(p,p_{0}))_{U}=-(w,B^{*}p)_{U}.$

If for some $\tilde{\beta}_{1}\in L^{2}(I;V^{*})$ the pair $(p,p_{1})\in
L^{2}(I;V)\times Y$ is a solution to
$\tilde{E}_{1}^{*}(p,p_{1})=\tilde{\beta}_{1}$ then for all $v\in W_{\infty}$:

$\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(V^{*}),L^{2}(V)}+(v(0),p_{1})_{Y}-(w,B^{*}p)_{U}=\langle\tilde{\beta}_{1},v\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}.$
(3.8)

Now we assume that ${\mathcal{F}}^{\prime}(y)$ is not only an element of
$\displaystyle{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$ but rather that it can
be extended to an operator
${{\mathcal{F}}^{\prime}(y)}\in{\mathcal{L}}(L^{2}(I;V),L^{2}(I;V^{*}))$. This
is guaranteed by (A5) at minimizers $\bar{y}$. Then (3.8) implies that $p\in
W_{\infty}$, and $p_{1}=p(0)$. In particular
$(p,p_{1})=(p,p(0))\in\widetilde{W}_{\infty}$, and (3.8) can equivalently be
expressed as

$\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}=\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}=\langle
v,\widetilde{\beta}_{1}\rangle_{L^{2}(I;V),L^{2}(I;V^{*})},$ (3.9)

for all $v\in L^{2}(I;V)$, where we assumed that $\widetilde{\beta}_{1}\in
L^{2}(I;V^{*})$. Conversely, of course, if $p\in\widetilde{W}_{\infty}$, then
$\displaystyle\widetilde{E}^{*}_{1}(p,p(0))=-p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(y)^{*}p\in
L^{2}(I;V^{*})$.

From now on, let $q_{0}=\bar{y}_{0}$ denote a reference (or nominal) parameter
with associated solution $x_{0}=(\bar{y},\bar{u})$. In Proposition 4.1 we
shall argue that the regular point condition Assumption (H2) is satisfied and
that consequently there exists a Lagrange multiplier $(\bar{p},\bar{p}_{1})$
such that the pair $(x_{0},\lambda_{0})=(\bar{y},\bar{u},\bar{p},\bar{p}_{1})$
satisfies (2.3). Moreover, it will turn out that $\bar{p}\in W_{\infty}$,
$\bar{p}_{1}=p(0)$, and that $\bar{u}\in U\cap C(I;{\mathcal{U}})$. For
convenience let us present (2.3) for the present case

$\displaystyle 0\in\begin{cases}\bar{y}+E^{*}_{1}(\bar{p},\bar{p}(0)),\\\
\alpha\bar{u}-B^{*}\bar{p}+\partial\mathbf{I}_{U_{ad}}(\bar{u}),\\\
\bar{y}_{t}-{\mathcal{A}}\bar{y}-{\mathcal{F}}(\bar{y})-B\bar{u},\\\
{\bar{y}(0)}-y_{0},\end{cases}$ (3.10)

where $\displaystyle E=\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}=e^{\prime}(\bar{y},\bar{u},\bar{y}_{0})$. We stress that
while the Lagrange multiplier $\bar{p}\in W_{\infty}$, the operator
$E^{*}_{1}$ in (3.9) is still considered as an element of
${\mathcal{L}}(L^{2}(I;V)\times Y,W^{*}_{\infty})$.

We are now prepared to specify the multivalued operators

$\displaystyle{\mathcal{T}}:$ $\displaystyle W_{\infty}\times U\times
L^{2}(I;V)\times Y\longrightarrow W_{\infty}^{*}\times U\times
L^{2}(I;V^{*})\times Y,\quad\text{and}$ (3.11)
$\displaystyle\underline{{\mathcal{T}}}:$ $\displaystyle
W_{\infty}\times(U\cap
C(I;{\mathcal{U}}))\times\widetilde{W}_{\infty}\longrightarrow
L^{2}(I;V^{*})\times(U\cap C(I;{\mathcal{U}}))\times L^{2}(I;V^{*})\times Y$
(3.12)

corresponding to (2.7) and (2.14) by

${\mathcal{T}}\begin{pmatrix}y\\\ u\\\ p\\\
p_{1}\end{pmatrix}=\begin{pmatrix}E^{*}_{1}(p,p_{1})+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}p]^{\prime}(y-\bar{y})\\\
\alpha u-B^{*}p\\\
y_{t}-{\mathcal{A}}y-Bu-{\mathcal{F}}^{\prime}(\bar{y})(y-\bar{y})-{\mathcal{F}}(\bar{y})\\\
y(0)-{\bar{y}_{0}}\end{pmatrix}+\begin{pmatrix}0\\\
\partial\mathbf{I}_{U_{ad}}(u)\\\ 0\\\ 0\end{pmatrix},$ (3.13)

and

$\underline{{\mathcal{T}}}\begin{pmatrix}y\\\ \underline{u}\\\
\underline{p}\\\
\underline{p}(0)\end{pmatrix}=\begin{pmatrix}-\underline{p}_{t}-{\mathcal{A}}^{*}\underline{p}-\underline{{\mathcal{F}}^{\prime}(\bar{y})^{*}}\,\underline{p}+y-[\underline{{\mathcal{F}}^{\prime}(\bar{y})^{*}}\,\underline{\bar{p}}]^{\prime}(y-\bar{y})\\\
\alpha\underline{u}-B^{*}\underline{p}\\\
y_{t}-{\mathcal{A}}y-B\underline{u}-{\mathcal{F}}^{\prime}(\bar{y})(y-\bar{y})-{\mathcal{F}}(\bar{y})\\\
y(0)-{\bar{y}_{0}}\end{pmatrix}+\begin{pmatrix}0\\\
\underline{\partial\mathbf{I}_{U_{ad}}}(\underline{u})\\\ 0\\\
0\end{pmatrix}.$ (3.14)

where

$\underline{\partial\mathbf{I}_{U_{ad}}}(\underline{u})=\left\\{\widetilde{u}\in
U\cap C(I;{\mathcal{U}}):\
(\widetilde{u}(t),v-\underline{u}(t))_{\mathcal{U}}\leq 0,\ \forall t\in I,\
v\in B_{\eta}(0)\right\\},$ (3.15)

with $\displaystyle B_{\eta}(0)=\left\\{v\in{\mathcal{U}};\left\lVert
v\right\rVert_{{\mathcal{U}}}\leq\eta\right\\}$. In (3.14), we underline the
elements which are taken from different domains when compared to (3.13). The
range of the first two coordinates of $\underline{{\mathcal{T}}}$ is smaller
than that of ${\mathcal{T}}$. Accordingly we can make use of (3.9) when moving
from the first row of (3.13) to the first row of (3.14).

For convenience for the subsequent work we recall that the strong regularity
condition introduced below (2.14) requires us to find neighborhoods of $0$ and
$(\bar{y},\bar{u},\bar{p},\bar{p}(0))$ of the form $\hat{V}\subset
L^{2}(I;V^{*})\times(U\cap C(\overline{I};{\mathcal{U}}))\times
L^{2}(I;V^{*})\times Y$ and $\hat{U}\subset W_{\infty}\times(U\cap
C(\overline{I};{\mathcal{U}}))\times\widetilde{W}_{\infty}$, such that for all
$\boldsymbol{\beta}=(\beta_{1},\beta_{2},\beta_{3},\beta_{4})\in\hat{V}$ the
equation

$\underline{{\mathcal{T}}}\left(y,u,\underline{p},\underline{p}(0)\right)^{T}=\left(\beta_{1},\beta_{2},\beta_{3},\beta_{4}\right)^{T}$
(3.16)

admits a unique solution
$(y,\underline{u},\underline{p},\underline{p}(0))\in\hat{U}$ depending
Lipschitz-continuously on $\boldsymbol{\beta}$.

######  Remark 3.3.

We observe that as a consequence of (A3) and Remark 3.2 the operator
$\underline{{\mathcal{T}}}$ is continuous.

Subsequently we shall frequently refrain from the underline-notation since the
meaning should be clear from the context.

###  3.3 Main Theorems.

In this subsection, we present the main theorems of this paper. The first
theorem asserts local continuous differentiability of the value function
$\mathcal{V}$ w.r.t. $y_{0}$, with $y_{0}$ small enough. The second theorem
establishes that $\mathcal{V}$ satisfies the HJB equation in the classical
sense. The proof of the first theorem is based on Theorem 2.1. It will be
given in Section 4 below. For this purpose it will be shown that assumptions A
imply (H1)-(H7). Moreover we need to assert the underlying assumption that
problem ($\mathcal{P}$) is well-posed. This will lead to a smallness
assumption on the initial states $y_{0}$. Consequently it would suffice to
assume that (A3) and (A4) only hold locally in the neighborhood of the origin.
Concerning (A5) observe that it is not implied by (A3). It is vacuously
satisfied for $\bar{y}=0$, which is the case for $y_{0}=0$, since then
$\mathcal{F}^{\prime}(0)=0$, see (3.3).

We invoke Theorem 2.1 to assert the Lipschitz continuity of the state, the
adjoint state, and the control with respect to the initial condition $y_{0}\in
Y$ in the neighborhood of a locally optimal solution $(\bar{y},\bar{u})$
corresponding to a sufficiently small reference initial state $\bar{y}_{0}$.
This will imply the differentiability of the value function associated to
local minima. We shall refer to the value function associated to local minima
as ’local value function’.

######  Theorem 3.1.

Let the assumptions (A) hold. Then associated to each local solution
$(\bar{y}(y_{0}),\bar{u}(y_{0}))$ of ($\mathcal{P}$) there exists a
neighborhood of $U(y_{0})$ such that the local value function
${\mathcal{V}}:U(y_{0})\subset Y\to\mathbb{R}$ is continuously differentiable,
provided that $y_{0}$ is sufficiently close to the origin in $Y$.

To obtain a HJB equation we require additionally that
$t\to({\mathcal{F}}(\bar{y}))(t)$ is continuous with values in $Y$ for global
solutions $(\bar{y},\bar{u})$ to ($\mathcal{P}$), with
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$. In view of the fact that for $y_{0}\in
V$ we can typically expect that the solution of semilinear parabolic equations
satisfy $y\in L^{2}(I;{\mathcal{D}}({\mathcal{A}}))\cap W^{1,2}(I;Y)\subset
C([0,\infty),V)$ this is not a restrictive assumption beyond that what is
already assumed in (A3).

######  Theorem 3.2.

Let the assumptions (A) hold, and let $(\bar{y}(y_{0}),\bar{u}(y_{0}))$ denote
a global solution of ($\mathcal{P}$), for
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$ with sufficiently small norm in $Y$.
Assume that there exists $T_{y_{0}}>0$ such that ${\mathcal{F}}(\bar{y})\in
C([0,T_{y_{0}});Y)$. Then the following Hamilton-Jacobi-Bellman equation holds
at $y_{0}$:

$\mathcal{V}^{\prime}(y)({\mathcal{A}}y+{\mathcal{F}}(y))+\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rangle_{Y}=0.$
(3.17)

Moreover the optimal feedback law is given by

${\bar{u}}(0)=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}({\bar{y}}(0))\right).$
(3.18)

The condition on the smallness of $y_{0}$ will be discussed in Remark 4.2
below. Roughly it involves well-posedness of the optimality system and second
order sufficient optimality at local solutions. A more detailed, respectively
stronger statement of Theorem 3.1 and Theorem 3.2, will be given in Theorem
4.1 and Theorem 5.1 below. The regularity assumptions
${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$ of Theorem 3.2 will be
addressed in Section 6.

##  4 Proof of Theorem 3.1.

In this section we give the proof for Theorem 3.1. Many of the technical
difficulties arise from the fact that we are working with an infinite horizon
optimal control problem. In this respect we can profit from techniques which
were developed in [BKP3], which, however, do not include the case of
constraints on the norm. Throughout we assume that assumptions (A1) - (A4)
hold.

