remarkRemark hypothesisHypothesis claimClaim Optimal Control of Perfect
PlasticityC. Meyer and S. Walther

# Optimal Control of Perfect Plasticity
Part II: Displacement Tracking††thanks: Submitted to the editors DATE. This
research was supported by the German Research Foundation (DFG) under grant
number ME 3281/9-1 within the priority program Non-smooth and Complementarity-
based Distributed Parameter Systems: Simulation and Hierarchical Optimization
(SPP 1962).

Christian Meyer TU Dortmund, Faculty of Mathematics, Vogelpothsweg 87, 44227
Dortmund, Germany (, http://www.mathematik.tu-
dortmund.de/de/personen/person/Christian+Meyer.html, ,
http://www.mathematik.tu-dortmund.de/de/personen/person/Stephan+Walther.html).
<EMAIL_ADDRESS><EMAIL_ADDRESS>Stephan
Walther22footnotemark: 2

###### Abstract

The paper is concerned with an optimal control problem governed by the rate-
independent system of quasi-static perfect elasto-plasticity. The objective is
optimize the displacement field in the domain occupied by the body by means of
prescribed Dirichlet boundary data, which serve as control variables. The
arising optimization problem is nonsmooth for several reasons, in particular,
since the control-to-state mapping is not single-valued. We therefore apply a
Yosida regularization to obtain a single-valued control-to-state operator.
Beside the existence of optimal solutions, their approximation by means of
this regularization approach is the main subject of this work. It turns out
that a so-called reverse approximation guaranteeing the existence of a
suitable recovery sequence can only be shown under an additional smoothness
assumption on at least one optimal solution.

###### keywords:

Optimal control of variational inequalities, perfect plasticity, rate-
independent systems, Yosida regularization, reverse approximation

49J20, 49J40, 74C05

##  1 Introduction

In this paper, we investigate the following optimal control problem governed
by the equations of _quasi-static perfect plasticity_ at small strain:

(P) $\left\\{\quad\begin{aligned} &\min\text{ \ \
}J(u,u_{D}):=\Psi(u)+\frac{\alpha}{2}\|u_{D}\|_{H^{1}(0,T;H^{2}(\Omega;\mathbb{R}^{n}))}^{2}\\\
&\;\begin{aligned} &\text{s.t.}\quad&-\operatorname{div}\sigma&=0&&\text{ in
}\Omega,\\\ &&\sigma&=\mathbb{C}(\nabla^{s}u-z)&&\text{ in }\Omega,\\\
&&\overset{\boldsymbol{.}}{z}&\in\partial
I_{\mathcal{K}(\Omega)}(\sigma)&&\text{ in }\Omega,\\\ &&u&=u_{D}&&\text{ on
}\Gamma_{D},\\\ &&\sigma\nu&=0&&\text{ on }\Gamma_{N},\\\
&&u(0)&=u_{0},\quad\sigma(0)=\sigma_{0}&&\text{ in }\Omega,\\\\[4.30554pt]
&\text{and}\quad&u_{D}(0)&=u_{0}&&\text{ on
}\Gamma_{D}.\end{aligned}\end{aligned}\right.$

Herein, $u:(0,T)\times\Omega\to\mathbb{R}^{n}$, $n=2,3$, is the displacement
field, while $\sigma,z:(0,T)\times\Omega\to\mathbb{R}^{n\times n}$ are the
stress tensor and the plastic strain. The boundary of $\Omega$ is split in two
disjoint parts $\Gamma_{D}$ and $\Gamma_{N}$ with outward unit normal $\nu$.
Moreover, $\mathbb{C}$ is the elasticity tensor and $\mathcal{K}(\Omega)$
denotes the set of feasible stresses. The initial data $u_{0}$ and
$\sigma_{0}$ are given and fixed. The Dirichlet data $u_{D}$ represent the
control variable and $\alpha>0$ a fixed Tikhonov regularization parameter. The
objective $\Psi$ only contains the displacement field. Objectives involving
the stress are considered in a companion paper [21]. This is the reason for
calling (P) _displacement tracking problem_. A mathematically rigorous version
of (P) involving the function spaces and a rigorous notion of solutions for
the state equation will be formulated in Section 3 and 4 below. The precise
assumptions on the data are given in Section 2. Regarding to a more detailed
description of the plasticity model, we refer to [25] and the references
therein.

Some words concerning our choice of the control variable are in order: In
general, Dirichlet control problems provide particular difficulties due to
regularity issues, when control functions in $L^{2}(\partial\Omega)$ are
considered, see e.g. [18]. Nonetheless, we consider the Dirichlet displacement
as control variables instead of distributed loads or forces on the Neumann
boundary due to the _safe load condition_. It is well known that the existence
of solutions for the perfect plasticity system can only be shown under this
additional condition (see e.g. [30, 8]), which would lead to rather complex
control constraints and it is a completely open question how to incorporate
these constraints in the analysis of (P). For this reason, we focus on the
Dirichlet control problem. A possible realization of these controls by means
of an additional linear elasticity equation avoiding the $H^{2}$-norm in the
objective is elaborated in the companion paper [21].

Beside the safe-load condition, problem (P) exhibits several additional
particular challenges. First of all, it is obviously nonsmooth due to the
convex subdifferential appearing in the state equation. Moreover, the state
equation is in general not uniquely solvable and its solutions significantly
lack regularity, see [30, 8]. Therefore, there is no single-valued control-to-
state mapping and (P) should rather be regarded as an optimization problem in
Banach space rather than an optimal control problem. Beside the existence of
optimal solutions, our main goal is to approximate (P) via replacing $\partial
I_{\mathcal{K}(\Omega)}$ by its Yosida regularization. This is of course a
classical procedure and, in order to show that the approximation works, i.e.,
that optimal solutions of the regularized problems converge to solutions of
(P) (in a certain topology), the following steps have to be performed:

  1. 1.

The existence of (weak) accumulation points of sequences of optimal solutions
of the regularized problems have to be verified.

  2. 2.

Weak limits have to be feasible for the original problem (P).

  3. 3.

In order to show the optimality of the weak limit, one has to construct a
_recovery sequence_ for at least one optimal solution of the original problem.

The last item is also known as _reverse approximation_ and might become a
challenging task in the context of optimization of rate-independent systems,
see [22]. This also happens to be the case here: In contrast to the perfect
plasticity system, its regularized counterpart admits a unique solution with
full regularity. It is therefore very unlikely that one can approximate
_every_ solution of the perfect plasticity system by means of regularization
and indeed, as classical examples demonstrate, this is in fact not true, see
e.g. [30] and Example 3.11 below. However, in the context of optimal control
and optimization, respectively, we have the control as an additional variable
at hand and, in order to construct a recovery sequence, we have to find a
sequence of _tuples of state and control_ feasible for the regularized
problems so that the associated objective function values converge to the
optimal value of (P). This leads to much more flexibility in the construction
of recovery sequences, provided that the set of controls is sufficiently rich.
Unfortunately, this is not the case for our Dirichlet control and we need an
additional control variable in terms of _distributed loads_ for the
construction of a recovery sequence. The idea is thus to introduce an
additional load in the balance of momentum of the regularized problems and to
drive this load to zero for vanishing regularization parameter. Our
regularization procedure therefore does not only replace the convex
subdifferential by its Yosida regularization, but also introduces a new
additional control variable. To the best of our knowledge, this is a
completely new idea.

