# Linearized Controllability Analysis of Semilinear Partial Differential
Equations

Bernd Kolar  Markus Schöberl  Institute of Automatic Control and Control
Systems Technology, Johannes Kepler University Linz, Altenbergerstrasse 66,
4040 Linz, Austria (e-mail<EMAIL_ADDRESS>markus.schoeberl@jku.at).

###### Abstract

It is well-known that the controllability of finite-dimensional nonlinear
systems can be established by showing the controllability of the linearized
system. However, this classical result does not generalize to infinite-
dimensional nonlinear systems. In this paper, we restrict ourselves to
semilinear infinite-dimensional systems, and show that the exact
controllability of the linearized system implies exact controllability of the
nonlinear system. The restrictions concerning the nonlinear operator are
similar to those that can be found in the literature about the linearized
stability analysis of semilinear systems.

###### keywords:

nonlinear systems, infinite-dimensional systems, linearization, exact
controllability

††thanks: This work has been supported by the Austrian Science Fund (FWF)
under grant number P 29964-N32.

##  1 Introduction

Studying system-theoretic properties like controllability or observability for
nonlinear infinite-dimensional systems is in general a very difficult task.
Roughly speaking, systems theory for partial differential equations can be
divided into formal (algebraic or geometric) methods like e.g. in Pommaret
(1994) and Schöberl (2014) that are based on the structure of the equations,
and functional-analytic methods that are rather based on the solutions.
Whereas formal methods have been proven to be very successful for finite-
dimensional nonlinear systems, in the infinite-dimensional case they suffer
from the drawback that the function spaces for e.g. the state and the input
cannot be properly specified. Thus, formal methods seem to be rather suited
for proving negative results like non-controllability or non-observability,
that can possibly be shown directly from the structure of the equations, see
e.g. Kolar et al. (2018) or Kolar and Schöberl (2019). For proving positive
results, in contrast, a functional-analytic approach seems to be
indispensable.

For finite-dimensional nonlinear systems, it is well-known that stability,
controllability, or observability can be established in a straightforward way
by proving the corresponding property for the linearized system, see e.g.
Nijmeijer and van der Schaft (1990) or Khalil (2002). Unfortunately, these
classical results do not generalize to infinite-dimensional nonlinear systems.
So far, the existing literature deals mainly with the linearized stability
analysis of infinite-dimensional nonlinear systems, see e.g. Desch and
Schappacher (1986), Smoller (1994), Kato (1995), Al Jamal et al. (2014), or Al
Jamal and Morris (2018). Particularly, it is shown in Smoller (1994) that for
semilinear systems with a nonlinear operator that is subject to certain
restrictions the stability of the linearized system implies – like in the
finite-dimensional case – the (local) stability of the original system. In the
present paper, we pursue a similar approach with respect to the exact
controllability problem for infinite-dimensional semilinear systems with
distributed input. Exact controllability means that the controllability map of
the system is surjective, and the basic idea of our approach is to apply the
local surjectivity theorem to this controllability map in order to establish a
connection between the exact controllability of the linearized system and the
(local) exact controllability of the nonlinear system. Even though the
controllability problem is quite different from the stability problem, we need
conditions on the nonlinear operator of the semilinear system that are very
similar to those in Smoller (1994). It should also be noted that in contrast
to the stability analysis we are dealing here with non-autonomous systems,
and, as mentioned in Schmid et al. (2019), there exist only very few papers on
non-autonomous semilinear systems in the context of control theory.

##  2 Preliminaries

Throughout the paper, we need the concept of a Fréchet derivative of maps
between infinite-dimensional spaces.

###### Definition 1

(Fréchet Derivative) A map $f:X\rightarrow Y$ from a Banach space $X$ to a
Banach space $Y$ is Fréchet differentiable at $x\in X$, if there exists a
bounded linear operator $Df(x):X\rightarrow Y$ such that

$\lim_{\left\|h\right\|_{X}\rightarrow
0}\frac{\left\|f(x+h)-f(x)-Df(x)h\right\|_{Y}}{\left\|h\right\|_{X}}=0\,.$ (1)

The map is Fréchet differentiable if it is Fréchet differentiable at every
$x\in X$, and it is continuously Fréchet differentiable if the Fréchet
derivative $Df(x)$ depends continuously on $x$.