###  4.1 Well-posedness of problem ($\mathcal{P}$).

Here we prove well-posedness for ($\mathcal{P}$) with small initial data.
First, we recall two consequences of the assumption that ${\mathcal{A}}$ is
the generator of an analytic semigroup.

######  Consequence 1.

Since ${\mathcal{A}}$ generates a strongly continuous analytic semigroup on
$Y$, there exist $\rho\geq 0$ and $\theta>0$ such that

$\langle(\rho I-{\mathcal{A}})v,v\rangle_{V^{*},V}\geq\theta\left\lVert
v\right\rVert^{2}_{V}$

See [BPDM, Part II, Chaptor 1, p 115], [Paz, Theorem 4.2, p14].

######  Consequence 2.

For all $y_{0}\in Y,f\in L^{2}(0,T;V^{*})$, and $T>0$, there exists a unique
solution $y\in W(0,T)$ to

$\dot{y}={\mathcal{A}}y+f,\quad y(0)=y_{0}.$ (4.1)

Furthermore, $y$ satisfies

$\left\lVert y\right\rVert_{W(0,T)}\leq c(T)\Big{(}\left\lVert
y_{0}\right\rVert_{Y}+\left\lVert f\right\rVert_{L^{2}(0,T;V^{*})}\Big{)}$
(4.2)

for a continuous function $c$. Assuming that $y\in L^{2}(0,\infty;Y)$,
consider the equation

$\dot{y}=\underbrace{({\mathcal{A}}-\rho
I)y}_{{\mathcal{A}}_{\rho}}+\underbrace{\rho y+f}_{f_{\rho}},\quad
y(0)=y_{0},$

where $f_{\rho}\in L^{2}(I;V^{*})$. Then the operator ${\mathcal{A}}_{\rho}$
generates a strongly continuous analytic semigroup on $Y$ which is
exponentially stable, see [BPDM, p 115, Theorem II.1.2.12]. It follows that
$y\in W_{\infty}$, that there exists $M_{\rho}$ such that

$\left\lVert y\right\rVert_{W_{\infty}}\leq M_{\rho}\Big{(}\left\lVert
y_{0}\right\rVert_{Y}+\left\lVert
f_{\rho}\right\rVert_{L^{2}(I;V^{*})}\Big{)},$ (4.3)

and that $y$ is the unique solution to (4.1) in $W_{\infty}$, see [BKP3,
Section 2.2] .

######  Lemma 4.1.

There exists a constant $C>0$, such that for all $\delta<(0,1]$ and for all
$y_{1}$ and $y_{2}$ in $W_{\infty}$ with $\displaystyle\left\lVert
y_{1}\right\rVert_{W_{\infty}}\leq\delta$ and $\displaystyle\left\lVert
y_{2}\right\rVert_{W_{\infty}}\leq\delta$, it holds that

$\left\lVert{\mathcal{F}}(y_{1})-{\mathcal{F}}(y_{2})\right\rVert_{L^{2}(I;V^{*})}\leq\delta
C\left\lVert y_{1}-y_{2}\right\rVert_{W_{\infty}}.$ (4.4)

###### Proof.

Let $y_{1},y_{2}$ be as in the statement of the lemma. Using Remark 3.1 we
obtain the estimate

$\displaystyle\left\lVert{\mathcal{F}}(y_{1})-{\mathcal{F}}(y_{2})\right\rVert_{L^{2}(I,V^{*})}\leq\int_{0}^{1}\left\lVert{\mathcal{F}}^{\prime}(y_{1}+t(y_{2}-y_{1}))-{\mathcal{F}}^{\prime}(0)\right\rVert_{{\mathcal{L}}(W_{\infty},L^{2}(I,V^{*}))}\,dt\,\|y_{2}-y_{1}\|_{W_{\infty}}$
$\displaystyle\leq\int_{0}^{1}\int_{0}^{1}\left\lVert{\mathcal{F}}^{\prime\prime}(s(y_{1}+t(y_{2}-y_{1})))(ty_{2}+(1-t)y_{1})\right\rVert_{{\mathcal{L}}(W_{\infty},L^{2}(I,V^{*}))}\,dsdt\,\|y_{2}-y_{1}\|_{W_{\infty}}.$

Now the claim follows by assumption (A3). ∎

######  Lemma 4.2.

Let $\displaystyle{\mathcal{A}}_{s}$ be the generator of an exponentially
stable analytic semigroup $\displaystyle e^{{\mathcal{A}}_{s}t}$ on $Y$. Let
$C$ denote the constant from Lemma 4.1. Then there exists a constant $M_{s}$
such that for all $y_{0}\in Y$ and $f\in L^{2}(I;V^{*})$ with

$\tilde{\gamma}=\left\lVert y_{0}\right\rVert_{Y}+\left\lVert
f\right\rVert_{L^{2}(I;V^{*})}\leq\frac{1}{4CM^{2}_{s}}$

the system

$y_{t}={\mathcal{A}}_{s}y+{\mathcal{F}}(y)+f,\quad y(0)=y_{0}$ (4.5)

has a unique solution $y\in W_{\infty}$, which satisfies

$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{s}\tilde{\gamma}.$

With Lemma 4.1 holding, this lemma can be verified in the same manner as
[BKP3, Lemma 5, p 6]. In the following corollary we shall use Lemma 4.2 with
${\mathcal{A}}_{s}={\mathcal{A}}-BK$, and the constant corresponding to
$M_{s}$ will be denoted by $M_{K}$. Further $\|{\mathcal{I}}\|$ denotes the
norm of the embedding constant of $W_{\infty}$ into $C(I;Y)$, $\|i\|$ is the
norm of the embedding $V$ into $Y$, and we recall the constant $\eta$ from
(3.1).

######  Corollary 4.3.

For all $\displaystyle y_{0}\in Y$ with

$\left\lVert
y_{0}\right\rVert_{Y}\leq\min\left\\{\frac{1}{4CM^{2}_{K}},\frac{\eta}{2M_{K}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\}$

there exists a control $u\in{U_{ad}}$ such that the system

$y_{t}={\mathcal{A}}y+{\mathcal{F}}(y)+Bu,\quad y(0)=y_{0}$ (4.6)

has a unique solution $y\in W_{\infty}$ satisfying

$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{K}\|y_{0}\|_{Y}\quad\text{ and
}\quad\left\lVert
u\right\rVert_{U}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|\|y\|_{W_{\infty}}\leq
2M_{K}\|y_{0}\|_{Y}\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|.$
(4.7)

###### Proof.

By Assumption (A2), there exists $K$ such that ${\mathcal{A}}-BK$ generates an
exponentially stable analytic semigroup on $Y$. Taking $u=-Ky$, equation (4.6)
becomes

$y_{t}=({\mathcal{A}}-BK)y+{\mathcal{F}}(y),\quad y(0)=y_{0}.$ (4.8)

Then by Lemma 4.2 with $\tilde{\gamma}=\|y_{0}\|_{Y}$ there exists $M_{K}$
such that (4.8) has a solution $y\in W_{\infty}$ satisfying

$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{K}\|y_{0}\|_{Y},$

and thus the first inequality in (4.7) holds. For every $t\in I$ we have

$\|u\|_{U}=\|Ky\|_{U}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|y\|_{Y}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|\|y\|_{W_{\infty}}\leq
2M_{K}\|y_{0}\|_{Y}\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|,$
(4.9)

and thus the second inequality in (4.7) holds.We still need to assert that
$u\in U_{ad}$. This follows from the second smallness condition on
$\left\lVert y_{0}\right\rVert_{Y}$ and (4.9). ∎

######  Remark 4.1.

In the above proof stabilization was achieved by the feedback control $u=-Ky$.
For this $u$ to be admissible it is needed that ${\mathcal{U}}_{ad}$ has
nonempty interior. The upper bound $\eta$ could be allowed to be time
dependent as long as it satisfies $\displaystyle\inf_{t\geq 0}|\eta(t)|>0$.

######  Corollary 4.4.

Let $\displaystyle y_{0}\in Y$ and let $u\in U_{ad}$ be such that the system

$y_{t}={\mathcal{A}}y+{\mathcal{F}}(y)+Bu,\quad y(0)=y_{0}$ (4.10)

has a unique solution $y\in L^{2}(I;Y)$. If

$\displaystyle\gamma:=\left\lVert y_{0}\right\rVert_{Y}+\left\lVert\rho
y+Bu\right\rVert_{L^{2}(I;V^{*})}\leq\min\left\\{\frac{1}{4CM^{2}_{\rho}},\frac{\eta}{2M_{\rho}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\},$

then $y\in W_{\infty}$ and it holds that

$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{\rho}\gamma.$

###### Proof.

Since $y\in L^{2}(I;Y)$, we can apply Lemma 4.2 to the equivalent system

$y_{t}=({\mathcal{A}}-\rho I)y+{\mathcal{F}}(y)+\tilde{f},$

where $\tilde{f}=\rho y+Bu$. This proves the assertion. ∎

######  Lemma 4.5.

There exists $\delta_{1}>0$ such that for all $\displaystyle y_{0}\in
B_{Y}(\delta_{1})$, problem ($\mathcal{P}$) possesses a solution
$(\bar{y},\bar{u})\in W_{\infty}\times U_{ad}$. Moreover, there exists a
constant $M>0$ independent of $y_{0}$ such that

$\max\big{\\{}\left\lVert\bar{y}\right\rVert_{W_{\infty}},\left\lVert\bar{u}\right\rVert_{U}\big{\\}}\leq
M\left\lVert y_{0}\right\rVert_{Y}$ (4.11)

###### Proof.

The proof of this lemma follows with analogous argumentation as provided in
[BKP3, Lemma 8]. Let us choose,
$\delta_{1}\leq\min\left\\{\frac{1}{4CM^{2}_{K}},\frac{\eta}{2M_{K}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\}$,
where $C$ as in Lemma 4.1 and $M_{K}$ denotes the constant from the Corollary
4.3. We obtain that for each $y_{0}\in B_{Y}(\delta_{1})$, there exists a
control $u\in U_{ad}$ with associated state $y$ satisfying

$\max\big{\\{}\left\lVert u\right\rVert_{U},\left\lVert
y\right\rVert_{W_{\infty}}\big{\\}}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y},$ (4.12)

where $\displaystyle\tilde{M}=2M_{K}\text{ max }\big{(}1,\left\lVert
i\right\rVert\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\big{)}$. We can thus consider
a minimizing sequence $\displaystyle(y_{n},u_{n})_{n\in\mathbb{N}}\in
W_{\infty}\times U_{ad}$ with $\displaystyle
J(y_{n},u_{n})\leq\frac{1}{2}M^{2}\left\lVert
y_{0}\right\rVert^{2}_{Y}(1+\alpha)$. For all $n\in\mathbb{N}$ that

$\left\lVert y_{n}\right\rVert_{L^{2}(I;Y)}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y}\sqrt{1+\alpha}\quad\text{and}\quad\left\lVert
u_{n}\right\rVert_{L^{2}(I;\,{\mathcal{U}})}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y}\sqrt{\frac{1+\alpha}{\alpha}}.$ (4.13)

We set $\eta(\alpha,\tilde{M})=\Big{[}1+\tilde{M}\|i\|\
\sqrt{(1+\alpha)}\Big{(}\rho+\frac{\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}}{\sqrt{\alpha}}\Big{)}\Big{]}$.
Then we have $\left\lVert y_{0}\right\rVert+\left\lVert\rho
y_{n}+Bu_{n}\right\rVert_{L^{2}(I;V^{*})}\leq\eta(\alpha,\tilde{M})\left\lVert
y_{0}\right\rVert_{Y}$. After further reduction of $\delta_{1}$, we obtain
with $M_{\rho}$ from Corollary 4.4:

$\left\lVert y_{0}\right\rVert+\left\lVert\rho
y_{n}+Bu_{n}\right\rVert_{L^{2}(I;V^{*})}\leq\frac{1}{4CM^{2}_{\rho}}.$

It follows from this corollary that the sequence
$\\{y_{n}\\}_{n\in\mathbb{N}}$ is bounded in $W_{\infty}$ with

$\sup_{n\in\mathbb{N}}\left\lVert y_{n}\right\rVert_{W_{\infty}}\leq
2M_{\rho}(1+\eta(\alpha,\tilde{M}))\left\lVert y_{0}\right\rVert_{Y}.$ (4.14)

Extracting if necessary a subsequence, there exists
$\displaystyle(\bar{y},\bar{u})\in W_{\infty}\times U$ such that
$\displaystyle({y}_{n},{u}_{n})\rightharpoonup(\bar{y},\bar{u})\in
W_{\infty}\times U$, and $(\bar{y},\bar{u})$ satisfies (4.12).