Nevertheless, even with this additional control variable, we are only able to
construct a recovery sequence under a fairly restrictive assumption. This
assumption is caused by additional smoothness constraints as part of the
regularized optimal control problems, which in turn are needed to pass to the
limit in the regularized plasticity system, when the regularization parameter
is driven to zero. If we assume that at least one optimal solution of the
original (i.e., unregularized) optimization problem admits an admittedly high
regularity, then we are able to construct a recovery sequence for this
particular solution, which meets the smoothness constraints and is therefore
feasible for the regularized optimal control problems. We thus obtain the
desired approximation result under the assumption that there exists at least
one “smooth” solution of (P).

Let us put our work into perspective: Quasi-static perfect plasticity is a
rate-independent system. Optimization and optimal control of such systems have
been considered by various authors and we only refer to [4, 5, 1, 6, 7, 29,
24, 2, 14] and the references therein. Albeit still nonsmooth, optimization
problems of this type substantially simplify, if the energy underlying the
rate-independent system is uniformly convex. In quasi-static plasticity, this
is the case, if hardening is present. In this case, the plasticity system
admits a unique solution in the energy space, which makes the construction of
recovery sequences almost trivial. Nevertheless, the derivation of optimality
conditions is still an intricate issue, see [32, 33, 34]. While all
contributions mentioned so far deal with uniformly convex energies, the
literature becomes rather scarce, when it comes to energies that lack strict
convexity. In [26, 28, 11, 10] the existence of optimal solutions for problems
with non-convex energies are shown. To the best of our knowledge, the
approximation of such problems has only been investigated in [22, 27], where a
time-discretization instead of a regularization is considered. The
approximation via discretization can however be hardly compared to our
situation, since the discrete rate-independent systems are still not uniquely
solvable so that there is still no (discrete) control-to-state map in contrast
to the regularized setting. Therefore, the discrete optimization problems are
still all but straight forward to solve, whereas the regularized optimal
control problems are amenable for standard adjoint-based optimization methods.

The paper is organized as follows: After introducing our notation and standard
assumptions in Section 2, we introduce a rigorous notion of solution to the
perfect plasticity system and recall the known results concerning the
existence of solutions and the lack of uniqueness in Section 3. Then, Section
4 is devoted to the existence of at least one (globally) optimal solution of
(P). In Section 5, we lay the foundations for our reverse approximation
argument for the construction of a recovery sequence, which is a basic
ingredient for our main result in Theorem 6.5. The last Section 6 covers this
result and shows that solutions of (P) can indeed be approximated via Yosida
regularization provided the mentioned regularity assumption is fulfilled.

##  2 Notation and Standing Assumptions

We start with a short introduction in the notation used throughout the paper
and in parallel list our standing assumptions. The latter are tacitly assumed
for the rest of the paper without mentioning them every time.

### General notation

Given two vector spaces $X$ and $Y$, we denote the space of linear and
continuous functions from $X$ into $Y$ by $\mathcal{L}(X,Y)$. If $X=Y$, we
simply write $\mathcal{L}(X)$. The dual space of $X$ is denoted by
$X^{*}=\mathcal{L}(X,\mathbb{R})$. If $H$ is a Hilbert space, we denote its
scalarproduct by $\left(\cdot,\cdot\right)_{H}$. For the whole paper, we fix
the final time $T>0$. To shorten the notation, Bochner-spaces are abbreviated
by $L^{p}(X):=L^{p}(0,T;X)$, $W^{1,p}(X):=W^{1,p}(0,T;X)$ ($p\in[1,\infty]$),
and $C(X):=C([0,T];X)$. Note that functions in $C(X)$ are continuous on the
whole time interval. When $G\in\mathcal{L}(X;Y)$ is a linear and continuous
operator, we can define an operator in $\mathcal{L}(L^{p}(X);L^{p}(Y))$ by
$G(u)(t):=G(u(t))$ for all $u\in L^{p}(X)$ and for almost all $t\in[0,T]$, we
denote this operator also by $G$, that is,
$G\in\mathcal{L}(L^{p}(X);L^{p}(Y))$, and analog for Bochner-Sobolev spaces,
i.e., $G\in\mathcal{L}(W^{1,p}(X);W^{1,p}(Y))$.

Given a coercive operator $G\in\mathcal{L}(H)$ in a Hilbert space $H$, we
denote its coercivity constant by $\gamma_{G}$, i.e.,
$\left(Gh,h\right)_{H}\geq\gamma_{G}\|h\|_{H}^{2}$ for all $h\in H$. With this
operator we can define a new scalar product, which induces an equivalent norm,
by $H\times
H\ni(h_{1},h_{2})\mapsto\left(Gh_{1},h_{2}\right)_{H}\in\mathbb{R}$. We denote
the Hilbert space equipped with this scalar product by $H_{G}$, that is
$\left(h_{1},h_{2}\right)_{H_{G}}=\left(Gh_{1},h_{2}\right)_{H}$ for all
$h_{1},h_{2}\in H$.

If $p\in[1,\infty]$, then we denote its conjugate exponent by $p^{\prime}$,
that is $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Furthermore, $c,C>0$ are generic
constants.

### Matrices

Given a matrix $\tau\in\mathbb{R}^{n\times n}$, we define its deviatoric
(i.e., trace-free) part as

$\tau^{D}:=\tau-\tfrac{1}{n}\operatorname{tr}(\tau)I$

and use the same notation for matrix-valued functions. The Frobenius norm is
denoted by $|A|_{F}^{2}=\sum_{i,j=1}^{n}A_{ij}^{2}$ for
$A\in\mathbb{R}^{n\times n}$ and for the associated scalar product, we write
$A:B=\sum_{i,j=1}^{n}A_{ij}B_{ij}$, $A,B\in\mathbb{R}^{n\times n}$. By
$\mathbb{R}^{n\times n}_{\textup{sym}}$, we denote the space of symmetric
matrices.

### Domain

The domain $\Omega\subset\mathbb{R}^{n}$, $n\in\mathbb{N}$, $n\geq 2$, is
bounded of class $C^{1}$. The boundary consists of two disjoint measurable
parts $\Gamma_{N}$ and $\Gamma_{D}$ such that
$\Gamma=\Gamma_{N}\cup\Gamma_{D}$. While $\Gamma_{N}$ is a relatively open
subset, $\Gamma_{D}$ is a relatively closed. We moreover suppose that
$\Gamma_{D}$ has a nonempty relative interior. In addition, the set
$\Omega\cup\Gamma_{N}$ is regular in the sense of Gröger, cf. [15]. Throughout
the article, $\nu:\partial\Omega\to\mathbb{R}^{n}$ denotes the outward unit
normal vector.

Thanks to the regularity of $\Omega$, the harmonic extension $\mathfrak{E}$
maps $C^{1}(\Gamma)$ to $W^{1,p}(\Omega)$ for some $p>n$. Moreover, the
maximum principle implies that

(1)
$\|\mathfrak{E}\varphi\|_{L^{\infty}(\Omega)}\leq\|\varphi\|_{L^{\infty}(\Gamma)}\quad\forall\,\varphi\in
C^{1}(\Gamma).$

######  Remark 2.1.

The $C^{1}$-regularity of $\Omega$ and its boundary, respectively, is required
for the trace theorem and the formula of integration by parts for
$\mathrm{BD}$-functions in [31, Chap. II, Theorem 2.1], which will be used
several times throughout the paper. In [12, Section 6], it is claimed that
this formula integration by parts also holds in Lipschitz domains, but no
proof is provided. Since the minimal regularity of the boundary is not in the
focus of this paper and would go beyond the scope of our work, we restrict to
domains of class $C^{1}$.