For bounded linear operators between Banach spaces, we use the usual operator
norm

$\left\|T\right\|=\sup_{x\in D(T),\,x\neq
0}\frac{\left\|Tx\right\|_{Y}}{\left\|x\right\|_{X}}\,,$

and denote it by $\left\|\cdot\right\|$ without a subscript.

Our approach to the linearized controllability analysis is based on the local
surjectivity theorem, that can be found e.g. in Abraham et al. (1988).

###### Theorem 2

(Local Surjectivity Theorem) Let $X$ and $Y$ be Banach spaces and $V\subset X$
be open. If the map $f:V\subset X\rightarrow Y$ is continuously Fréchet
differentiable and $Df(x_{0})$ is surjective for some $x_{0}\in V$, then $f$
is locally surjective. That is, there exist open neighborhoods $V_{1}$ of
$x_{0}$ and $W_{1}$ of $f(x_{0})$ such that
$\left.f\right|_{V_{1}}:V_{1}\rightarrow W_{1}$ is surjective.

We also make frequent use of Gronwall’s lemma in the form presented in Zeidler
(1986).

###### Lemma 3

(Gronwall) Let $f,g:[t_{0},\tau]\rightarrow\mathbb{R}$ be continuous
functions, with $g$ nondecreasing, and which, for fixed $K>0$, satisfy the
inequality

$f(t)\leq g(t)+K\intop_{t_{0}}^{t}f(s)ds\,,\quad\forall t\in[t_{0},\tau]\,.$

Then

$f(t)\leq g(t)e^{K(t-t_{0})}\,,\quad\forall t\in[t_{0},\tau]\,.$

##  3 Semilinear Systems

We consider semilinear systems of the form

$\dot{x}(t)=Ax(t)+f(x(t))+Bu(t)\,,\quad x(t_{0})=x_{0}\,,$ (2)

where $A$ is the infinitesimal generator of a strongly continuous semigroup
$T(t)$ on a Hilbert space $X$, the linear operator $B:U\rightarrow X$ is
bounded, and the nonlinear map $f:X\rightarrow X$ is continuously Fréchet
differentiable. Furthermore, we assume that $f(0)=0$ and $Df(0)=0$. With these
assumptions,

$(x_{s},u_{s})=(0,0)$

is an equilibrium of the system (2), and the linearization about this
equilibrium is given by

$\Delta\dot{x}(t)=A\Delta x(t)+B\Delta u(t)\,,\quad\Delta x(t_{0})=\Delta
x_{0}\,,$ (3)

which is just the linear part of the system.

###### Remark 4

It should be noted that the linearization of the system is based on the
Gâteaux derivative, see e.g. Al Jamal et al. (2014) or Al Jamal and Morris
(2018). Since the operator $A$ is typically unbounded, the right-hand side of
(2) does not possess a Fréchet derivative with respect to $x$.

As discussed in Pazy (1983), classical solutions of the semilinear system (2)
satisfy the integral equation

$x(t)=T(t)x(t_{0})+\intop_{t_{0}}^{t}T(t-s)\left(f(x(s))+Bu(s)\right)ds\,.$
(4)

Therefore, continuous solutions of the integral equation (4) are called mild
solutions of (2), see also Curtain and Zwart (1995) for the linear case. For
the controllability problem with $t_{0}=0$ and initial condition $x(0)=0$, we
have

$x(t)=\intop_{0}^{t}T(t-s)\left(f(x(s))+Bu(s)\right)ds\,.$ (5)

Throughout the paper, we assume that for some $\tau>0$ and all inputs $u(t)$
in an open neighborhood

$\mathcal{U}\subset L_{2}([0,\tau];U)$

of $u(t)=0$ the semilinear system (2) with $x(0)=0$ has a unique mild solution
on the interval $[0,\tau]$, i.e., a unique continuous solution of (5).

###### Remark 5

Proving the existence and uniqueness of solutions would require a (repeated)
application of Banach’s fixed-point theorem.