Let us prove that $(\bar{y},\bar{u})$ is feasible and optimal. Since $U_{ad}$
is weakly sequentially closed and $u_{n}\in U_{ad}$, we find $\bar{u}\in
U_{ad}$. For each fixed $T>0$ and arbitrary $z\in
L^{\infty}(0;T;{\mathcal{H}})\subset L^{2}(0,T;V)$, see (A4), we have for all
$\displaystyle n\in\mathbb{N}$ that

$\int_{0}^{T}\langle\dot{y}_{n}(t),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{A}}y_{n}(t)+{\mathcal{F}}(y_{n}(t))+Bu_{n}(t),z(t)\rangle_{V^{*},V}dt.$
(4.15)

Since $\dot{y}_{n}\rightharpoonup\dot{y}$ in $L^{2}(0,T;V^{*})$, we can pass
to the limit in the l.h.s. of the above equality. Moreover, since
${\mathcal{A}}y_{n}\rightharpoonup{\mathcal{A}}y$ in $L^{2}(0,T;V^{*})$,

$\int_{0}^{T}\langle{\mathcal{A}}y_{n}(t),z(t)\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}\int_{0}^{T}\langle{\mathcal{A}}\bar{y}(t),z(t)\rangle_{V^{*},V}dt.$

Analogously, we obtain that

$\int_{0}^{T}\langle
Bu_{n}(t),z(t)\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}\int_{0}^{T}\langle
B\bar{u}(t),z(t)\rangle_{V^{*},V}dt.$

If moreover $z\in L^{\infty}(0,T;{\mathcal{H}})\subset L^{2}(0,T;V)$, we use
(A4) to assert

$\int_{0}^{T}\langle{\mathcal{F}}(y_{n}(t))-{\mathcal{F}}(\bar{y}(t)),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{F}}(y_{n}(t))-{\mathcal{F}}(\bar{y}(t)),z(t)\rangle_{{\mathcal{H}}^{*},{\mathcal{H}}}dt\xrightarrow[n\rightarrow\infty]{}0.$

Thus we have for all $z\in L^{\infty}(0,T;{\mathcal{H}})$

$\int_{0}^{T}\langle\dot{y}(t)-{\mathcal{A}}y(t)-Bu(t),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{F}}(y(t)),z(t)\rangle_{V^{*},V}dt.$
(4.16)

Since $\displaystyle\dot{y}-{\mathcal{A}}y-Bu\in L^{2}(0,T;V^{*})$ and
$L^{\infty}(0,T;{\mathcal{H}})$ is dense in $L^{2}(0,T;V)$ we conclude that
(4.16) holds for all $z\in L^{2}(0,T;V)$ and $T>0$. This yields
$e(\bar{y},\bar{u})=(0,0)$, and thus $(\bar{y},\bar{u})$ is feasible. By weak
lower semicontinuity of norms it follows that $\displaystyle
J(\bar{y},\bar{u})\leq\liminf_{n\rightarrow\infty}J(\bar{y}_{n},\bar{u}_{n})$,
which proves the optimality of $(\bar{y},\bar{u})$, and (4.11) follows from
(4.13). ∎

For the derivation of the optimality system for ($\mathcal{P}$), we need the
following lemma which is taken from [BKP1, Lemma 2.5].

######  Lemma 4.6.

Let $\displaystyle G\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$ such that
$\displaystyle\left\lVert G\right\rVert<\frac{1}{M_{K}}$, where $\left\lVert
G\right\rVert$ denotes the operator norm of $G$. Then for all $\displaystyle
f\in L^{2}(I;V^{*})$ and $y_{0}\in Y$, there exists a unique solution to the
problem:

$y_{t}=({\mathcal{A}}-BK)y(t)+(Gy)(t)+f(t),\quad y=y_{0}.$

Moreover,

$\left\lVert
y\right\rVert_{W_{\infty}}\leq\frac{M_{K}}{1-{M_{K}\|G\|}}\left(\left\lVert
f\right\rVert_{L^{2}(I;V^{*})}+\left\lVert y_{0}\right\rVert_{Y}\right).$

We close this section by deriving the optimality conditions for
($\mathcal{P}$).

######  Proposition 4.1.

Let the assumptions (A1) - (A4) hold. Then there exists
$\delta_{2}\in(0,\delta_{1}]$ such that each local solution
$(\bar{y},\bar{u})$ with $y_{0}\in B_{Y}(\delta_{2})$ is a regular point, i.e.
(2.3) is satisfied, and there exists an adjoint state
$(\bar{p},\bar{p}_{1})\in L^{2}(I;V)\times Y$ satisfying

$\displaystyle\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(\bar{y})v,\bar{p}\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+(v(0),\bar{p}_{1})_{Y}+(\bar{y},\bar{p})_{L^{2}(I;V)}$
$\displaystyle=0,\quad\text{for all }v\in W_{\infty},$ (4.17)
$\displaystyle\langle\alpha\bar{u}-B^{*}\bar{p},u-\bar{u}\rangle_{U}$
$\displaystyle\geq 0,\quad\text{for all }u\in U_{ad}.$ (4.18)

If the assumption (A5) is satisfied, then

$-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}=-\bar{y}\quad\text{
in }L^{2}(I;V^{*}),$

and hence $\bar{p}\in W_{\infty}$ and

$\lim_{t\rightarrow\infty}\bar{p}(t)=0.$ (4.19)

Moreover, there exists $\widetilde{M}>0$, independent of $y_{0}\in
B_{Y}(\delta_{2})$, such that

$\left\lVert\bar{p}\right\rVert_{W_{\infty}}\leq\widetilde{M}\left\lVert
y_{0}\right\rVert_{Y},\text{ and }u\in C(\overline{I},{\mathcal{U}}).$ (4.20)

###### Proof.

To verify the regular point condition, we evaluate $e$ defined in (3.6) at
$(\bar{y},\bar{u},y_{0})$. To check the claim on the range of
$e^{\prime}(\bar{y},\bar{u},y_{0})$ we consider for arbitrary
$\displaystyle(r,s)\in L^{2}(I,V^{*})\times Y$ the equation

$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-\bar{u})=r,\
z(0)=s,$ (4.21)

for unknowns $(z,w)\in W_{\infty}\times{U_{ad}}$. By taking $w=-Kz\in U$ we
obtain

$z_{t}-({\mathcal{A}}-BK)z-{\mathcal{F}}^{\prime}(\bar{y})z+B\bar{u}=r,\
z(0)=s.$

We apply Lemma 4.6 to this equation with $\displaystyle
G=-{\mathcal{F}}^{\prime}(\bar{y})$ and $f=r-B\bar{u}$. By Lemma 4.5 and (3.3)
in Remark 3.1 there exists $\delta_{2}\in(0,\delta_{1}]$ such that
$\|{\mathcal{F}}^{\prime}(\bar{y})\|_{{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))}\leq\frac{1}{2}M_{K}$.
Consequently by Lemma 4.6 there exists $\widetilde{M}$ such that

$\displaystyle\left\lVert z\right\rVert_{W_{\infty}}$
$\displaystyle\leq\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}\left\lVert\bar{u}\right\rVert_{U}\big{)}$
$\displaystyle\leq\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)},$ (4.22)

with $M$ as in (4.11). We shall need to check whether $w=-Kz$ is feasible,
which will be the case if $w(t)\leq\eta$ for a.e. $t\in I$. Indeed we have

$\left\lVert w(t)\right\rVert_{Y}\leq\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\left\lVert
z(t)\right\rVert_{Y}\leq\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\left\lVert{\mathcal{I}}\right\rVert\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)}.$

Consequently, possibly after further reducing $\delta_{2}$, and choosing
$\tilde{\delta}>0$ sufficiently small we have

$\left\lVert w\right\rVert_{L^{\infty}(I;Y)}\leq\eta\text{ for all
}\displaystyle y_{0}\in B_{Y}(\delta_{2})\text{ and all }(r,s)\text{
satisfying }\left\lVert(r,s)\right\rVert_{L^{2}(I;V^{*})\times
Y}\leq\tilde{\delta}.$ (4.23)

Consequently the regular point condition is satisfied. Hence there exists a
multiplier $\displaystyle\lambda=(p,p_{1})\in L^{2}(I;V)\times Y$ satisfying,

$\langle{\mathcal{L}}_{y}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1}),v\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}=0,\quad\langle{\mathcal{L}}_{u}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1}),u-\bar{u}\rangle_{U}\geq
0,\ \forall u\in U_{ad}$ (4.24)

where

${\mathcal{L}}(y,u,y_{0},p,p_{1})=J(y,u)+\int_{0}^{\infty}\langle
p,y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\rangle_{V,V^{*}}dt+\langle
p_{1},y(0)-y_{0}\rangle_{Y}.$

This implies that (4.17) holds.

Now, if we impose the additional assumption (A5), we have
${\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}\in L^{2}(I;V^{*})$. Thus
$-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}+\bar{y}\in
L^{2}(I;V^{*})$ and the previous identity implies that $\bar{p}\in
W_{\infty}$. Thus we derive

$-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}=-\bar{y}\
\text{in }L^{2}(I;V^{*})\text{ and }\lim_{t\rightarrow\infty}\bar{p}(t)=0,$

and (4.17)-(4.19). Testing the first identity in (4.24) with $v\in L^{2}(I;V)$
we also have $\bar{p}_{1}=\bar{p}(0)$, which is well-defined since $\bar{p}\in
W_{\infty}\subset C(I;Y)$. The second identity in (4.24) gives (4.18). It
remains to estimate $\displaystyle\bar{p}\in W_{\infty}$.