### Spaces

Throughout the paper, by $L^{p}(\Omega;M)$ we denote Lebesgue spaces with
values in $M$, where $p\in[1,\infty]$ and $M$ is a finite dimensional space.
To shorten notation, we abbreviate

$\mathbf{L}^{p}(\Omega):=L^{p}(\Omega;\mathbb{R}^{n})\quad\text{and}\quad\mathbb{L}^{p}(\Omega):=L^{p}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}}).$

Given $s\in\mathbb{N}$ and $p\in[1,\infty]$, the Sobolev spaces of vector-
resp. tensor-valued functions are denoted by

$\displaystyle\mathbf{W}^{s,p}(\Omega)$
$\displaystyle:=W^{s,p}(\Omega;\mathbb{R}^{n}),\quad$
$\displaystyle\mathbf{H}^{s}(\Omega)$
$\displaystyle:=\mathbf{W}^{s,2}(\Omega),$
$\displaystyle\mathbb{W}^{s,p}(\Omega)$
$\displaystyle:=W^{s,p}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}}),\quad$
$\displaystyle\mathbb{H}^{s}(\Omega)$
$\displaystyle:=\mathbb{W}^{s,2}(\Omega).$

Furthermore, set

(2) $\mathbf{W}_{D}^{1,p}(\Omega):=\overline{\\{\psi|_{\Omega}:\psi\in
C^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n}),\;\operatorname{supp}(\psi)\cap\Gamma_{D}=\emptyset\\}}^{\mathbf{W}^{1,p}(\Omega)}$

and define $\mathbf{H}^{1}_{D}(\Omega)$ analogously. The dual of
$\mathbf{H}^{1}_{D}(\Omega)$ is denoted by $\mathbf{H}^{-1}_{D}(\Omega)$. The
space of _bounded deformation_ is abbreviated by

$\mathrm{BD}(\Omega):=\\{u\in\mathbf{L}^{1}(\Omega):\tfrac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})\in\mathfrak{M}(\Omega)\;\forall\,i,j=1,...,n\\},$

where $\mathfrak{M}(\Omega)$ denotes the space of regular Borel measures on
$\Omega$ and the (partial) derivatives are of course understood in a
distributional sense. Equipped with the norm

$\|u\|_{\mathrm{BD}(\Omega)}:=\|u\|_{\mathbf{L}^{1}(\Omega)}+\sum_{i,j=1}^{n}\tfrac{1}{2}\|\partial_{i}u_{j}+\partial_{j}u_{i}\|_{\mathfrak{M}(\Omega)},$

it becomes a Banach space.

### Coefficients

The elasticity tensor satisfies $\mathbb{C}\in\mathcal{L}(\mathbb{R}^{d\times
d}_{\textup{sym}})$ and is symmetric and coercive. In addition we set
$\mathbb{A}:=\mathbb{C}^{-1}$ and note that $\mathbb{A}$ is symmetric and
coercive, too. Let us note that $\mathbb{C}$ could also depend on space,
however, to keep the discussion concise, we restrict ourselves to constant
elasticity tensors.

### Yield condition

The set defining the yield condition is denoted by
$K\subset\mathbb{R}^{n\times n}_{\textup{sym}}$ and is closed and convex and
there exists $0<\rho<R$ such that

(3) $\overline{B_{\mathbb{R}^{n\times n}}(0;\varrho)}\subset
K\subset\overline{B_{\mathbb{R}^{n\times n}}(0;R)}.$

Given this set, we define the _set of admissible stresses_ as

$\mathcal{K}(\Omega):=\\{\tau\in\mathbb{L}^{2}(\Omega):\tau^{D}(x)\in K\text{
f.a.a. }x\in\Omega\\}.$

######  Remark 2.2.

The boundedness of the set $K$ is not really needed for our analysis. It is
only required for the formula of integration by parts in (13), which we only
need to compare our notion of solution to the one in [8]. Nevertheless, we
kept the boundedness assumption on the set $K$, since it is fulfilled in all
practically relevant examples such as e.g. the von Mises or the Tresca yield
condition.

### Operators

Throughout the paper,
$\nabla^{s}:=\frac{1}{2}(\nabla+\nabla^{\top}):\mathbf{W}^{1,p}(\Omega)\to\mathbb{L}^{p}(\Omega)$
denotes the linearized strain. Its restriction to
$\mathbf{W}_{D}^{1,p}(\Omega)$ is denoted by the same symbol and, for the
adjoint of this restriction, we write
$-\operatorname{div}:=(\nabla^{s})^{*}:\mathbb{L}^{p^{\prime}}(\Omega)\to\mathbf{W}^{1,p}_{D}(\Omega)^{*}$.

Let $\mathcal{K}\subset\mathbb{L}^{2}(\Omega)$ be a closed and convex set. We
denote the indicator function by

$I_{\mathcal{K}}:\mathbb{L}^{2}(\Omega)\rightarrow\\{0,\infty\\},\text{ \ \ \
\ \ \ }\tau\mapsto\begin{cases}0,&\tau\in\mathcal{K},\\\
\infty,&\tau\notin\mathcal{K}.\end{cases}$

By $\partial I_{\mathcal{K}}:\mathbb{L}^{2}(\Omega)\rightarrow
2^{\mathbb{L}^{2}(\Omega)}$ we denote the subdifferential of the indicator
function. For $\lambda>0$, the Yosida regularization is given by

(4) $I_{\lambda}:\mathbb{L}^{2}(\Omega)\rightarrow\mathbb{R},\text{ \ \ \ \ \
\
}\tau\mapsto\frac{1}{2\lambda}\|\tau-\pi_{\mathcal{K}}(\tau)\|_{\mathbb{L}^{2}(\Omega)}^{2},$

where $\pi_{\mathcal{K}}$ is the projection onto $\mathcal{K}$ in
$\mathbb{L}^{2}(\Omega)$, and its Fréchet derivative is

$\displaystyle\partial
I_{\lambda}(\tau)=\frac{1}{\lambda}(\tau-\pi_{\mathcal{K}}(\tau)).$

When $\lambda=0$ we define $I_{\lambda}=I_{0}:=I_{\mathcal{K}}$. For a
sequence $\\{\lambda_{n}\\}_{n\in\mathbb{N}}\subset(0,\infty)$ we abbreviate
$I_{n}:=I_{\lambda_{n}}$.

### Initial data

For the initial stress field $\sigma_{0}$, we assume that
$\sigma_{0}\in\mathbf{W}^{1,p}(\Omega)$ with some $p>n$. Moreover,
$\sigma_{0}$ satisfies the equilibrium condition, i.e.,
$\operatorname{div}\sigma_{0}=0$ a.e. in $\Omega$, and the yield condition,
i.e., $\sigma_{0}\in\mathcal{K}(\Omega)$. The initial displacement $u_{0}$ is
supposed to be an element of $\mathbf{H}^{2}(\Omega)$ and we require
$\operatorname{tr}(\nabla^{s}u_{0}-\mathbb{A}\sigma_{0})=0$ a.e. in $\Omega$
in order to obtain a purely deviatoric initial plastic strain.

######  Remark 2.3.

The high regularity of $u_{0}$ is just needed to ensure that the feasible set
of (P) is nonempty. For the mere discussion of the state system, this is not
necessary. The same holds for the assumption
$\sigma_{0}\in\mathbf{W}^{1,p}(\Omega)$, which will be needed to construct a
recovery sequence for the optimal control problem.