In the following, we denote the solution for an input $u(t)\in\mathcal{U}$ by

$x(t)=S_{t}(u)\,,\quad t\in[0,\tau]\,.$

The map

$S_{\tau}(u):\mathcal{U}\rightarrow X$ (6)

with $t=\tau$ is the controllability map of the semilinear system on
$[0,\tau]$, and we call the system locally exactly controllable on $[0,\tau]$
if this map is locally surjective. In other words, for every final state
$x(\tau)\in X$ in an open neighborhood of the origin there must exist an input
trajectory $u(t)$ on the time interval $[0,\tau]$ such that

$x(\tau)=S_{\tau}(u)\,.$

The basic idea is now to use the local surjectivity theorem in order to
establish a connection between the local exact controllability of the
semilinear system (2) and the exact controllability of the linearized system
(3).

###### Theorem 6

Assume that the controllability map (6) of the semilinear system (2) satisfies
the following conditions:

  1. (A1)

$S_{\tau}(u)$ is continuously Fréchet differentiable with respect to $u$ in an
open neighborhood of $u=0$.

  2. (A2)

The Fréchet derivative $DS_{\tau}(0)$ at $u=0$ coincides with the
controllability map

$\intop_{0}^{\tau}T(\tau-s)B\Delta u(s)ds:L_{2}([0,\tau];U)\rightarrow X$

of the linearized system (3) on $[0,\tau]$.

Then exact controllability of the linearized system (3) on $[0,\tau]$ implies
local exact controllability of the semilinear system (2) on $[0,\tau]$.

###### Proof 3.1

The condition (A1) is a prerequisite for the application of the local
surjectivity theorem. With condition (A2) and the local surjectivity theorem,
surjectivity of the controllability map of the linearized system implies local
surjectivity of the controllability map of the semilinear system.
Consequently, exact controllability of the linearized system implies local
exact controllability of the semilinear system.

The practical use of this theorem is of course limited, since it would require
knowledge of the controllability map (6) of the semilinear system. Thus, in
the remainder of the paper, we translate the conditions (A1) and (A2) of
Theorem 6 into (sufficient) conditions on the nonlinear term $f$ of the system
(2). The main difficulty consists in proving the continuous Fréchet
differentiability of $S_{\tau}(u)$.

##  4 Fréchet Differentiability of the Controllability Map

In this section, we show that the conditions (A1) and (A2) of Theorem 6 are
satisfied for all semilinear systems (2) where the nonlinear operator $f$
satisfies the following additional assumptions.

  1. (B1)

For every bounded set $\mathcal{B}\subset X$, there exist positive constants
$\alpha,\gamma\in\mathbb{R}$ such that

$\left\|f(x_{1})-f(x_{2})-Df(x_{2})(x_{1}-x_{2})\right\|_{X}\leq\alpha\left\|x_{1}-x_{2}\right\|_{X}^{1+\gamma}$

for all $x_{1},x_{2}\in\mathcal{B}$.

  2. (B2)

The Fréchet derivative $Df$ is locally Lipschitz continuous.

The assumption (B1) with $\gamma=1$ is also used in Smoller (1994) for the
linearized stability analysis of semilinear autonomous systems.

We proceed in several steps. First, we show that for all $t\in[0,\tau]$ the
map $S_{t}(u):\mathcal{U}\rightarrow X$ is locally Lipschitz continuous (Lemma
7). Based on this result, we prove that $S_{t}(u)$ is also Fréchet
differentiable with respect to $u$, and that the Fréchet derivative coincides
with the solution operator (controllability map) of the linearized system
(Theorem 8). Finally, we prove that $S_{t}(u)$ is even continuously Fréchet
differentiable (Theorem 9).

###### Lemma 7

There exists a constant $c$ such that

$\left\|S_{t}(u_{1})-S_{t}(u_{2})\right\|_{X}\leq
c\left\|u_{1}-u_{2}\right\|_{L_{2}}$

for all $t\in[0,\tau]$ and $u_{1},u_{2}\in\mathcal{U}$.