Let $\displaystyle r\in L^{2}(I;V^{*})$ with $\displaystyle\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}\leq\tilde{\delta}$ and consider

$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-\bar{u})=-r,\
z(0)=0.$ (4.25)

Arguing as in (4.21)-(4.22) there exists a solution to (4.25) with $w=-Kz$
such that

$\left\lVert
z\right\rVert_{W_{\infty}}\leq\widetilde{M}\big{(}\tilde{\delta}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)}\leq\widetilde{M}\big{(}\tilde{\delta}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\delta_{2}\big{)}=:C_{1}.$
(4.26)

From (4.1) we have that $\displaystyle\left\lVert
w\right\rVert_{L^{\infty}(I,\,{\mathcal{U}})}\leq\eta$. Let us now observe
that

$\displaystyle\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}$
$\displaystyle=\langle\bar{p},-z_{t}+{\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}+\langle\bar{p},B(w-\bar{u})\rangle_{L^{2}(I;Y)}$
$\displaystyle=\langle\bar{p}_{t}+{\mathcal{A}}^{*}\bar{p}+{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p},z\rangle+\langle
B^{*}\bar{p},w-\bar{u}\rangle_{U},$

where we have used that $z(0)=0$ and
$\displaystyle\lim_{t\rightarrow\infty}\bar{p}(t)=0$, since $\bar{p}\in
W_{\infty}$. We next estimate using (4.17), (4.19) and (4.26)

$\displaystyle\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}$
$\displaystyle\leq\left\lVert\bar{y}\right\rVert_{L^{2}(I,V^{*})}\left\lVert
z\right\rVert_{L^{2}(I,V)}+\alpha\langle\bar{u},w-\bar{u}\rangle_{U}$
$\displaystyle\leq\left(\left\lVert\bar{y}\right\rVert_{L^{2}(I,V^{*})}+\alpha\left\lVert\bar{u}\right\rVert_{U}\right)\left(C_{1}+\eta+\left\lVert\bar{u}\right\rVert_{U}\right).$

By (4.11), this implies the existence of a constant $C_{2}$ such that

$\sup_{\left\lVert
r\right\rVert_{L^{2}(I,V^{*})}\leq\tilde{\delta}}\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}\leq
C_{2}\left\lVert y_{0}\right\rVert_{Y}$

and thus

$\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}\leq\frac{C_{2}}{\tilde{\delta}}\left\lVert
y_{0}\right\rVert_{Y},\quad\text{for all }y_{0}\in B_{Y}(\delta_{2}).$ (4.27)

Now we estimate, again using (A5)

$\displaystyle\left\lVert\bar{p}_{t}\right\rVert_{L^{2}(I;V^{*})}$
$\displaystyle\leq\left\lVert{\mathcal{A}}^{*}\bar{p}+{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}-\bar{y}\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}+C_{4}\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}+\left\lVert\bar{y}\right\rVert_{L^{2}(I;V^{*})}.$

By (4.11) and (4.27) we obtain
$\left\lVert\bar{p}_{t}\right\rVert_{L^{2}(I;V^{*})}\leq C_{5}\left\lVert
y_{0}\right\rVert_{Y}.$ Combining this estimate with (4.27) yields (4.20).
Finally, by (4.18) we find
$\displaystyle\bar{u}(t)=\mathbb{P}_{{\mathcal{U}}_{ad}}\left(\frac{1}{\alpha}B^{*}\bar{p}(t)\right)$.
Since $\bar{p}\in C(\bar{I};Y)$ and $B^{*}\in{\mathcal{L}}(Y,U)$ this implies
that $u\in C(\bar{I};{\mathcal{U}})$. ∎

###  4.2 Verification of (H1)-(H6).

In this section we specialize the previously proved abstract results in
Section 2 to the semilinear parabolic setting. We start with the following
lemma which shows that assumptions A imply (H1)-(H6).

######  Lemma 4.7.

Consider problem ($\mathcal{P}$) with assumptions (A1)-(A4) holding. Then
(H1)–(H4), (H6) are satisfied for ($\mathcal{P}$) uniformly for all $y_{0}\in
B_{Y}(\widetilde{\delta}_{2})$ for some
$\displaystyle\widetilde{\delta}_{2}\in(0,\delta_{2}]$. If moreover, (A5)
holds, then (H5) holds as well.

###### Proof.

Throughout $y_{0}\\!\in\\!B_{Y}(\delta_{2})$, $(\bar{y},\bar{u})$ denotes a
local solution to ($\mathcal{P}$), and $(p,p_{1})\in L^{2}(I;V^{*})\times Y$
the associated Lagrange multiplier.

  1. (i)

Verification of (H1): The initial condition $y_{0}$ is our nominal reference
parameter $q$. Lemma 4.5 guarantees the existence of a local solution
$(\bar{y},\bar{u})\sim x_{0}$ to ($\mathcal{P}$)$\sim$ ($P_{q_{0}}$). Clearly
$f$ defined in (3.5) satisfies the required regularity assumptions. Moreover
$e$ satisfies the regularity assumptions as a consequence of (A3).

  2. (ii)

Verification of (H2): Proposition 4.1 implies that $(\bar{y},\bar{u})$ is a
regular point.

  3. (iii)

Verification of (H3): The second derivative of $e$ is given by

$e^{\prime\prime}(\bar{y},\bar{u},y_{0})((v_{1},w_{1}),(v_{2},w_{2}))=\begin{pmatrix}{\mathcal{F}}^{\prime\prime}(\bar{y})(v_{1},v_{2})\\\
0\end{pmatrix},\quad\forall\ v_{1},v_{2}\in W_{\infty},\ \forall
w_{1},w_{2}\in U.$ (4.28)

For the second derivative of ${\mathcal{L}}$ w.r.t. $(y,u)$, we find

${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v_{1},w_{1}),(v_{2},w_{2}))=\int_{0}^{\infty}(v_{1},v_{2})_{Y}dt+\alpha\int_{0}^{\infty}(w_{1},w_{2})_{Y}dt\\\
+\int_{0}^{\infty}\langle\bar{p},{\mathcal{F}}^{\prime\prime}(\bar{y})(v_{1},v_{2})\rangle_{V,V^{*}}dt.$
(4.29)

By (A3) for ${\mathcal{F}}^{\prime\prime}$ and Lemma 4.5 , there exists
$\widetilde{M}_{1}$ such that

$\int_{0}^{\infty}\langle\bar{p},{\mathcal{F}}^{\prime\prime}(\bar{y})(v,v)\rangle_{V,V^{*}}dt\leq\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}},\quad\forall\ v\in W_{\infty},$ (4.30)

for each solution $(\bar{y},\bar{u})$ of ($\mathcal{P}$) with $y_{0}\in
B_{Y}(\delta_{2})$. Then we obtain

${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))\geq\int_{0}^{\infty}\left\lVert
v\right\rVert_{Y}^{2}dt+\alpha\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{{\mathcal{U}}}dt-\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}}.$ (4.31)

Now let $0\neq(v,w)\in\ker E\subset W_{\infty}\times U_{ad}$, where $E$ as
defined in (3.7) is evaluated at $(\bar{y},\bar{u})$. Then,

$v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(\bar{y})v-Bw=0,\quad v(0)=0.$

Next choose $\rho>0$, such that the semigroup generated by
$({\mathcal{A}}-\rho I)$ is exponentially stable. This is possible due to
(A1). We equivalently write the system in the previous equation as,

$v_{t}-({\mathcal{A}}-\rho I)v-{\mathcal{F}}^{\prime}(\bar{y})v-\rho
v-Bw=0,\quad v(0)=0.$

Now, we invoke Lemma 4.6 with ${\cal A}-BK$ replaced by ${\mathcal{A}}-\rho
I$, $\displaystyle G={\mathcal{F}}^{\prime}(\bar{y})$, and $f(t)=\rho
v(t)+Bw(t)$, and the role of the constant $M_{K}$ will now be assumed by a
parameter $M_{\rho}$. By selecting $\widetilde{\delta}_{2}\in(0,\delta_{2}]$
such that $\left\lVert\bar{y}\right\rVert_{W_{\infty}}$ sufficiently small, we
can guarantee that
$\displaystyle\left\lVert{\mathcal{F}}^{\prime}(\bar{y})\right\rVert_{{\mathcal{L}}(W_{\infty};L^{2}(I;V^{*}))}\leq{}^{1}\\!/_{2M_{\rho}}$,
see (4.11) and (3.3) in Remark 3.1. Then the following estimate holds,

$\left\lVert v\right\rVert_{W_{\infty}}\leq 2{M_{\rho}}\left\lVert
v+Bw\right\rVert_{L^{2}(I;V^{*})}.$

This implies that

$\left\lVert v\right\rVert^{2}_{W_{\infty}}\leq\widetilde{M}_{2}(\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert w\right\rVert^{2}_{L^{2}(I;Y)}).$
(4.32)

for a constant $\widetilde{M}_{2}$ depending on $M_{\rho},\|B\|$, and the
embedding of $Y$ into $V^{*}$. These preliminaries allow the following lower
bound on ${\mathcal{L}}^{\prime\prime}$:

$\displaystyle{\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))$
$\displaystyle\geq\int_{0}^{\infty}\left\lVert
v\right\rVert_{Y}^{2}dt+\alpha\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{Y}dt-\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}}$ $\displaystyle\text{by ~{} \eqref{ker_vw}
~{}}\geq\int_{0}^{\infty}\left\lVert v\right\rVert_{Y}^{2}+\alpha$
$\displaystyle\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{Y}-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]$
$\displaystyle=\left(1-\widetilde{M}_{1}\widetilde{M}_{2}\right.$
$\displaystyle\left.\|\bar{p}\|_{L^{2}(I;V)}\right)\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left(\alpha-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\right)\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}$
$\displaystyle\geq\tilde{\gamma}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]$ (4.33)

where
$\displaystyle\tilde{\gamma}=\min\left\\{1-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)},\alpha-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\right\\}$.
By possible further reduction of $\tilde{\delta}_{2}$ it can be guaranteed
that $\tilde{\gamma}>0$, see (4.27). Then by (4.32), we obtain,

$\displaystyle{\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))$
$\displaystyle\geq\frac{\tilde{\gamma}}{2}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]+\frac{\tilde{\gamma}}{2\widetilde{M}_{2}}\left\lVert
v\right\rVert^{2}_{W_{\infty}},$
$\displaystyle\geq\frac{\tilde{\gamma}}{2\widetilde{M}_{2}}\left\lVert
v\right\rVert^{2}_{W_{\infty}}+\frac{\tilde{\gamma}}{2}\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}.$

By selecting
$\displaystyle\bar{\gamma}=\min\left\\{\frac{\tilde{\gamma}}{2\widetilde{M}_{2}},\frac{\tilde{\gamma}}{2}\right\\}$,
we obtain the positive definiteness of ${\mathcal{L}}^{\prime\prime}$, i.e.

${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))\geq\bar{\gamma}\left\lVert(v,w)\right\rVert^{2}_{W_{\infty}\times
U},\ y_{0}\in B_{Y}(\widetilde{\delta}_{2}),\ (v,w)\in\text{ker }E.$ (4.34)

Thus (H3) is satisfied.

  4. (iv)

Verification of (H4): It can easily be checked that $f^{\prime}(y,u)$ can be
extended to an element in $\underline{X^{*}}=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}}))$ for each $(y,u)\in X=W_{\infty}\times U$. We refer
to Remark 3.2 to show that the restriction of $e^{\prime}(y,u,y_{0})^{*}$ to
$\underline{W^{*}}$ satisfies
$\underline{e^{\prime}(y,u,y_{0})^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})={\mathcal{L}}(W_{\infty}\times
Y,L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}})))$.

  5. (v)

Verification of (H6): This is trivially satisfied.

Thus we have proved that assumptions (A1)-(A4) imply (H1)-(H4), and (H6) for
all $\displaystyle y_{0}\in B_{Y}(\widetilde{\delta}_{2})$.

  6. (vi)

Verification of (H5): Here we use (A5) and have
$(p,p_{1})=(p,p(0))\in\widetilde{W}_{\infty}$. Observe that
$\underline{{\mathcal{L}}^{\prime}}:W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times Y\times W_{\infty}\to
L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}}))$ evaluated at
$(\bar{y},\bar{u},y_{0},p)$ is given by

$\underline{{\mathcal{L}}^{\prime}}(\bar{y},\bar{u},y_{0},\bar{p})=\begin{pmatrix}\bar{y}+\underline{e^{\prime}(\bar{y},\bar{u},y_{0})^{*}}\bar{p}\\\
\alpha\bar{u}-B^{*}\bar{p}\end{pmatrix}=\begin{pmatrix}\bar{y}-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}\\\
\alpha\bar{u}-B^{*}\bar{p}\end{pmatrix}.$

Further for $(v.w)\in W_{\infty}\times U$ we have

$(\underline{{\mathcal{L}}^{\prime}})^{\prime}(\bar{y},\bar{u},y_{0},\bar{p})(v,w)=\begin{pmatrix}v-[{\mathcal{F}}^{\prime}(\bar{y})^{*}]^{\prime}(\bar{p},v)\\\
\alpha w\end{pmatrix}\in L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})).$

By (A3) and Remark 3.2 we have that $\underline{{\mathcal{L}}^{\prime}}$ and
$(\underline{{\mathcal{L}}^{\prime}})^{\prime}$ are continuous as mappings
from $W_{\infty}\times(U\cap C(\bar{I};{\mathcal{U}}))\times Y\times
W_{\infty}$ to $L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}}))$,
respectively to ${\mathcal{L}}(W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}));L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})))$.