### Optimization Problem

The Tikhonov parameter $\alpha$ is a positive constant and $\Psi$ is a
functional that is bounded from below and satisfies a certain lower
semicontinuity assumption w.r.t. weak convergence in the displacement space,
which will be made precise in Section 4 below, see (23).

##  3 State Equation

We start our investigations with the analysis of the state system and recall
some known results concerning quasi-static perfect plasticity. Already since
the pioneering work of Suquet [30], it is well known that a precise definition
of a solution to the system of perfect plasticity is all but straight forward,
since a solution of the system in its “natural” form (below termed strong
solution) does in general not exist due to a lack of regularity of the
displacement and the plastic strain, respectively. We start with the
definition of the function spaces already indicating this lack of regularity:

######  Definition 3.1 (State spaces).

  1. 1.

_Stress space:_

$\Sigma(\Omega):=\\{\tau\in\mathbb{L}^{2}(\Omega):\operatorname{div}\tau\in\mathbf{L}^{n}(\Omega),\;\tau^{D}\in\mathbb{L}^{\infty}(\Omega)\\}$

  2. 2.

_Displacement space:_

$\mathcal{U}:=\\{u\in
H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega)):\nabla^{s}\overset{\boldsymbol{.}}{u}\in
L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}}))\\}.$

Herein, $L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}}))$ is the space of weakly measurable functions with values
in $\mathfrak{M}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}})$, for which
$t\mapsto\|\mu(t)\|_{\mathfrak{M}}$ is an element of $L^{2}(0,T;\mathbb{R})$.
For the definition of weak measurability, we refer to [9, Section 8].

We say that a sequence $\\{u_{n}\\}\subset\mathcal{U}$ converges weakly in
$\mathcal{U}$ to $u$ and write $u_{n}\rightharpoonup u$ in $\mathcal{U}$, iff

(5) $u_{n}\rightharpoonup u\;\text{ in
}H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega)),\quad\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}\rightharpoonup^{*}\nabla^{s}\overset{\boldsymbol{.}}{u}\;\text{
in }L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}})).$

Note that, by [9, Theorem 8.20.3],
$L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}}))=L^{2}(C_{0}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}}))^{*}$, which gives a meaning to the weak-$\ast$ convergence
in (5).

######  Remark 3.2.

Unfortunately, $\mathrm{BD}(\Omega)$ does not admit the Radon-Nikodým property
and therefore weak measurability does not imply Bochner-measurability.

######  Definition 3.3 (Equilibrium condition).

We define the set of stresses which fulfill the _equilibrium condition_ as

$\displaystyle\mathcal{E}(\Omega):=\operatorname{ker}(\operatorname{div})=\\{\tau\in\mathbb{L}^{2}(\Omega):\left(\tau,\nabla^{s}\varphi\right)_{\mathbb{L}^{2}(\Omega)}=0\;\;\forall\,\varphi\in\mathbf{H}^{1}_{D}(\Omega)\\}.$

Note that $\sigma\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega)$ implies
$\sigma\in\Sigma(\Omega)$.

With the above definitions at hand, we can now define a hierarchy of three
different solutions:

######  Definition 3.4 (Notions of solutions).

Let $u_{D}\in H^{1}(\mathbf{H}^{1}(\Omega))$ with $u_{D}(0)=u_{0}$ a.e. on
$\Gamma_{D}$ be given. Then we define the following notions of solutions to
the perfect plasticity system:

  1. 1.

_Reduced solution:_ A function $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$ is
called _reduced solution_ of the state equation, if, for almost all
$t\in(0,T)$, the following holds true:

$\bullet$ Equilibrium and yield condition: (6b)
$\displaystyle\sigma(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$
$\bullet$ Reduced flow rule inequality: (6c)
$\displaystyle\int_{\Omega}\big{(}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t)-\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}(t)\big{)}:\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x\geq
0\quad\forall\,\tau\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$ $\bullet$
Initial condition: (6d) $\displaystyle\sigma(0)=\sigma_{0}.$

  2. 2.

_Weak solution:_ A tuple $(u,\sigma)\in\mathcal{U}\times
H^{1}(\mathbb{L}^{2}(\Omega))$ is called _weak solution_ of the state
equation, if, for almost all $t\in(0,T)$, there holds

$\bullet$ Equilibrium and yield condition: (7b)
$\displaystyle\sigma(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$
$\bullet$ Weak flow rule inequality: (7c) $\displaystyle\begin{aligned}
\int_{\Omega}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t):\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x+\int_{\Omega}\overset{\boldsymbol{.}}{u}(t)\cdot\operatorname{div}\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x\qquad\qquad&\\\
\geq\int_{\Omega}\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}(t):\big{(}\tau-\sigma(t)\big{)}+\overset{\boldsymbol{.}}{u}_{D}(t)\cdot\operatorname{div}\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x\qquad&\\\\[-4.30554pt]
\forall\,\tau\in\Sigma(\Omega)\cap\mathcal{K}(\Omega),&\end{aligned}$
$\bullet$ Initial condition: (7d) $\displaystyle
u(0)=u_{0},\quad\sigma(0)=\sigma_{0}.$

  3. 3.

_Strong solution:_ A tuple $(u,\sigma)\in H^{1}(\mathbf{H}^{1}(\Omega))\times
H^{1}(\mathbb{L}^{2}(\Omega))$ is called _strong solution_ of the state
equation, if, for almost all $t\in(0,T)$, there holds

$\bullet$ Equilibrium and yield condition: (8b)
$\displaystyle\sigma(t)\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$
$\bullet$ Strong flow rule inequality: (8c) $\displaystyle\begin{aligned}
\int_{\Omega}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t):\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x+\int_{\Omega}\nabla^{s}\overset{\boldsymbol{.}}{u}(t):\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x\geq
0\qquad&\\\\[-4.30554pt] \forall\,\tau\in\mathcal{K}(\Omega),&\end{aligned}$
$\bullet$ Dirichlet boundary condition: (8d) $\displaystyle
u(t)-u_{D}(t)\in\mathbf{H}^{1}_{D}(\Omega)$ $\bullet$ Initial condition: (8e)
$\displaystyle u(0)=u_{0},\quad\sigma(0)=\sigma_{0}.$

Some words concerning this definition are in order. First, let us shortly
investigate the relationship between the three different solution concepts. By
restricting the test functions in (7c) to functions in $\mathcal{E}(\Omega)$,
one immediately observes that every weak solution is also a reduced solution.
Moreover, by integration by parts, it is evident that (8c) and (8d) imply
(7c). On the other hand, if a weak solution satisfies $u\in
H^{1}(\mathbf{H}^{1}(\Omega))$ and the Dirichlet boundary conditions in (8d),
then integration by parts yields (8c), provided that
$\Sigma(\Omega)\cap\mathcal{K}(\Omega)$ is dense in $\mathcal{K}(\Omega)$,
which is a direct consequence of Lemma A.1 proven in the appendix. Thus, we
have the following relations between the three different solution concepts:

######  Corollary 3.5 (Relations between the solution concepts).

  1. 1.

If $(u,\sigma)$ is a weak solution, then $\sigma$ is automatically a reduced
solution.

  2. 2.

A weak solution $(u,\sigma)$ is a strong solution, if and only if $u\in
H^{1}(\mathbf{H}^{1}(\Omega))$ and $(u-u_{D})(t)\in\mathbf{H}^{1}_{D}(\Omega)$
for all $t\in[0,T]$.