###### Proof 4.1

Let $x_{1}(t)=S_{t}(u_{1})$ and $x_{2}(t)=S_{t}(u_{2})$ denote the solutions
for two input trajectories $u_{1},u_{2}\in\mathcal{U}$. These solutions
satisfy the integral equations

$x_{1}(t)=\intop_{0}^{t}T(t-s)\left(f(x_{1}(s))+Bu_{1}(s)\right)ds$

and

$x_{2}(t)=\intop_{0}^{t}T(t-s)\left(f(x_{2}(s))+Bu_{2}(s)\right)ds\,.$

For the norm of the difference $x_{1}(t)-x_{2}(t)$, we get the estimate

$\displaystyle\left\|x_{1}(t)-x_{2}(t)\right\|_{X}$
$\displaystyle\leq\left\|\intop_{0}^{t}T(t-s)\left(f(x_{1}(s))-f(x_{2}(s))\right)ds\right\|_{X}$
$\displaystyle\phantom{\leq}+\left\|\intop_{0}^{t}T(t-s)B(u_{1}(s)-u_{2}(s))ds\right\|_{X}$
$\displaystyle\leq\intop_{0}^{t}\left\|T(t-s)\right\|\left\|f(x_{1}(s))-f(x_{2}(s))\right\|_{X}ds$
$\displaystyle\phantom{\leq}+k\left\|u_{1}-u_{2}\right\|_{L_{2}}$
$\displaystyle\leq ML\intop_{0}^{t}\left\|x_{1}(s)-x_{2}(s)\right\|_{X}ds$
$\displaystyle\phantom{\leq}+k\left\|u_{1}-u_{2}\right\|_{L_{2}}$ (7)

with $M=\sup_{t\in[0,\tau]}\left\|T(t)\right\|$ and some $k>0$. Since the
continuous Fréchet differentiability of $f$ guarantees only local Lipschitz
continuity, the Lipschitz constant $L$ depends of course on

$\sup_{u\in\mathcal{U},t\in[0,\tau]}\left\|S_{t}(u)\right\|_{X}\,,$

i.e., on the maximal “size” of the solutions with input trajectories
$u\in\mathcal{U}$. We have also used the fact that the term

$\intop_{0}^{t}T(t-s)Bu(s)ds$

is just the controllability map of the linear part of the system and therefore
bounded with some constant $k$, see e.g. Curtain and Zwart (1995). Applying
Gronwall’s lemma to (7) yields

$\left\|x_{1}(t)-x_{2}(t)\right\|_{X}\leq
k\left\|u_{1}-u_{2}\right\|_{L_{2}}e^{MLt}$

and finally

$\left\|x_{1}(t)-x_{2}(t)\right\|_{X}\leq\underbrace{ke^{ML\tau}}_{c}\left\|u_{1}-u_{2}\right\|_{L_{2}}\,,\quad\forall
t\in[0,\tau]\,,$

which completes the proof.

With help of Lemma 7, we can now show that the solutions $x(t)=S_{t}(u)$ are
Fréchet differentiable with respect to $u$. The proof is divided in fact into
two parts. First, we show that if the Fréchet derivative exists it coincides
with the solution operator of the linearized system. Subsequently, we prove
that the solution operator of the linearized system satisfies the condition
(1) for a Fréchet derivative.

###### Theorem 8

For all $t\in[0,\tau]$, the map

$S_{t}(u):L_{2}([0,\tau];U)\rightarrow X$

is Fréchet differentiable with respect to $u$ at every
$\bar{u}\in\mathcal{U}$. The Fréchet derivative $DS_{t}(\bar{u})$ coincides
with the solution operator (controllability map)

$L_{t}\Delta u:L_{2}([0,\tau];U)\rightarrow X$

of the linearized, time-variant system111The system (8) is the linearization
of (2) along the trajectory $x(t)=S_{t}(\bar{u})$. For $\bar{u}=0$ we have
$S_{t}(\bar{u})=0$, and (8) becomes the linear, time-invariant system (3).
Thus, the condition (A2) of Theorem 6 is contained as a special case.

$\Delta\dot{x}(t)=\left(A+Df(S_{t}(\bar{u}))\right)\Delta x(t)+B\Delta u(t)$
(8)

with $\Delta x(0)=0$.

###### Proof 4.2

If $S_{t}(u)$ is Fréchet differentiable at $\bar{u}\in\mathcal{U}$ for all
$t\in[0,\tau]$, then we can differentiate the integral equation (5) and
observe that the Fréchet derivative $DS_{t}(\bar{u})$ must satisfy the
integral equation222It should be noted that the integrals in this equation are
in general not Lebesgue integrals but Pettis integrals, see also Curtain and
Zwart (1995).