This proves the lemma. ∎

######  Remark 4.2.

Let us summarize our findings so far. There exists $\tilde{\delta}_{2}$ such
that for each $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ problem ($\mathcal{P}$)
posesses a solution $(\bar{y},\bar{u})\in W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))$, with an adjoint
$\bar{p}\in\widetilde{W}_{\infty}$. Further (A1)-(A5) imply (H1)-(H6) for
($\mathcal{P}$) with $y_{0}\in B_{Y}(\tilde{\delta}_{2})$. As a consequence
for each $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ and each associated local
solution $(\bar{y},\bar{u})$ there exists a neighborhood $\hat{V}$ of the
origin in ${\mathcal{Y}}:=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}}))\times L^{2}(I;V^{*})\times Y$ such that for each
$\boldsymbol{\beta}\in\hat{V}$ there exists a unique solution
$\displaystyle\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})},{p_{1}}_{(\boldsymbol{\beta})}\right)\in
W_{\infty}\times U\times L^{2}(I;V)\times Y$ to
${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$, see step (ii) of the proof of
Theorem 2.1. – To verify the remaining assumption (H7) we need to argue that
$\left(u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)\in(U\cap
C(\bar{I};{\mathcal{U}}))\times W_{\infty}$ and that
$\displaystyle\boldsymbol{\beta}\mapsto\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$
is Lipschitz continuous from $\hat{V}\subset\mathcal{Y}$ to
$W_{\infty}\times(U\cap C(\bar{I};{\mathcal{U}}))\times W_{\infty}$ .

######  Remark 4.3.

Here we remark on the smallness assumption on $y_{0}$ expressed by
$\delta_{2}$, respectively $\tilde{\delta}_{2}$. The condition $y_{0}\in
B_{Y}(\delta_{2})$ guarantees the well-posedness of ($\mathcal{P}$), existence
and boundedness of adjoint states as expressed in Proposition 4.1. The
additional condition $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ implies that the
second order optimality condition (H3) is satisfied, for each local solution
associated to an initial condition $y_{0}\in B_{Y}(\delta_{2})$. In what
follows formulate the results for all $y_{0}\in B_{Y}(\tilde{\delta}_{2})$.
Alternatively we could narrow down the claims to neighborhoods of single local
solutions $(\bar{y},\bar{u})$ with $y_{0}\in B_{Y}(\delta_{2})$ and
additionally assuming that the second order condition is satisfies at
$(\bar{y},\bar{u})$. Concerning the second order condition itself, in some
publications, see e.g. [Gri], it is required to hold only for elements
$x=(y,u)\in\text{ker}E$ and $u=u_{1}-u_{2}$, with $u_{1},u_{2}$ in $U_{ad}$.
By a scaling argument it can easily be seen that this condition is equivalent
to the one we use.

###  4.3 Verification of (H7) and Lipschitz stability of the linearized
problem.

Throughout the remainder, we assume that (A1)-(A5) are satisfied and that
$y_{0}\in B_{Y}(\tilde{\delta}_{2})$ so that Proposition 4.1 and Lemma 4.7 are
applicable. In the following, the triple $(y,u,p)$ refers to the solution
${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$. Throughout without loss of
generality, we also assume that $\hat{V}$ is bounded.

######  Lemma 4.8.

Let assumptions (A) hold and let $(\bar{y},\bar{u})$, and $\bar{p}$ denote a
local solution and associated adjoint state to ($\mathcal{P}$) corresponding
to an initial datum $y_{0}\in B_{Y}(\tilde{\delta}_{2})$. Then the mapping
$\boldsymbol{\beta}\mapsto p_{(\boldsymbol{\beta})}$ is continuous from
$\hat{V}$ to $W_{\infty}$.

###### Proof.

Step 1: For $\boldsymbol{\beta}\in\hat{V}$, with $\hat{V}$ as in Remark 4.2,
let
$\displaystyle(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})},{p_{1}}_{(\boldsymbol{\beta})})$
be the solution to ${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$. As a
consequence of (A5) it is also a solution to
$\underline{{\mathcal{T}}}(y,u,p,p(0))=\boldsymbol{\beta}$ with $p\in
W_{\infty}$. Thus the first two equations of this latter equality can be
expressed as

$\displaystyle-
p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(\bar{y})^{*}p+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})$
$\displaystyle=\beta_{1}\quad\text{ in }L^{2}(I;V^{*}),$ (4.35a)
$\displaystyle\langle\alpha u-B^{*}p-\beta_{2},w-u\rangle_{U}$
$\displaystyle\geq 0\quad\text{for all }w\in U_{ad},$ (4.35b)

where we dropped the dependence of
$\displaystyle\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$
on $\displaystyle\boldsymbol{\beta}$. Since $p\in\widetilde{W}_{\infty}\subset
C(\bar{I};Y)$ inequality (4.35b) implies that $u\in C(\bar{I};{\mathcal{U}})$.

Step 2: (Boundedness of
$\\{p_{(\boldsymbol{\beta})}:\boldsymbol{\beta}\in\hat{V}\\}$). Since
$\hat{V}$ is assumed to be bounded, the discussion in Remark 4.2 shows that
there exists a constant $M>0$ such that

$\left\lVert y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\boldsymbol{\beta})}\right\rVert_{U}\leq M\quad\text{for all
}\boldsymbol{\beta}\in\hat{V}.$

To argue the boundedness of $p_{(\boldsymbol{\beta})}$, we use similar
techniques as in the proof of Proposition 4.1. In the following
$\tilde{\delta},z$ and $C_{1}$ are taken as in the proof of that proposition.
For arbitrary $r\in R=\left\\{r\in L^{2}(I;V^{*}):\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}\leq\tilde{\delta}\right\\}$, $z$ denotes the
solution to

$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-u)=-r,\ z(0)=0,\
w=-Kz.$ (4.36)

We know that $\left\lVert z\right\rVert_{W_{\infty}}\leq C_{1}$ and
$\left\lVert w\right\rVert_{L^{\infty}(I;{\mathcal{U}})}\leq\eta$
independently of $r\in R$. Due to (4.35a), we have

$\displaystyle\langle p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}$
$\displaystyle=\langle
p,-z_{t}+{\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}+\langle
B^{*}p,w-u\rangle_{U},$ $\displaystyle=\langle
y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})-\beta_{1},z\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\alpha
u-\beta_{2},w-u\rangle_{U},$ (4.37)

where we used (4.35b) and the feasibility of $w\in U_{ad}$. Consequently

$\langle p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}\leq\left\lVert
z\right\rVert_{L^{2}(I;V)}\left(\left\lVert
y\right\rVert_{L^{2}(I;V^{*})}+\left\lVert\beta_{1}\right\rVert_{L^{2}(I;V^{*})}+\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})\right\rVert_{L^{2}(I;V^{*})}\right)\\\
+\left(\alpha\left\lVert
u\right\rVert_{U}+\left\lVert\beta_{2}\right\rVert_{U}\right)\left\lVert
w-u\right\rVert_{U}.$

The right hand side is uniformly bounded for $\boldsymbol{\beta}$ in the
bounded set $\hat{V}$ and w.r.t. $r\in R$. Hence taking the supremum w.r.t.
$r\in R$ we verified that $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{L^{2}(I;V^{*})}:\boldsymbol{\beta}\in\hat{V}\right\\}$
is bounded. Boundedness of $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}:\boldsymbol{\beta}\in\hat{V}\right\\}$
follows from (4.35a).

Step 3: (Continuity of $p_{(\boldsymbol{\beta})}$ in $W_{\infty}$). Let
$\\{\boldsymbol{\beta}_{n}\\}$ be a convergent sequence in $\hat{V}$ with
limit $\boldsymbol{\beta}$. Since $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}:n\in\mathbb{N}\right\\}$ is
bounded, there exists a subsequence $\\{\boldsymbol{\beta}_{n_{k}}\\}$ such
that $\displaystyle p_{(\boldsymbol{\beta}_{n_{k}})}\rightharpoonup\tilde{p}$
weakly in $W_{\infty}$ and strongly $L^{2}(I;Y)$. Here we need the compactness
of $W_{\infty}$ in $L^{2}(I;Y)$, see e.g. [Emm, Satz 8.1.12, pg 213]. Passing
to the limit in the variational form of

$-\partial_{t}p_{(\boldsymbol{\beta}_{n_{k}})}-{\mathcal{A}}^{*}p_{(\boldsymbol{\beta}_{n_{k}})}-{\mathcal{F}}^{\prime}(\bar{y})^{*}p_{(\boldsymbol{\beta}_{n_{k}})}+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}\left(y_{(\boldsymbol{\beta}_{n_{k}})}-\bar{y}\right)=(\boldsymbol{\beta}_{n_{k}})_{1},$

we obtain

$-\partial_{t}\tilde{p}-{\mathcal{A}}^{*}\tilde{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\tilde{p}+y_{(\boldsymbol{\beta})}-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}\left(y_{(\boldsymbol{\beta})}-\bar{y}\right)=(\boldsymbol{\beta})_{1}.$
(4.38)

Since the solution to this equation is unique we have $\displaystyle
p_{(\boldsymbol{\beta}_{n})}\rightharpoonup p_{(\boldsymbol{\beta})}$ weakly
in $W_{\infty}$. To obtain strong convergence we set
$\delta\boldsymbol{\beta}=\boldsymbol{\beta}_{n}-\boldsymbol{\beta},\ \delta
p=p_{(\boldsymbol{\beta}_{n})}-p_{(\boldsymbol{\beta})},\ \delta
y=y_{(\boldsymbol{\beta}_{n})}-y_{(\boldsymbol{\beta})}$. From (4.35a) we
derive that

$-\partial_{t}(\delta p)-{\mathcal{A}}^{*}(\delta
p)-{\mathcal{F}}^{\prime}(\bar{y})^{*}(\delta
p)+(I-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p})]^{\prime}(\delta
y)=(\delta\boldsymbol{\beta})_{1}.$ (4.39)

Consider again (4.36) with $r\in R$. Then we obtain

$\langle\delta
p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}=\langle(I-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime})(\delta
y)-(\delta\boldsymbol{\beta})_{1},z\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\delta
p,BKz-\bar{u}\rangle_{U}.$

This implies that for some $C_{2}>0$,

$\sup_{r\in R}\ \langle\delta p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}\leq
C_{2}\left(\left\lVert\delta y\right\rVert_{W_{\infty}}+\left\lVert\delta
p\right\rVert_{L^{2}(I;Y)}+\left\lVert(\delta\boldsymbol{\beta})_{1}\right\rVert_{L^{2}(I;V^{*})}\right).$
(4.40)

Since $\left\lVert\delta y\right\rVert_{W_{\infty}}\to 0,\ \left\lVert\delta
p\right\rVert_{L^{2}(I;Y)}\to 0,\
\left\lVert(\delta\boldsymbol{\beta})_{1}\right\rVert_{L^{2}(I;V^{*})}\to 0$
for $n\to 0$ this implies that $\left\lVert\delta
p\right\rVert_{L^{2}(I;V)}\to 0$. Together with (4.39) this implies that
$\displaystyle\lim_{n\rightarrow\infty}\left\lVert\delta
p\right\rVert_{W_{\infty}}=0$. ∎

######  Proposition 4.2.

Let assumptions (A) hold and let $(\bar{y},\bar{u})$, and $\bar{p}$ denote a
local solution and associated adjoint state state to ($\mathcal{P}$)
corresponding to an initial condition $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Then there exists $\varepsilon>0$ and $C>0$ such
that for all $\boldsymbol{\hat{\beta}}$ and $\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$

$\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{C(\bar{I};{\mathcal{U}})}\\!\\\
\leq
C\left(\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right)$
(4.41)

holds.