One may further ask why no Dirichlet boundary conditions appear in the
definition of a weak solution. In fact, from a mechanical point of view, it is
reasonable that no boundary conditions are imposed, since plastic slips may
well develop on the Dirichlet part on the boundary, too, in form of tangential
jumps of the displacement perpendicular to the outward normal $\nu$. This
observation is implicitly contained in the above definition as demonstrated in
[8, Theorem 6.1]. For convenience of the reader, we shortly sketch the
underlying arguments. To this end, suppose that a weak solution is given and
let us define the _plastic strain_ $z\in
L^{2}_{w}(\mathfrak{M}(\Omega\cup\Gamma_{D};\mathbb{R}^{n\times
n}_{\textup{sym}}))$ by

(9) $z\lfloor_{\Omega}:=\nabla^{s}(u)-\mathbb{A}\sigma\,\textup{d}x,\quad
z\lfloor_{\Gamma_{D}}:=(u-u_{D})\odot\nu\mathcal{H}^{n-1},$

where $\odot$ refers to the symmetrized dyadic product, i.e., $a\odot
b=1/2(a_{i}b_{j}+a_{j}b_{i})_{i,j=1}^{n}$ for $a,b\in\mathbb{R}^{n}$. Note
that functions in $\mathrm{BD}(\Omega)$ admit traces in
$L^{1}(\partial\Omega;\mathbb{R}^{n})$ (see e.g. [31, Chap. II, Thm. 2.1]) so
that $z\lfloor_{\Gamma_{D}}$ is well defined. According to [8, Lemma 5.5],
these equations carry over to the time derivatives for almost all $t\in(0,T)$,
i.e.,

(10)
$\overset{\boldsymbol{.}}{z}\lfloor_{\Omega}:=\nabla^{s}(\overset{\boldsymbol{.}}{u})-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}\,\textup{d}x,\quad\overset{\boldsymbol{.}}{z}\lfloor_{\Gamma_{D}}:=(\overset{\boldsymbol{.}}{u}-\overset{\boldsymbol{.}}{u}_{D})\odot\nu\mathcal{H}^{n-1}.$

Let us prove that the trace of $p$ vanishes. For this purpose, we need the
following formula of integration by parts:

######  Lemma 3.6 ([31, Chap. II, Thm. 2.1]).

For every $v\in\mathrm{BD}(\Omega)$ and every $\varphi\in W^{1,p}(\Omega)$,
$p>n$, there holds

(11)
$\int_{\Omega}\tfrac{1}{2}(v_{i}\partial_{j}\varphi+v_{j}\partial_{i}\varphi)\,\textup{d}x+\int_{\Omega}\varphi\,\textup{d}(\nabla^{s}v)_{ij}=\int_{\partial\Omega}\varphi\,\tfrac{1}{2}(v_{i}\nu_{j}+v_{j}\nu_{i})\,\textup{d}s$

for all $i,j=1,...,n$.

######  Remark 3.7.

The result in [31] is only stated for test functions in $C^{1}(\bar{\Omega})$.
However, the embeddings
$\mathrm{BD}(\Omega)\hookrightarrow\mathbf{L}^{\frac{n}{n-1}}(\Omega)$ and
$W^{1,p}(\Omega)\hookrightarrow C(\bar{\Omega})$, $p>n$, along with the trace
theorem for $\mathrm{BD}$-functions and the density of $C^{1}(\bar{\Omega})$
in $W^{1,p}(\Omega)$ imply that the integration by parts also holds for test
functions in $W^{1,p}(\Omega)$.

Now, let $\varphi\in C^{\infty}_{c}(\Omega)$ be arbitrary. Then, since
$\mathcal{K}(\Omega)$ just acts on the deviatoric part,
$\varphi\,\delta_{ij}+\sigma_{ij}(t)\in\Sigma(\Omega)\cap\mathcal{K}(\Omega)$
for all $t\in[0,T]$ and therefore (7c) and the above formula of integration by
parts give

$\sum_{i}\Big{(}\int_{\Omega}(\mathbb{A}\overset{\boldsymbol{.}}{\sigma})_{ii}\,\varphi\,\textup{d}x+\int_{\Omega}\varphi\,\textup{d}(\nabla^{s}\overset{\boldsymbol{.}}{u})_{ii}\Big{)}=0\quad\forall\,\varphi\in
C^{\infty}_{c}(\Omega)$

and therefore $\operatorname{tr}\overset{\boldsymbol{.}}{z}\lfloor_{\Omega}=0$
f.a.a. $t\in(0,T)$. Since
$\operatorname{tr}(\nabla^{s}(u_{0})-\mathbb{A}\sigma_{0})=0$, [8, Theorem
7.1] yields $\operatorname{tr}z\lfloor_{\Omega}=0$ for all $t\in[0,T]$.
Similarly, we choose an arbitrary test function $\psi\in C^{\infty}(\Gamma)$
with $\operatorname{supp}(\psi)\subset\Gamma_{D}$ and test (7c) with
$\mathfrak{E}\psi\,\delta_{ij}+\sigma_{ij}(t)\in\Sigma(\Omega)\cap\mathcal{K}(\Omega)$.
Note that $\mathfrak{E}\psi\,\delta_{ij}\in\Sigma(\Omega)$, since the harmonic
extension maps into $W^{1,p}(\Omega)$ with $p>n$. Applying then again the
formula of integration by parts implies, in view of
$\operatorname{tr}\overset{\boldsymbol{.}}{z}\lfloor_{\Omega}=0$, that

(12)
$(\overset{\boldsymbol{.}}{u}-\overset{\boldsymbol{.}}{u}_{D})\cdot\nu=0\quad\text{a.e.~{}on
}\Gamma_{D}.$

As $u_{0}=u_{D}(0)$ a.e. on $\Gamma_{D}$, this yields $(u-u_{D})\cdot\nu=0$
a.e. on $\Gamma_{D}$, giving in turn
$\operatorname{tr}z\lfloor_{\Gamma_{D}}=0$ for all $t\in[0,T]$. Now that we
know that $z$ is deviatoric, the formula of integration by parts from [8,
Proposition 2.2] is applicable, which yields

(13)
$\langle\tau^{D},\overset{\boldsymbol{.}}{z}(t)\rangle+\int_{\Omega}\tau:\big{(}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t)-\nabla^{s}(\overset{\boldsymbol{.}}{u}_{D}(t))\big{)}\,\textup{d}x=\int_{\Omega}\operatorname{div}\tau\cdot(\overset{\boldsymbol{.}}{u}(t)-\overset{\boldsymbol{.}}{u}_{D}(t))\,\textup{d}x$

for almost all $t\in(0,T)$ and all $\tau\in\Sigma(\Omega)$. It is to be noted
that the duality product $\langle\tau^{D},\overset{\boldsymbol{.}}{z}\rangle$
has to be treated with care, since, in general, $\tau^{D}\notin
C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}})$, but
$\overset{\boldsymbol{.}}{z}$ is only a measure. For a detailed and rigorous
discussion of this issue, we refer to [8, Section 2.3]. Inserting (13) in the
flow rule inequality (7c) then results in

(14) $\langle\tau^{D}-\sigma^{D}(t),\overset{\boldsymbol{.}}{z}(t)\rangle\geq
0\quad\forall\,\tau\in\Sigma(\Omega)\cap\mathcal{K}(\Omega),$

which is just the maximum plastic work inequality illustrating that $z$ as
defined in (9) is indeed the correct object for the plastic strain. As a
byproduct, we obtain the second equation in (9) as boundary condition on
$\Gamma_{D}$ indicating that the Dirichlet boundary condition in (8d) as part
of the definition of a strong solution is in general too restrictive as
already mentioned above. Accordingly, a strong solution does in general not
exist, while we have the following result for a weak solution:

######  Proposition 3.8 (Existence of weak solutions, [30, Résultat 2]).

For all $u_{D}\in H^{1}(\mathbf{H}^{1}(\Omega))$, there exists a weak solution
in the sense of Definition 3.4.