$DS_{t}(\bar{u})=\intop_{0}^{t}T(t-s)Df(S_{s}(\bar{u}))DS_{s}(\bar{u})ds\\\
+\intop_{0}^{t}T(t-s)Bds\,.$

The solution of the linearized system (8) meets

$L_{t}\Delta u=\intop_{0}^{t}T(t-s)Df(S_{s}(\bar{u}))L_{s}\Delta uds\\\
+\intop_{0}^{t}T(t-s)B\Delta u(s)ds\,,$ (9)

and because of $D(L_{t}\Delta u)=L_{t}$ the solution operator $L_{t}$
satisfies exactly the same integral equation

$L_{t}=\intop_{0}^{t}T(t-s)Df(S_{s}(\bar{u}))L_{s}ds+\intop_{0}^{t}T(t-s)Bds\,.$
(10)

Thus, if the Fréchet derivative $DS_{t}(\bar{u})$ exists, it coincides with
the operator $L_{t}$.333Provided that the integral equations have a unique
solution, but it is well-known that the Fréchet derivative is unique if it
exists. Consequently, we must prove that $L_{t}$ satisfies the condition

$\lim_{\left\|\Delta u\right\|_{L_{2}}\rightarrow
0}\frac{\left\|S_{t}(\bar{u}+\Delta u)-S_{t}(\bar{u})-L_{t}\Delta
u\right\|_{X}}{\left\|\Delta u\right\|_{L_{2}}}=0$ (11)

for a Fréchet derivative of $S_{t}(u)$ at $\bar{u}$, cf. (1). For this
purpose, we introduce the abbreviation

$\sigma(t)=S_{t}(\bar{u}+\Delta u)-S_{t}(\bar{u})-L_{t}\Delta u$

for the sum in the numerator of (11). Substituting

$\displaystyle S_{t}(\bar{u}+\Delta u)$
$\displaystyle=\intop_{0}^{t}T(t-s)f(S_{s}(\bar{u}+\Delta u))ds$
$\displaystyle\phantom{=}+\intop_{0}^{t}T(t-s)B(\bar{u}(s)+\Delta u(s))ds$
$\displaystyle S_{t}(\bar{u})$
$\displaystyle=\intop_{0}^{t}T(t-s)f(S_{s}(\bar{u}))ds$
$\displaystyle\phantom{=}+\intop_{0}^{t}T(t-s)B\bar{u}(s)ds$ $\displaystyle
L_{t}\Delta u$
$\displaystyle=\intop_{0}^{t}T(t-s)Df(S_{s}(\bar{u}))L_{s}\Delta uds$
$\displaystyle\phantom{=}+\intop_{0}^{t}T(t-s)B\Delta u(s)ds$

according to the integral equations (5) and (9) yields

$\sigma(t)=\intop_{0}^{t}T(t-s)\left(f(S_{s}(\bar{u}+\Delta
u))-f(S_{s}(\bar{u}))\right.\\\ \left.-Df(S_{s}(\bar{u}))L_{s}\Delta
u\right)ds\,.$ (12)

Now we write $f(S_{s}(\bar{u}+\Delta u))$ as

$f(S_{s}(\bar{u}+\Delta u))=\\\
f(S_{s}(\bar{u}))+Df(S_{s}(\bar{u}))\left(S_{s}(\bar{u}+\Delta
u)-S_{s}(\bar{u})\right)+r(s)$ (13)

with some rest $r(s)$, and substituting

$f(S_{s}(\bar{u}+\Delta u))-f(S_{s}(\bar{u}))=\\\
Df(S_{s}(\bar{u}))\left(S_{s}(\bar{u}+\Delta u)-S_{s}(\bar{u})\right)+r(s)$

into (12) yields

$\sigma(t)=\intop_{0}^{t}T(t-s)Df(S_{s}(\bar{u}))\sigma(s)ds+\intop_{0}^{t}T(t-s)r(s)ds\,.$