###### Proof.

Since $\bar{p}\in W_{\infty}$, there exists $T>0$ such that

$\frac{1}{\alpha}\left\lVert
B^{*}\bar{p}\right\rVert_{Y}\leq\frac{\eta}{2},\quad\forall t>T.$

Since $p(0)=\bar{p}$ and since by Lemma 4.8,
$\displaystyle\boldsymbol{\beta}\in{\mathcal{Y}}\mapsto
p_{(\boldsymbol{\beta})}\in W_{\infty}\subset C(\bar{I};Y)$ is continuous,
there exists $\varepsilon>0$ such that

$\frac{1}{\alpha}\left\lVert
B^{*}p_{(\boldsymbol{\beta})}(t)+\beta_{2}(t)\right\rVert_{Y}\leq\frac{\eta}{4},\quad\forall
t\geq T,\ \forall\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon).$

Consequently the constraints are inactive for these parameter values, i.e. we
have

$u_{(\boldsymbol{\beta})}(t)=\frac{1}{\alpha}\left[B^{*}p_{(\boldsymbol{\beta})}(t)+\beta_{2}(t)\right],\
\left\lVert u_{(\boldsymbol{\beta})}(t)\right\rVert_{Y}\leq\eta,\quad\forall
t\geq T,\ \forall\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon).$ (4.42)

Let us henceforth set
$(y,u,p)=\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$,
and
$\left(\hat{y},\hat{u},\hat{p}\right)=\left(\hat{y}_{(\boldsymbol{\hat{\beta}})},\hat{u}_{(\boldsymbol{\hat{\beta}})},\hat{p}_{(\boldsymbol{\hat{\beta}})}\right)$.
We shall use that

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}=\sup_{\left\lVert
r\right\rVert_{L^{2}(T,\infty;V^{*})}\leq
1}\int_{T}^{\infty}\langle\hat{p}(t)-p(t),r(t)\rangle_{V,V^{*}}dt.$

Let $z\in W(T,\infty)$ be such that,

$z_{t}-({\mathcal{A}}-BK)z-{\mathcal{F}}^{\prime}(\bar{y})z=r,\ z(T)=0,$

From Lemma 4.6, see also the proof of Proposition 4.1, we know that there
exists a constant $C_{1}>0$ such that $\displaystyle\left\lVert
z\right\rVert_{W(T,\infty)}\leq C_{1}\left\lVert
r\right\rVert_{L^{2}(T,\infty;V^{*})}$. Then we can estimate

$\displaystyle\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}$
$\displaystyle=\sup_{\left\lVert r\right\rVert_{L^{2}(T,\infty;V^{*})}\leq
1}\int_{T}^{\infty}\langle\hat{p}-p,r\rangle_{V,V^{*}}dt$
$\displaystyle\leq\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}-\langle(\hat{p}_{t}-p_{t})+{\mathcal{A}}^{*}(\hat{p}-p)+{\mathcal{F}}^{\prime}(\bar{y})^{*}(\hat{p}-p),z\rangle_{V^{*},V}dt$
$\displaystyle\hskip 199.16928pt+\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}\langle B^{*}(\hat{p}-p),Kz\rangle_{V,V^{*}}dt.$

In the following, $C_{i}$ denote constants independent of
$\boldsymbol{\hat{\beta}}\text{ and }\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$. From (4.35a) and (4.42) we obtain, for
$C_{2}>0$,

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}\leq
C_{2}\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}\left[\left\lVert\hat{y}-y\right\rVert_{V^{*}}+\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(\hat{y}-y)\right\rVert_{V^{*}}+\left\lVert\hat{\beta}_{1}-\beta_{1}\right\rVert_{V^{*}}\right.\\\
\left.+\left\lVert\hat{\beta}_{2}-\beta_{2}\right\rVert_{U\cap
C(I;{\mathcal{U}})}+\alpha\left\lVert B^{*}\right\rVert\left\lVert
K\right\rVert\left\lVert\hat{u}-u\right\rVert\right]\left\lVert
z\right\rVert_{V}dt.$

From (A3) recall that
$\displaystyle\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(\hat{y}-y)\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}$. This gives the following
estimate for $C_{4}>0$,

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}\leq
C_{4}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.43)

By (4.35a) we have $\displaystyle\left(\hat{p}_{t}-p_{t}\right)\in
L^{2}(T,\infty;V^{*})$. Then we obtain $\hat{p}-p\in W(T,\infty)$. Then there
exists $C_{5}>0$ independent of $\boldsymbol{\hat{\beta}}\text{ and
}\boldsymbol{\beta}\in\hat{V}\cap B_{{\mathcal{Y}}}(\varepsilon)$ such that,

$\left\lVert\hat{p}-p\right\rVert_{W(T,\infty)}\leq
C_{5}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.44)

By the embedding $\displaystyle W(T,\infty)\subset C(T,\infty;Y)$, there
exists a constant $C_{6}>0$:

$\left\lVert\hat{p}-p\right\rVert_{C([T,\infty);Y)}\leq
C_{6}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.45)

Similarly, we estimate on $[0,T]$:

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}=\sup_{\left\lVert
r\right\rVert_{L^{2}(0,T;V^{*})}\leq
1}\int_{0}^{T}\langle\hat{p}-p,r\rangle_{V,V^{*}}dt.$ (4.46)

Choose $z$ as

$z_{t}-\left({\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\right)=r,\
z(0)=0,$

Then there exists $\displaystyle C_{7}>0$ such that $\displaystyle\left\lVert
z\right\rVert_{W(0,T)}\leq C_{7}\left\lVert r\right\rVert_{L^{2}(0,T;V^{*})}$
by Lemma 4.6. Note that $C_{7}$ depends on $T$, but $T$ is fixed. We obtain
the following estimate,

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}\leq\sup_{\left\lVert
r\right\rVert\leq
1}\int_{0}^{T}-\langle(\hat{p}_{t}-p_{t})+{\mathcal{A}}^{*}(\hat{p}-p)+{\mathcal{F}}^{\prime}(\bar{y})^{*}(\hat{p}-p),z\rangle_{V^{*},V}dt\\\
+\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert z(T)\right\rVert_{Y}.$

Then by a similar computation to that for $t\in[T,\infty)$ case, we obtain,

$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}\leq
C_{8}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{\beta}_{1}-\beta_{1}\right\rVert_{L^{2}(I;V^{*})}\right)+\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert
z(T)\right\rVert_{Y}.$ (4.47)

By (4.45) with $\left\lVert z(T)\right\rVert_{Y}\leq C_{9}$, we obtain

$\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert
z(T)\right\rVert_{Y}\leq
C_{7}C_{9}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$

Combining this estimate with (4.43) and (4.47), we obtain for some $C_{10}>0$

$\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}\leq
C_{10}\left(\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.48)

We also have

$u_{(\boldsymbol{\beta})}=\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}p_{(\boldsymbol{\beta})}+\beta_{2}\right)\right]\in
U\cap C(\bar{I};{\mathcal{U}}),$

and thus

$\displaystyle\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}(t)-u_{(\boldsymbol{\beta})}(t)\right\rVert_{{\mathcal{U}}}$
$\displaystyle\leq\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}\hat{p}_{(\hat{\boldsymbol{\beta}})}(t)+\hat{\beta}_{2}(t)\right)\right]-\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}p(t)_{(\boldsymbol{\beta})}+\beta_{2}(t)\right)\right]\right\rVert_{{\mathcal{U}}}$
$\displaystyle\leq\frac{1}{\alpha}\left(\left\lVert
B^{*}\right\rVert\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}(t)-p(t)_{(\boldsymbol{\beta})}\right\rVert_{Y}+\left\lVert\hat{\beta}_{2}(t)-\beta_{2}(t)\right\rVert_{{\mathcal{U}}}\right).$

This yields

$\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{C(\bar{I};{\mathcal{U}})}\leq
C_{11}\left(\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{\beta}_{2}-\beta_{2}\right\rVert_{C(\bar{I};{\mathcal{U}})}\right),$
(4.49)

and (4.41) follows. Combining Remark 4.2 and (4.41) there exists a constant
$L$ such that

$\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U\cap
C(\bar{I};Y)}\leq
L\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}$
(4.50)

for all $\displaystyle\boldsymbol{\hat{\beta}}$ and
$\displaystyle\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$. Here and in the following the $p_{1}$
coordinate of the adjoint state coincides with $p(0)$. Therefore it is not
indicated. ∎

With the above proposition, the verification of (H1)–(H7) is concluded. We
also obtain the following corollary to Theorem 2.1.

######  Corollary 4.9.

Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be a local solution of
($\mathcal{P}$) corresponding to an initial datum $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Then there exist $\delta_{3}>0$, a neighborhood
$\hat{U}=\hat{U}(\bar{y},\bar{u},p)\subset W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times W_{\infty}$, and a constant $\mu>0$ such that
for each $y_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$ there exists a unique
$(y(y_{0}),u(y_{0}),p(y_{0}))\in\hat{U}$ satisfying the first order condition,
and

$\left\lVert\left(y(\hat{y}_{0}),u(\hat{y}_{0})),p(\hat{y}_{0}))\right)-\left(y(\tilde{y}_{0}),u(\tilde{y}_{0}),p(\tilde{y}_{0}))\right)\right\rVert_{W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times
W_{\infty}}\leq\mu\left\lVert\hat{y}_{0}-\tilde{y}_{0}\right\rVert_{Y},$
(4.51)

for all $\hat{y}_{0},\tilde{y}_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$, and
$\left(y(y_{0}),u(y_{0})\right)$ is a local solution of ($\mathcal{P}$).

Next we obtain one of the main results of this paper, the Fréchet
differentiability of the local value function associated to ($\mathcal{P}$).
By local value function we refer to the fact that for some $y_{0}\in
B_{Y}(\tilde{\delta}_{2})$ may not be unique. But since due to the second
order optimality condition local solutions are locally unique under small
perturbations of $y_{0}$, there is a well-defined local value function. We
continue to use the notation for $\hat{U}$ and $B_{Y}(\bar{y}_{0},\delta_{3})$
of Corollary 4.9.

######  Theorem 4.1.

(Sensitivity of Cost) Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be
a local solution of ($\mathcal{P}$) corresponding to an initial datum
$\bar{y}_{0}\in B_{Y}(\tilde{\delta}_{2})$. Then for each $y_{0}\in
B_{Y}(\bar{y}_{0},\delta_{3})$ the local value function $\mathcal{V}$
associated to ($\mathcal{P}$) is Fréchet differentiable with derivative given
by

$\mathcal{V}^{\prime}(y_{0})=-p(0;y_{0}).$ (4.52)

###### Proof.

Let $\bar{y}_{0}\in B_{Y}(\tilde{\delta}_{2}),y_{0}\in
B_{Y}(\bar{y}_{0},\delta_{3})$, and choose $\delta y_{0}$ sufficiently small
so that $y_{0}+\delta y_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$ as well.
Following Corollary 4.9 let $(\tilde{y}(y_{0}+s(\delta
y_{0})),\tilde{u}(y_{0}+s(\delta y_{0})),\tilde{p}(y_{0}+s(\delta
y_{0})))\in\hat{U}$ for $s\in[0,1]$ be solutions of the optimality system with
$(\tilde{y}(y_{0}+s(\delta y_{0})),\tilde{u}(y_{0}+s(\delta y_{0})))$ local
solutions to ($\mathcal{P}$). We obtain

$\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})=\left(\frac{1}{2}\left\lVert\tilde{y}\right\rVert^{2}_{L^{2}(I,Y)}+\frac{\alpha}{2}\left\lVert\tilde{u}\right\rVert^{2}_{U}\right)-\left(\frac{1}{2}\left\lVert
y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{\alpha}{2}\left\lVert
u\right\rVert^{2}_{U}\right)\\\ =\langle
y,\tilde{y}-y\rangle_{L^{2}(I,Y)}+\alpha\langle
u,\tilde{u}-u\rangle_{U}+\frac{1}{2}\left\lVert\tilde{y}-y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{{\alpha}}{2}\left\lVert\tilde{u}-u\right\rVert^{2}_{U}.$
(4.53)