######  Proof 3.9.

Using the Yosida regularization, Suquet showed in [30] the existence of a
functions $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$ and $v\in
L^{2}_{w}(\mathrm{BD}(\Omega))$ so that, for almost all $t\in(0,T)$,

(15) $\displaystyle-\operatorname{div}\sigma(t)$
$\displaystyle\in\mathcal{E}(\Omega)\cap\mathcal{K}(\Omega),$
$\displaystyle\int_{\Omega}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}(t):$
$\displaystyle\,\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x+\int_{\Omega}v(t)\cdot\operatorname{div}\big{(}\tau-\sigma(t)\big{)}\,\textup{d}x$
$\displaystyle\geq\langle\overset{\boldsymbol{.}}{u}_{D}(t),(\tau-\sigma(t))\nu\rangle_{H^{1/2}(\Gamma_{D}),H^{-1/2}(\Gamma_{D})}\quad\forall\,\tau\in\Sigma(\Omega)\cap\mathcal{K}(\Omega),$
$\displaystyle\sigma(0)$ $\displaystyle=\sigma_{0}.$

Due to the continuous embedding
$\mathrm{BD}(\Omega)\hookrightarrow\mathbf{L}^{\frac{n}{n-1}}(\Omega)$ (see
e.g. [31, Chap. II, Theorem 2.2]) and the Radon-Nikodym property of
$\mathbf{L}^{\frac{n}{n-1}}(\Omega)$, we have that $v\in
L^{2}(\mathbf{L}^{\frac{n}{n-1}}(\Omega))$. Therefore,

$u(t):=u_{0}+\int_{0}^{t}v(r)\,\textup{d}r$

is an element of $H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega))$ and satisfies the
initial condition in (7d). Inserting this in (15) and integrating the right
hand side by parts gives the desired flow rule inequality (7c). The claimed
regularity of $u$ directly follows from the regularity of
$v=\overset{\boldsymbol{.}}{u}$.

######  Remark 3.10 (Other equivalent notions of solutions).

Beside the reformulation of the flow rule in terms of the maximum plastic work
inequality (14), there are other solutions concepts, which are equivalent to
the definition of a weak solution, such as the notion of a _quasi-static
evolution_ , which in essence corresponds to a global energetic solution in
the sense of [23]. For an overview over the various notions of solutions and a
rigorous proof of their equivalence, we refer to [8, Section 6]. A slightly
sloppy, but very illustrating derivation of the flow rule out of the quasi-
static evolution can also be found in [13].

Unfortunately, the weak solution is not unique, as the following example
shows:

######  Example 3.11 ([30, Section 2.1]).

We choose $\Omega=(0,1)$, $\Gamma_{D}=\partial\Omega=\\{0,1\\}$, $T=1$,
$K=[-1,1]$, $\mathbb{C}=1$, $(\sigma_{0},u_{0})=0$, and $u_{D}(t,x):=2tx$. One
easily verifies that the stress does only depend on the time with
$\sigma(t)=2t$ for $t\in(0,\frac{1}{2})$ and $\sigma(t)=1$ for
$t\in(\frac{1}{2},1)$. For the displacement one obtains $u(t,x)=2tx$ for
$(t,x)\in(0,\frac{1}{2})\times(0,1)$ so that it is unique for
$t\in(0,\frac{1}{2})$. For $t\in(\frac{1}{2},0)$ there are more than one
solution, for example

$\displaystyle u(t,x)$ $\displaystyle=2tx,\text{\, if
\,}(t,x)\in(\tfrac{1}{2},1)\times(0,1),$ $\displaystyle u(t,x)$
$\displaystyle=\left\\{\begin{array}[]{l
l}\frac{2tx}{\beta}+x-\frac{x}{\beta},&\text{\, if
\,}(t,x)\in(\frac{1}{2},1)\times[0,\beta],\\\\[2.15277pt] 2t+x-1,&\text{\, if
\,}(t,x)\in(\frac{1}{2},1)\times[\beta,1],\end{array}\right.$ $\displaystyle
u(t,x)$ $\displaystyle=\left\\{\begin{array}[]{l l}x,&\text{\, if
\,}(t,x)\in(\frac{1}{2},1)\times[0,\beta],\\\\[2.15277pt] \alpha
t+x-\frac{\alpha}{2},&\text{\, if
\,}(t,x)\in(\frac{1}{2},1)\times[\beta,1],\end{array}\right.$

where $\alpha\in[0,2]$ and $\beta\in[0,1]$ can be freely chosen. Note that the
last solution just provides the minimal regularity, i.e.,
$\partial_{x}\overset{\boldsymbol{.}}{u}(t)\in\mathfrak{M}(\Omega)$.

The uniqueness of the stress field observed in the above example is a general
result:

######  Lemma 3.12 (Uniqueness of the stress, [17, Theorem 1], [21, Lemma
3.5]).

Assume that $\sigma_{1},\sigma_{2}\in H^{1}(\mathbb{L}^{2}(\Omega))$ are two
reduced solutions. Then $\sigma_{1}=\sigma_{2}$.

######  Remark 3.13 (Optimal control vs. optimization).

Since the displacement field as part of a weak solution is not unique in
general, there is no (single-valued) control-to-state operator mapping $u_{D}$
to $u$. Therefore, one might argue that (P) is actually no real optimal
control problem. Strictly speaking, one should rather regard it as an
_optimization problem_ with the triple $(u,\sigma,u_{D})$ as optimization
variables.

##  4 Existence of Optimal Solutions

Before we come to the main point of our analysis, which concerns the
approximation of (P) by means of regularized optimal control problems, let us
address the existence of optimal solutions. The proof in principle follows the
classical direct method, for which we need the following boundedness and
continuity results:

######  Lemma 4.1 ([21, Lemma 3.6]).

Let $u_{D}\in H^{1}(\mathbf{H}^{1}(\Omega))$ be given and $\sigma$ be the
associated reduced solution. Then there holds

(16)
$\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}\leq\gamma_{\mathbb{A}}^{-1}\,\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))},$

where $\gamma_{\mathbb{A}}$ is the coercivity constant of $\mathbb{A}$.
Consequently, there is a constant $C>0$ such that
$\|\sigma\|_{H^{1}(\mathbb{L}^{2}(\Omega))}\leq
C(\|\sigma_{0}\|_{\mathbb{L}^{2}(\Omega)}+\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))})$.

######  Lemma 4.2 (Continuity of reduced solutions, [21, Proposition 3.10]).

Let $\\{u_{D,n}\\}\subset H^{1}(\mathbf{H}^{1}(\Omega))$ be a sequence such
that

(17) $\begin{gathered}u_{D,n}\rightharpoonup u_{D}\;\text{ in
}H^{1}(\mathbf{H}^{1}(\Omega)),\quad u_{D,n}\rightarrow u_{D}\;\text{ in
}L^{2}(\mathbf{H}^{1}(\Omega)),\\\ u_{D,n}(T)\rightarrow u_{D}(T)\;\text{ in
}\mathbf{H}^{1}(\Omega)\end{gathered}$

and denote the (unique) reduced solution associated with $u_{D,n}$ by
$\sigma_{n}$. Then $\sigma_{n}\rightharpoonup\sigma$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$, where $\sigma$ is the reduced solution w.r.t.
$u_{D}$.