For the norm of $\sigma(t)$ we get the estimate

$\displaystyle\left\|\sigma(t)\right\|_{X}$
$\displaystyle\leq\intop_{0}^{t}\left\|T(t-s)\right\|\left\|Df(S_{s}(\bar{u}))\right\|\left\|\sigma(s)\right\|_{X}ds+$
$\displaystyle\phantom{\leq}+\intop_{0}^{t}\left\|T(t-s)\right\|\left\|r(s)\right\|_{X}ds$
$\displaystyle\leq
M\bar{c}\intop_{0}^{t}\left\|\sigma(s)\right\|_{X}ds+M\intop_{0}^{t}\left\|r(s)\right\|_{X}ds$

with

$\bar{c}=\sup_{t\in[0,\tau]}\left\|Df(S_{t}(\bar{u}))\right\|\,,$

and applying Gronwall’s lemma yields

$\left\|\sigma(t)\right\|_{X}\leq\left(M\intop_{0}^{t}\left\|r(s)\right\|_{X}ds\right)e^{M\bar{c}t}\,.$
(14)

Now we make use of the additional assumption (B1) to obtain an inequality for
$\left\|r(s)\right\|_{X}$. Applying (B1) to the right-hand side of

$r(s)=f(S_{s}(\bar{u}+\Delta u))-f(S_{s}(\bar{u}))\\\
-Df(S_{s}(\bar{u}))\left(S_{s}(\bar{u}+\Delta u)-S_{s}(\bar{u})\right)$

yields

$\left\|r(s)\right\|_{X}\leq\alpha\left\|S_{s}(\bar{u}+\Delta
u)-S_{s}(\bar{u})\right\|_{X}^{1+\gamma}\,,$

and with Lemma 7 we get

$\left\|r(s)\right\|_{X}\leq\alpha c^{1+\gamma}\left\|\Delta
u\right\|_{L_{2}}^{1+\gamma}\,,\quad\forall s\in[0,\tau]\,.$

Substituting this inequality into (14) results in

$\left\|\sigma(t)\right\|_{X}\leq M\alpha
c^{1+\gamma}te^{M\bar{c}t}\left\|\Delta u\right\|_{L_{2}}^{1+\gamma}\,,\quad
t\in[0,\tau]\,,$

and because of

$\lim_{\left\|\Delta u\right\|_{L_{2}}\rightarrow
0}\frac{\left\|\sigma(t)\right\|_{X}}{\left\|\Delta
u\right\|_{L_{2}}}\leq\lim_{\left\|\Delta u\right\|_{L_{2}}\rightarrow
0}M\alpha c^{1+\gamma}te^{M\bar{c}t}\left\|\Delta
u\right\|_{L_{2}}^{\gamma}=0$

the condition (11) for a Fréchet derivative is indeed satisfied.

However, for the application of the local surjectivity theorem, it is not
enough to prove the Fréchet differentiability. We have to show that
$S_{\tau}(u)$ is continuously Fréchet differentiable, i.e., that the Fréchet
derivative $DS_{\tau}(\bar{u})$ depends continuously on the point
$\bar{u}\in\mathcal{U}$.

###### Theorem 9

For all $t\in[0,\tau]$, the map

$S_{t}(u):L_{2}([0,\tau];U)\rightarrow X$

is continuously Fréchet differentiable on $\mathcal{U}$.

###### Proof 4.3

We have to show that the map $u\rightarrow DS_{t}(u)$ is continuous. If
$DS_{t}(u_{1})$ and $DS_{t}(u_{2})$ are the Fréchet derivatives of $S_{t}(u)$
at $u_{1},u_{2}\in\mathcal{U}$, then they satisfy the integral equations

$DS_{t}(u_{1})=\intop_{0}^{t}T(t-s)\left(Df(S_{s}(u_{1}))DS_{s}(u_{1})+B\right)ds$

and

$DS_{t}(u_{2})=\intop_{0}^{t}T(t-s)\left(Df(S_{s}(u_{2}))DS_{s}(u_{2})+B\right)ds\,.$

The difference of these equations can be written as

$\begin{array}[]{l}DS_{t}(u_{1})-DS_{t}(u_{2})=\\\
\intop_{0}^{t}T(t-s)\left(Df(S_{s}(u_{1}))DS_{s}(u_{1})\right.-\left.Df(S_{s}(u_{2}))DS_{s}(u_{2})\right)ds\\\
=\intop_{0}^{t}T(t-s)\left(Df(S_{s}(u_{1}))\left(DS_{s}(u_{1})\right.\right.\\\
\phantom{=}-\left.DS_{s}(u_{2})\right)+\left(Df(S_{s}(u_{1}))\right.-\left.\left.Df(S_{s}(u_{2}))\right)DS_{s}(u_{2})\right)ds\,.\end{array}$