Observe the identity

$\langle y,\tilde{y}-y\rangle_{L^{2}(I,Y)}+\alpha\langle
u,\tilde{u}-u\rangle_{U}={-}(p(0),s(\delta
y_{0}))_{Y}{-}\langle(\tilde{y}_{t}-y_{t})-{\mathcal{A}}(\tilde{y}-y)-{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle+\alpha\langle
u,\tilde{u}-u\rangle_{U}\\\ ={-}(p(0),s(\delta
y_{0}))_{Y}{-}\langle{\mathcal{F}}(\tilde{y})-{\mathcal{F}}(y)-{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U},$

where $p=p(y_{0})$. Now we have for $\displaystyle\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})$,

$\mathcal{V}(y_{0}+{s}(\delta y_{0}))-\mathcal{V}(y_{0})={-}(p(0),{s}(\delta
y_{0}))_{Y}+\langle{\mathcal{F}}(y)-{\mathcal{F}}(\tilde{y})+{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I,V^{*}),L^{2}(I,V)}\\\
+\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U}+\frac{1}{2}\left\lVert\tilde{y}-y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{{\alpha}}{2}\left\lVert\tilde{u}-u\right\rVert^{2}_{U}.$
(4.54)

Since $p\in L^{2}(I;V)$,
$\left\lVert\tilde{y}-y\right\rVert_{W_{\infty}}=O(s)$, and by the continuous
Fréchet differentiability of ${\mathcal{F}}^{\prime}$ due to (A3) we have

$\left\lvert\langle{\mathcal{F}}(\tilde{y})-{\mathcal{F}}(y)+{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I,V^{*}),L^{2}(I,V)}\right\rvert=o(s)$
(4.55)

Let $s_{n}\to 0$ be an arbitrary convergent sequence. By Corollary 4.9 we have
that

$\left\lVert\tilde{u}(y_{0}+s_{n}(\delta
y_{0}))-u(y_{0})\right\rVert_{U}\leq\mu s_{n}(\delta y_{0})$

for all $s_{n}$ sufficiently small. Hence there exists a subsequence, denoted
by the same notation and some $\dot{u}$ such that

$s_{n}^{-1}\left(\tilde{u}(y_{0}+s_{n}(\delta
y_{0}))-u(y_{0})\right)\rightharpoonup\dot{u}\text{ weakly in }U.$

Using (4.18), we have

$\lim_{n\rightarrow\infty}s_{n}^{-1}\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U}=\langle\alpha u-B^{*}p,\dot{u}\rangle_{U}\geq
0.$

Analogously

$\lim_{n\rightarrow\infty}s_{n}^{-1}\langle\alpha\tilde{u}-B^{*}p,u-\tilde{u}\rangle_{U}=\langle\alpha
u-B^{*}p,\dot{u}\rangle_{U}\leq 0.$

and hence $\langle\alpha u-B^{*}p,\dot{u}\rangle_{U}=0$. Since the sequence
$\\{s_{n}\\}$ is arbitrary, we obtain

$\langle\alpha u-B^{*}p,\tilde{u}-u\rangle_{U}=o(s).$ (4.56)

Corollary 4.9 yields,

$\left\lVert\tilde{y}(y_{0}+{s_{n}}(\delta
y_{0}))-y(y_{0})\right\rVert^{2}_{L^{2}(I;Y)}+\alpha\left\lVert\tilde{u}(y_{0}+{s_{n}}(\delta
y_{0}))-u(y_{0})\right\rVert^{2}_{L^{2}(I;Y)}=o(s_{n}).$ (4.57)

Combining (4.55), (4.56), and (4.57) we obtain

$\lim_{s\rightarrow 0^{+}}{s^{-1}}\left(\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})\right)={-}(p(0),(\delta y_{0}))_{Y}.$ (4.58)

This implies the Gateaux differentiability. Since $y_{0}\to p(y_{0})$ is
continuous from $B_{Y}(\bar{y}_{0};\delta_{3})$ to $C(\bar{I},Y)$ the mapping
$y_{0}\to\mathcal{V}(y_{0})$ is Fréchet differentiable in
$B_{Y}(\bar{y}_{0};\delta_{3})$. ∎

######  Remark 4.4 (Sensitivity w.r.t. other parameters).

We have developed a technique to verify the continuous differentiability of
the local value function $\mathcal{V}$ pertaining to a semilinear parabolic
equation on infinite time horizon subject to control constraints with respect
to small initial data $y_{0}\in Y$. Thus the parameter $q$ in ($P_{q}$) is the
initial condition $y_{0}$. The reason to focus on this case is due to feedback
control. Without much additional effort the sensitivity analysis of the value
function could be carried out with respect to other parameters as for instance
additive noise on the right hand side of the state equation. The papers cited
in the introduction, see e.g. [GHH], [GV], consider such situations for the
finite horizon case.

##  5 Proof of Theorem 3.2: Derivation of the HJB Equation.

Utilizing the results established so far we now verify that the (global) value
function $\mathcal{V}$ (i.e. the value function associated to global minima)
is a solution to a Hamilton-Jacobi-Bellman equation. The initial conditions
will be chosen from the neighborhood $Y_{0}$ of the origin in $Y$ so that the
assertions of Theorem 4.1 and Corollary 4.9 are available. It will be
convenient to recall the dynamic programming principle for the infinite time
horizon problem: let $y_{0}$ be an initial condition for which a solution to
($\mathcal{P}$) exists. Then for all $\tau>0$, we have

$\mathcal{V}(y_{0})=\inf_{u\in
L^{2}(0,\tau;\mathcal{U}_{ad})}\int_{0}^{\tau}\ell(S(u,y_{0};t),u(t))dt+\mathcal{V}(S(u,y_{0};\tau)),$
(5.1)

where $\displaystyle\ell(y,u)=\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert
u\right\rVert^{2}_{{\mathcal{U}}}$, and $S(u,y_{0};t)$ denotes the solution to
(3.2b), (3.2c) on $(0,\tau]$.

For convenience we restate Theorem 3.2. Utilizing the notation that we have
already established we can now slightly ease the assumption on the regularity
of ${\mathcal{F}}(\bar{y})$.

######  Theorem 5.1.

Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be a global solution of
($\mathcal{P}$) corresponding to an initial datum $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Let $Y_{0}$ denote the subset of initial
conditions in $B_{Y}(\bar{y}_{0},\delta_{3})$ which allow global solutions in
$\hat{U}$, and assume that for each $y_{0}\in{\mathcal{D}}({\mathcal{A}})\cap
Y_{0}$ there exists $T_{y_{0}}>0$ such that ${\mathcal{F}}(\bar{y})\in
C([0,T_{y_{0}});Y)$. Then the following Hamilton-Jacobi-Bellman equation holds
at $y_{0}$:

$\mathcal{V}^{\prime}(y)({\mathcal{A}}y+{\mathcal{F}}(y))+\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rangle_{Y}=0.$
(5.2)

If for the optimal trajectory $\bar{y}(t)\in
B_{Y}(\bar{y}_{0},\delta_{3})\cap{\mathcal{D}}({\mathcal{A}})$ for a.a.
$t\in(0,\infty$) and $T_{y_{0}}=\infty$, then (5.2) holds at a.a.
$t\in(0,\infty$) and

${\bar{u}}(t)=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}({\bar{y}}(t))\right).$
(5.3)

###### Proof.

The proof is similar to that of [BKP1, Proposition 10]. For the sake of
completeness and since it also requires some changes we provide it here.
Choose and fix some $y_{0}\in{\mathcal{D}}({\mathcal{A}})\cap Y_{0}$. Then the
existence of a (globally) optimal pair $(\hat{y},\hat{u})\in W_{\infty}\times
U_{ad}$ to ($\mathcal{P}$) and of an associated adjoint state $\hat{p}\in
W_{\infty}$ with $(\hat{y},\hat{u},\hat{p})\in\hat{U}$ are guaranteed, see
Corollary 4.9. In particular we have that
$\displaystyle\hat{u}(t)=\mathbb{P}_{\mathcal{U}_{ad}}\left(\frac{1}{\alpha}B^{*}{\hat{p}}(t)\right)$,
and since $\displaystyle\hat{p}\in C([0,\infty);Y)$ we have that
$\displaystyle\hat{u}\in C([0,\infty);Y)$. Let $u_{0}$ denote the limit of
$\hat{u}$ as time $t$ tends to $0$. Since $\hat{y}\in C([0,\infty);Y)$ and
since $B_{Y}(y_{0},\delta_{3})$ is open there exists $\tau_{y_{0}}>0$ such
that $\hat{y}(t)\in B_{Y}(y_{0},\delta_{3})$, for all $t\in[0,\tau_{y_{0}})$.

Step 1: Let us first prove that

$\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\big{)}+\ell(y_{0},u_{0})=0.$
(5.4)

For this purpose we invoke the dynamic programing principle: We have

$\frac{1}{\tau}\int_{0}^{\tau}\ell(\hat{y}(s),\hat{u}(s))ds+\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=0,$
(5.5)

where we choose $\tau\in(0,\min(T_{y_{0}},\tau_{y_{0}}))$ . By continuity of
$\hat{y}$ and $\hat{u}$ at time $0$, the first term converges to
$\displaystyle\ell(y_{0},u_{0})$ as $\tau\to 0$. To take $\tau\to 0$ in the
second term we first consider

$\frac{1}{\tau}\big{(}\hat{y}(\tau)-y_{0}\big{)}=\frac{1}{\tau}\big{(}e^{{\mathcal{A}}\tau}y_{0}-y_{0}\big{)}+\frac{1}{\tau}\int_{0}^{\tau}e^{{\mathcal{A}}(\tau-s)}\big{[}{\mathcal{F}}(\hat{y}(s))+B\hat{u}(s)\big{]}ds.$
(5.6)

Using the facts that $y_{0}\in{\mathcal{D}}({\mathcal{A}})$, that the terms in
square brackets are continuous with values in $Y$, and that ${\mathcal{A}}$
generates a strongly continuous semigroup on $Y$, we can pass to the limit in
(5.6) to obtain that

$\lim_{\tau\to
0^{+}}\frac{1}{\tau}\big{(}\hat{y}(\tau)-y_{0}\big{)}={\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\text{
in }Y.$ (5.7)

Now we return to the second term in (5.5) which we express as

$\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=\int_{0}^{1}\mathcal{V}^{\prime}\big{(}y_{0}+s(\hat{y}(\tau)-y_{0})\big{)}ds\;\frac{1}{\tau}(\hat{y}(\tau)-y_{0}).$
(5.8)

Using (5.7) and since $y\to\mathcal{V}^{\prime}(y)$ is continuously
differentiable at $y_{0}$, we can pass to the limit in (5.8) to obtain

$\lim_{\tau\to
0^{+}}\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\big{)}.$
(5.9)

Now we can pass to the limit in (5.5) and obtain (5.4).

Step 2: For $u\in\mathcal{U}_{ad}$ we define $\tilde{u}\in U_{ad}$ by,

$\widetilde{u}(\tau,x)=\begin{cases}u\quad\text{for }\tau\in(0,1)\\\
0\quad\text{ for }\tau\in[1,\infty)\end{cases}$

and define $\tilde{y}=S(y_{0},\tilde{u})$ as the solution to (3.2b), (3.2c).
Then $\tilde{y}(t)\in B_{Y}(\bar{y}_{0},\delta_{3})$, for all $t$ sufficiently
small, and by (5.1) we have,

$\frac{1}{\tau}\int_{0}^{\tau}\ell(\tilde{y}(s),u(s))ds+\frac{1}{\tau}\big{(}\mathcal{V}(\tilde{y}(\tau))-\mathcal{V}(y_{0})\big{)}\geq
0,$

for all $\tau$ sufficiently small. We pass to the limit $\tau\to 0^{+}$ with
the same arguments as in Step 1 and obtain

$\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu\big{)}+\ell(y_{0},u)\geq
0.$ (5.10)

This inequality becomes an equality if $u=u_{0}$, and thus the quadratic
function on the left had side of (5.10) reaches its minimum $0$ at $u=u_{0}$.
This implies that

$u_{0}=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right).$

Inserting this expression into (5.4) we obtain

$\mathcal{V}^{\prime}(y_{0})({\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0}))+\frac{1}{2}\left\lVert
y_{0}\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y_{0}),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right)\right\rangle_{Y}=0.$
(5.11)

Under the additional assumptions on the trajectory, (5.3) follows.