######  Lemma 4.3.

There is a constant $C>0$, independent of $u_{D}$, such that every weak
solution w.r.t. $u_{D}$ fulfills

$\Big{(}\int_{0}^{T}\|\overset{\boldsymbol{.}}{u}(t)\|^{2}_{\mathrm{BD}(\Omega)}\,\textup{d}t\Big{)}^{1/2}\leq
C\,\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{(}1+\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{)}.$

######  Proof 4.4.

Let $\varphi\in C^{\infty}_{c}(\Omega)$ with
$\|\varphi\|_{L^{\infty}(\Omega)}\leq 1$ and $i,j\in\\{1,...,n\\}$ be
arbitrary. According to (3), the test function

$(\tau_{\varphi})_{ij}=(\tau_{\varphi})_{ji}:=-\frac{\varrho}{\sqrt{2}}\,\varphi,\quad(\tau_{\varphi})_{kl}=0\quad\forall\,(k,l)\notin\\{(i,j),(j,i)\\}$

is admissible for (7c). Using $\operatorname{div}\sigma=0$, we deduce

$\int_{\Omega}\varphi\,\textup{d}(\nabla^{s}\overset{\boldsymbol{.}}{u})_{ij}\leq\frac{\sqrt{2}}{\varrho}\Big{(}\int_{\Omega}\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}:\sigma\,\textup{d}x-\int_{\Omega}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}:(\tau_{\varphi}-\sigma)\,\textup{d}x\Big{)}$

and consequently, since $\varphi\in C^{\infty}_{c}(\Omega)$ with
$\|\varphi\|_{L^{\infty}(\Omega)}\leq 1$ was arbitrary,

$\displaystyle\|\nabla^{s}\overset{\boldsymbol{.}}{u}\|_{L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}}))}$ $\displaystyle\leq C\big{(}\begin{aligned}
&\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\,\|\sigma\|_{L^{\infty}(\mathbb{L}^{2}(\Omega))}\\\
&+\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}+\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}\|\sigma\|_{L^{\infty}(\mathbb{L}^{2}(\Omega))}\big{)}\end{aligned}$
$\displaystyle\leq
C\,\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{(}1+\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{)},$

where we used Lemma 4.1.

Since $\Gamma_{D}$ is assumed to have a nonempty relative interior, there is a
set $\Lambda\subset\Gamma_{D}$ and a constant $\delta>0$ such that $\Lambda$
has positive boundary measure and
$\operatorname{dist}(\Lambda,\partial\Gamma_{D})\geq\delta$. By [31, Chap. II,
Theorem 2.1], $\overset{\boldsymbol{.}}{u}(t)$ admits a trace in
$\mathbf{L}^{1}(\Gamma)$ for almost all $t\in(0,T)$. In the following, we
neglect the variable $t$ for the sake of readability. The restriction of this
trace to $\Lambda$ is denoted by $\overset{\boldsymbol{.}}{u}|_{\Lambda}$. We
extend $\operatorname{sign}(\overset{\boldsymbol{.}}{u}|_{\Lambda})$ (where
the sign is to be understood componentwise) to the whole boundary $\Gamma$ by
zero and apply convolution with a smoothing kernel to obtain a sequence of
functions $\\{\varphi_{n}\\}\subset C^{\infty}(\Gamma;\mathbb{R}^{n})$ with
$\operatorname{supp}(\varphi_{n})\subset\Gamma_{D}$ (thanks to
$\operatorname{dist}(\Lambda,\partial\Gamma_{D})\geq\delta$) and
$\|\varphi_{n}\|_{L^{\infty}(\Gamma;\mathbb{R}^{n})}\leq 1$ for all
$n\in\mathbb{N}$. Given these functions, let us define

$(\tau_{n})_{ij}=\frac{\varrho}{\sqrt{2}}\,\mathfrak{E}(\varphi_{n,i}\,\nu_{j}+\varphi_{n,j}\,\nu_{i}),$

where $\mathfrak{E}$ denoted the harmonic extension and $\nu$ is the outward
normal. Then, (1) implies
$\|\tau_{n}\|_{\mathbb{L}^{\infty}(\Omega)}\leq\varrho$ and, since in addition
$\tau_{n}$ vanishes on $\Gamma_{N}$ by construction, we have
$\tau_{n}\in\Sigma(\Omega)\cap\mathcal{K}(\Omega)$. Note that, by the mapping
properties of $\mathfrak{E}$,
$\tau_{n}\in\mathbb{W}^{1,p}(\Omega)\hookrightarrow\Sigma(\Omega)$. If we
insert this as test function in (7c) and apply again the integration by parts
from Lemma 3.6, then $\operatorname{div}\sigma=0$ and (12) imply

$\int_{\Gamma_{D}}\varphi_{n}\cdot\overset{\boldsymbol{.}}{u}\,\textup{d}s\leq\frac{\sqrt{2}}{\varrho}\Big{(}\int_{\Omega}\tau_{n}:\textup{d}\nabla^{s}(\overset{\boldsymbol{.}}{u})-\int_{\Omega}\nabla^{s}\overset{\boldsymbol{.}}{u}_{D}:\sigma\,\textup{d}x+\int_{\Omega}\mathbb{A}\overset{\boldsymbol{.}}{\sigma}:(\tau_{n}-\sigma)\,\textup{d}x\Big{)}.$

Now, since $\varphi_{n}\to\operatorname{sign}(\overset{\boldsymbol{.}}{u})$
a.e. in $\Lambda$, $\varphi_{n}\to 0$ a.e. in $\Gamma_{D}\setminus\Lambda$ and
$|\varphi_{n}\cdot\overset{\boldsymbol{.}}{u}|\leq|\overset{\boldsymbol{.}}{u}|$
a.e. on $\Gamma_{D}$, Lebesgue’s dominated convergence theorem along with our
previous estimate gives

$\displaystyle\|\overset{\boldsymbol{.}}{u}\|_{L^{2}(\mathbf{L}^{1}(\Lambda))}\leq
C\,\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{(}1+\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}\big{)}.$

Thanks to [31, Chap. II, Proposition 2.4], this completes the proof.

######  Remark 4.5.

A priori estimates for quasistatic evolutions (which is an equivalent notion
of solution as mentioned above) are already proven in [8, Thm. 5.2] in a
slightly different setting.

######  Lemma 4.6.

Let $\\{u_{n}\\}\subset\mathcal{U}$ be a sequence such that, for all
$n\in\mathbb{N}$,

(18)
$u_{n}(0)=u_{0}\quad\text{and}\quad\int_{0}^{T}\|\overset{\boldsymbol{.}}{u}_{n}(t)\|_{\mathrm{BD}(\Omega)}^{2}\,\textup{d}t\leq
C$

with a constant $C>0$. Then there exists a subsequence converging weakly in
$\mathcal{U}$ as defined in (5).