For the norm of the difference, we get the estimate

$\begin{array}[]{l}\left\|DS_{t}(u_{1})-DS_{t}(u_{2})\right\|\leq\\\
\intop_{0}^{t}\left\|T(t-s)\right\|\left\|Df(S_{s}(u_{1}))\right\|\left\|DS_{s}(u_{1})-DS_{s}(u_{2})\right\|ds\\\
+\intop_{0}^{t}\left\|T(t-s)\right\|\left\|Df(S_{s}(u_{1}))-Df(S_{s}(u_{2}))\right\|\left\|DS_{s}(u_{2})\right\|ds\\\
\leq Mc_{1}\intop_{0}^{t}\left\|DS_{s}(u_{1})-DS_{s}(u_{2})\right\|ds\\\
\phantom{\leq}+Mc_{2}\intop_{0}^{t}\left\|Df(S_{s}(u_{1}))-Df(S_{s}(u_{2}))\right\|ds\end{array}$
(15)

with

$c_{1}=\underset{u\in\mathcal{U},t\in[0,\tau]}{\sup}\left\|Df(S_{t}(u))\right\|$

and

$c_{2}=\underset{u\in\mathcal{U},t\in[0,\tau]}{\sup}\left\|DS_{t}(u)\right\|\,.$

With the Lipschitz continuity

$\left\|Df(S_{s}(u_{1}))-Df(S_{s}(u_{2}))\right\|\leq
c_{3}\left\|S_{s}(u_{1})-S_{s}(u_{2})\right\|_{X}$

of $Df$ according to assumption (B2) and Lemma 7, we also get

$\left\|Df(S_{s}(u_{1}))-Df(S_{s}(u_{2}))\right\|\leq\underbrace{c_{3}c}_{c_{4}}\left\|u_{1}-u_{2}\right\|_{L_{2}}$

$\forall s\in[0,t]$. Thus, (15) can be simplified to

$\left\|DS_{t}(u_{1})-DS_{t}(u_{2})\right\|\leq\\\
Mc_{1}\intop_{0}^{t}\left\|DS_{s}(u_{1})-DS_{s}(u_{2})\right\|ds+Mc_{2}c_{4}t\left\|u_{1}-u_{2}\right\|_{L_{2}}\,.$

Applying Gronwall’s lemma yields

$\left\|DS_{t}(u_{1})-DS_{t}(u_{2})\right\|\leq
Mc_{2}c_{4}te^{Mc_{1}t}\left\|u_{1}-u_{2}\right\|_{L_{2}}\,,$

which shows that the map $u\rightarrow DS_{t}(u)$ is continuous for all
$t\in[0,\tau]$.

With Theorem 8 and Theorem 9, we can finally state our main result.

###### Theorem 10

Consider a semilinear system (2) with a nonlinear term $f$ that meets the
conditions (B1) and (B2). If the linearized system (3) is exactly controllable
on $[0,\tau]$, then the original system is locally exactly controllable on
$[0,\tau]$.

###### Proof 4.4

We only have to show that the conditions (A1) and (A2) of Theorem 6 are
satisfied. Condition (A1) follows from Theorem 9, and condition (A2) from
Theorem 8 with $\bar{u}=0$.

##  5 Conclusion

We have shown for a class of semilinear infinite-dimensional systems that
exact controllability of the linearized system implies local exact
controllability of the original system. The assumptions on the nonlinear
operator are similar to those used in Smoller (1994) for the linearized
stability analysis, i.e., Lyapunov’s indirect method. Future research will
deal with the question whether these assumptions can be relaxed. A further
interesting question is whether the approach with the local surjectivity
theorem can also be applied to the approximate controllability problem. Such
an extension is of course not at all straightforward. Since the infinite-
dimensional spaces in the local surjectivity theorem are Banach spaces, i.e.,
complete, the intuitive idea of simply using the reachable subspace of the
linearized system as target space is not directly applicable.

The first author would like to thank Hans Zwart for interesting discussions
about semilinear systems during a research visit at the University of Twente
in March 2019.

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