∎

##  6 Some Applications

In this section we discuss the applicability of the framework in two specific
cases. It should be noted that even for linear state equations, the
sensitivity result for the constraint infinite horizon optimal control problem
may be new.

###  6.1 Fisher’s Equation

We consider the optimal stabilization problem for the Fisher equation in an
open connected bounded domain $\Omega$ in $\mathbb{R}^{d},\
d\in\\{1,2,3,4\\}$, with Lipschitzian boundary $\Gamma=\partial\Omega$:

$\displaystyle(\mathcal{P}_{Fis})\qquad\mathcal{V}(y_{0})=\min_{\begin{matrix}(y,u)\in
W_{\infty}\times U_{ad}\end{matrix}}\ \frac{1}{2}\int_{0}^{\infty}\left\lVert
y\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u\right\rVert^{2}_{{\mathcal{U}}}dt,$ (6.1a) subject to $\displaystyle y_{t}$
$\displaystyle=\Delta y+y(1-y)+Bu\quad$ $\displaystyle\text{ in
}Q=(0,\infty)\times\Omega$ (6.1b) $\displaystyle y$ $\displaystyle=0\quad$
$\displaystyle\text{ on }\Sigma=(0,\infty)\times\Gamma$ (6.1c) $\displaystyle
y(0)$ $\displaystyle=y_{0}\quad$ $\displaystyle\text{ in }\Omega.$ (6.1d)

where ${\mathcal{U}}$ and $U_{ad}$ are as in Section 3.1,
$B\in{\mathcal{L}}({{\mathcal{U}},Y})$, with $Y=L^{2}(\Omega)$ and
$V=H^{1}_{0}(\Omega)$. To further cast this problem in the framework of
Section 3, we define the operator

${\mathcal{A}}y=(\Delta+\mathbf{I})y\quad\text{and}\quad
y|_{\Gamma}=0,\quad{\mathcal{D}}({\mathcal{A}})=H^{2}(\Omega)\cap V.$

Clearly ${\mathcal{A}}$ has an extension as operator
${\mathcal{A}}\in{\mathcal{L}}(V,V^{*})$. Moreover it generates an analytic
semigroup on $Y$. Thus (A1) holds. For ${\mathcal{U}}=Y$ and $B=\mathbf{I}$,
condition (A2) is trivially satisfied. Feedback stabilization by finite
dimensional controllers was analyzed in [Tri], for example.

It can readily be checked that the nonlinearity ${\mathcal{F}}(y)=-y^{2}$ is
twice continuously differentiable as mapping ${\mathcal{F}}:W_{\infty}\to
L^{2}(I;V^{*})$. We note that the second derivative is constant, so the
required boundedness of the second derivative is automatic. For the sake of
illustration we verify the boundedness of the bilinear form of the second
derivative on $W_{\infty}\times W_{\infty}$. For this purpose, for arbitrary
$y\in W_{\infty},v_{1},v_{2}\in W_{\infty},\phi\in L^{2}(I;V)$ we estimate

$\displaystyle\int_{0}^{\infty}\langle{\mathcal{F}}^{\prime\prime}({y})(v_{1},v_{2}),\phi\rangle_{V^{*},V}dt$
$\displaystyle\leq 2\int_{0}^{\infty}\int_{\Omega}v_{1}v_{2}\phi\
dxdt\leq\int^{\infty}_{0}\left\lVert
v_{1}\right\rVert_{L^{2}(\Omega)}\left\lVert
v_{2}\right\rVert_{L^{4}(\Omega)}\left\lVert\phi\right\rVert_{L^{4}(\Omega)}\,dt,$
(6.2) $\displaystyle\leq C_{1}\left\lVert
v_{1}\right\rVert_{W_{\infty}}\int^{\infty}_{0}\left\lVert
v_{2}\right\rVert_{V}\left\lVert\phi\right\rVert_{V}\,dt\leq C_{2}\left\lVert
v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{L^{2}(I;V)}\left\lVert\phi\right\rVert_{L^{2}(I;V)},$
$\displaystyle\leq C_{3}\left\lVert v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{W_{\infty}}\left\lVert\phi\right\rVert_{L^{2}(I;V)},$

where $C_{i}$ are embedding constants, independent of $y\in W_{\infty},v\in
W_{\infty},\phi\in L^{2}(I;V)$. We use that $V$ embeds continuously into
$L^{4}(\Omega)$ in dimension up to 4. This implies that
$\left\lVert{\mathcal{F}}^{\prime\prime}({y})(v_{1},v_{2})\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{W_{\infty}}$. Finally we have
${\mathcal{F}}(0)={\mathcal{F}}^{\prime}(0)=0$ and thus (A3) and (3.3) are
satisfied.

Turning to (A4) we show that ${\mathcal{F}}(y):W(0,T)\rightarrow
L^{1}(0,T;V^{*})$ is continuous for every $T>0$. We consider the sequence
$\displaystyle y_{n}\rightharpoonup\hat{y}$ in $W_{\infty}$ and let $z\in
L^{\infty}(0,T;V)$ be given. Then we estimate

$\displaystyle\int_{0}^{T}\langle{\mathcal{F}}(y_{n})-{\mathcal{F}}(\hat{y}),z\rangle_{V^{*},V}dt$
$\displaystyle=\int_{0}^{T}\langle
y_{n}^{2}-\hat{y}^{2},z\rangle_{V^{*},V}=\int_{0}^{T}\int_{\Omega}(y_{n}-\hat{y})(y_{n}+\hat{y})z\
dxdt$ $\displaystyle\leq C_{4}\int_{0}^{T}\left\lVert
y_{n}-\hat{y}\right\rVert_{Y}\left\lVert
y_{n}+\hat{y}\right\rVert_{L^{4}(\Omega)}\left\lVert
z\right\rVert_{L^{4}(\Omega)}\ dt$ $\displaystyle\leq C_{4}\left\lVert
y_{n}-\hat{y}\right\rVert_{L^{2}(0,T;Y)}\left[\left\lVert
y_{n}\right\rVert_{L^{2}(0,T;V)}+\left\lVert\hat{y}\right\rVert_{L^{2}(0,T;V)}\right]\left\lVert
z\right\rVert_{L^{\infty}(0,T;V)}.$

Since $V$ is compactly embedded in $Y$, we obtain by the Aubin Lions lemma
that $\displaystyle\left\lVert y_{n}-\hat{y}\right\rVert_{L^{2}(0,T;Y)}\to 0$
for $n\to\infty$. This implies

$\int_{0}^{T}\langle{\mathcal{F}}(y_{n})-{\mathcal{F}}(\hat{y}),z\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}0,$

and (A4) follows. It is simple to check that
$\displaystyle{\mathcal{F}}^{\prime}(\bar{y})=2\bar{y}\in{\mathcal{L}}(L^{2}(I;V),L^{2}(I;V^{*}))$
and thus (A5) holds as well.

We turn to the assumption ${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$, for
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$ and some $T_{y_{0}}$, arising in
Theorem 3.2 for $y_{0}\in{\mathcal{D}}({\mathcal{A}})$. Utilizing the fact
that $V$ embeds continuously into $L^{4}(\Omega)$ in dimension $d\leq 4$ and
$\bar{y}\in L^{2}(I;V)$, we have ${\mathcal{F}}(\bar{y})\in L^{2}(I;Y)$. Hence
parabolic regularity theory implies that $\bar{y}\in C([0,\infty);V)$ for
$y_{0}\in V$, and ${\mathcal{F}}(\bar{y})\in C([0,\infty);Y)$ follows.

######  Remark 6.1.

The specificity of this example rests in the fact that the second derivative
is independent of the point were it is taken. Other nontrivial cases of
analogous structure are reaction diffusion systems with bilinear coupling, see
[Gri] where the finite horizon case was treated. Even the case of the Navier
Stokes equations falls in this category. Sensitivity for the infinite horizon
problems was treated by independent techniques in [BKP3].

###  6.2 Nonlinearities induced by functions with globally Lipschitz
continuous second derivative.

Consider the system ($\mathcal{P}$) with ${\mathcal{A}}$ associated to a
strongly elliptic second order operator with domain $H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$, so that (A1)-(A2) are satisfied. Let
${\mathcal{F}}:W_{\infty}\to L(I;V^{*})$ be the Nemytskii operator associated
to a mapping $\mathfrak{f}:\mathbb{R}\to\mathbb{R}$ which is assumed to be
$C^{2}(\mathbb{R})$ with first and second derivatives globally Lipschitz
continuous, and second derivative globally bounded. The regularity assumption
${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$ for $y_{0}\in
V=H^{1}_{0}(\Omega)$ is satisfied by parabolic regularity theory. We discuss
assumption (A3)-(A5) for such an ${\mathcal{F}}$, and show that they are
satisfied for dimensions $d\in\\{1,2\\}$. For the finite horizon problem it
will turn out that $d=3$ is also admissible. By direct calculation it can be
checked that ${\mathcal{F}}$ is continuously Fréchet differentiable for
$d\in\\{1,2,3\\}$. We leave this part to the reader and immediately turn to
the second derivative.

We proceed by considering the general dimension $d$ to highlight, how the
restrictions on the dimension arise. Thus let $d\in\mathbb{N}$ with $d>1$. The
case $d=1$ can be treated with minor modifications from those in the following
steps.

####  6.2.1 Second derivative of ${\mathcal{F}}(y)$.

For $y,h_{1},h_{2}\in W_{\infty}$ the relevant expression is given by

$\left\lVert{\mathcal{F}}^{\prime}(y+h_{2})h_{1}-{\mathcal{F}}^{\prime}(y)h_{1}-{\mathcal{F}}^{\prime\prime}(y)(h_{1},h_{2})\right\rVert_{L^{2}(I;V^{*})}\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\langle{\mathcal{F}}^{\prime}(y+h_{2})h_{1}-{\mathcal{F}}^{\prime}(y)h_{1}-{\mathcal{F}}^{\prime\prime}(y)(h_{1},h_{2}),\varphi\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}(\mathfrak{f}^{\prime}(y(t,x)+h_{2}(t,x))-\mathfrak{f}^{\prime}(y(t,x))-\mathfrak{f}^{\prime\prime}(y(t,x))h_{2}(t,x))h_{1}(t,x)\varphi(t,x)dxdt\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}g(t,x)\ h_{2}(t,x)h_{1}(t,x)\varphi(t,x)dxdt,$

where $\displaystyle
g(t,x)=\int_{0}^{1}(\mathfrak{f}^{\prime\prime}(y(t,x)+sh_{2}(t,x))-\mathfrak{f}^{\prime\prime}(y(t,x)))ds$.
Note that $g$ is bounded on $I\times\Omega$ and $g\in W_{\infty}$. Here we use
that $\mathfrak{f}^{\prime\prime}$ is globally Lipschitz continuous and that
$h_{1}\in W_{\infty}$. Henceforth we let $r\in(1,\frac{2d}{d-2}]$ so that
$W^{1,2}(\Omega)\subset L^{r}(\Omega)$ continuously. Let $r^{\prime}$ denote
the conjugate of $r$ so that $r^{\prime}\in[\frac{2d}{d+2},\infty)$ for $d>2$
and $r^{\prime}\in(1,\infty)$ for $d=2$. We further choose $\rho>1,\sigma>2$