######  Proof 4.7.

Owing to (18), $\\{\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}\\}$ is bounded in
$L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}}))$,
which, according to [9, Theorem 8.20.3], is the dual of
$L^{2}(C_{0}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}}))^{*}$. Thus, there
exists a subsequence such that

(19)
$\nabla^{s}\overset{\boldsymbol{.}}{u}_{n_{k}}\rightharpoonup^{*}w\quad\text{in
}L^{2}_{w}(\mathfrak{M}(\Omega;\mathbb{R}^{n\times n}_{\textup{sym}})).$

Due to $\mathrm{BD}(\Omega)\hookrightarrow\mathbf{L}^{\frac{n}{n-1}}(\Omega)$,
$\\{\overset{\boldsymbol{.}}{u}_{n_{k}}\\}$ is bounded in
$L^{2}(\mathbf{L}^{\frac{n}{n-1}}(\Omega))$ and, since all $u_{n}$ share the
same initial value, $\\{u_{n_{k}}\\}$ is bounded in
$H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega))$ so that, by reflexivity, there is
another subsequence (denoted w.l.o.g. by the same symbol) such that

(20) $u_{n_{k}}\rightharpoonup u\quad\text{in
}H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega)).$

Now, for every $\tau\in C^{\infty}_{c}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}})$ and every $\varphi\in C^{\infty}_{c}(0,T)$, (19) and (20)
imply

$\displaystyle\int_{0}^{T}\langle w(t),\tau\rangle\varphi(t)\,\textup{d}t$
$\displaystyle=\lim_{k\to\infty}\int_{0}^{T}\langle\nabla^{s}\overset{\boldsymbol{.}}{u}_{n_{k}}(t),\tau\rangle\varphi(t)\,\textup{d}t$
$\displaystyle=\lim_{k\to\infty}\int_{0}^{T}\int_{\Omega}\overset{\boldsymbol{.}}{u}_{n_{k}}(t)\cdot\operatorname{div}\tau\,\textup{d}x\,\varphi(t)\,\textup{d}t$
$\displaystyle=\int_{0}^{T}\int_{\Omega}\overset{\boldsymbol{.}}{u}(t)\cdot\operatorname{div}\tau\,\textup{d}x\,\varphi(t)\,\textup{d}t$

and hence $w(t)=\nabla^{s}\overset{\boldsymbol{.}}{u}(t)$ a.e. in $(0,T)$.

######  Proposition 4.8 (Continuity properties of weak solutions).

Let $\\{u_{D,n}\\}_{n\in\mathbb{N}}\subset H^{1}(\mathbf{H}^{1}(\Omega))$ be a
sequence fulfilling (17). Then, there is a subsequence of weak solutions
$\\{u_{n_{k}},\sigma_{n_{k}}\\}_{k\in\mathbb{N}}$ associated with
$\\{u_{D,n_{k}}\\}$ such that

$\displaystyle\sigma_{n_{k}}\rightharpoonup\sigma\;\text{ in
}H^{1}(\mathbb{L}^{2}(\Omega)),\quad u_{n_{k}}\rightharpoonup u\;\text{ in
}\mathcal{U},$

and the weak limit $(u,\sigma)$ is a weak solution associated with the limit
$u_{D}$.

######  Proof 4.9.

Since we already know that the stress component of every weak solution is also
a reduced one and the latter is unique by Lemma 3.12, the convergence of the
stresses follows from Lemma 4.2 (even for the whole sequence).

Owing to Lemma 4.3, $\\{\overset{\boldsymbol{.}}{u}_{n}\\}$ fulfills the
boundedness assumption in (18) so that, by Lemma 4.6, there is a subsequence
$\\{u_{n_{k}}\\}$ converging weakly in $\mathcal{U}$ to some limit
$u\in\mathcal{U}$. Due to $H^{1}(\mathbf{L}^{1}(\Omega))\hookrightarrow
C(\mathbf{L}^{1}(\Omega))$, the weak limit $u$ also satisfies the initial
condition.

It remains to prove that $(u,\sigma)$ fulfills the flow rule inequality (7c).
To this end, choose an arbitrary $\tau\in L^{2}(\mathbb{L}^{2}(\Omega))$ with
$\tau(t)\in\Sigma(\Omega)\cap\mathcal{K}(\Omega)$ for almost all $t\in[0,T]$.
Then, the flow rule inequality for $(u_{n_{k}},\sigma_{n_{k}})$ along with
$\operatorname{div}\sigma_{n_{k}}=0$ and the (weak) convergences of
$u_{D,n_{k}}$, $u_{n_{k}}$, and $\sigma_{n_{k}}$ yields

(21)
$\displaystyle\liminf_{k\to\infty}\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n_{k}},\sigma_{n_{k}}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\leq\lim_{k\to\infty}\begin{aligned}
\Big{[}&\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n_{k}}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n_{k}},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\\\
&+\int_{0}^{T}\int_{\Omega}(\overset{\boldsymbol{.}}{u}_{n_{k}}-\overset{\boldsymbol{.}}{u}_{D,n_{k}})\operatorname{div}\tau\,\textup{d}x\textup{d}t-\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n_{k}},\sigma_{n_{k}}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\Big{]}\end{aligned}$
$\displaystyle=\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{D},\tau\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}+\int_{0}^{T}\int_{\Omega}(\overset{\boldsymbol{.}}{u}-\overset{\boldsymbol{.}}{u}_{D})\operatorname{div}\tau\,\textup{d}x\textup{d}t-\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D},\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$

where we used Lemma 3.9 in our companion paper [21] for the convergence of the
last term. On the other hand, the weak lower semicontinuity of
$\|\cdot\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}$ together with
$H^{1}(\mathbb{L}^{2}(\Omega))\hookrightarrow C(\mathbb{L}^{2}(\Omega))$ gives

(22)
$\displaystyle\liminf_{k\rightarrow\infty}\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n_{k}},\sigma_{n_{k}}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\qquad=\frac{1}{2}\liminf_{k\rightarrow\infty}\|\sigma_{n_{k}}(T)\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}-\frac{1}{2}\|\sigma_{0}\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}$
$\displaystyle\qquad\geq\frac{1}{2}\|\sigma(T)\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}-\frac{1}{2}\|\sigma_{0}\|_{\mathbb{L}^{2}(\Omega)_{\mathbb{A}}}^{2}=\left(\mathbb{A}\overset{\boldsymbol{.}}{\sigma},\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}.$

Together with (21) and $\operatorname{div}\sigma=0$, this implies the flow
rule inequality for the weak limit.

Given these boundedness and continuity results, we can now establish the
existence of at least one optimal solution. Before we do so, let us recall our
optimization problem and state it in a rigorous manner:

(P) $\left\\{\,\begin{aligned} \min\text{ \ \
}&J(u,u_{D}):=\Psi(u)+\frac{\alpha}{2}\,\|u_{D}\|_{H^{1}(\mathbf{H}^{2}(\Omega))}^{2}\\\
\text{s.t.}\text{ \ \ }&u_{D}\in
H^{1}(\mathbf{H}^{2}(\Omega)),\quad(u,\sigma)\in\mathcal{U}\times
H^{1}(\mathbb{L}^{2}(\Omega)),\\\ &(u,\sigma)\text{ is a weak solution w.r.t.\
}u_{D},\quad\text{and}\quad
u_{D}(0)-u_{0}\in\mathbf{H}^{1}_{D}(\Omega),\end{aligned}\right.$

where $\Psi:\mathcal{U}\to\mathbb{R}$ is bounded from below and lower
semicontinous w.r.t. weak convergence in $\mathcal{U}$ as defined in (5),
i.e.,

(23) $u_{n}\rightharpoonup u\text{ in
}\mathcal{U}\quad\Longrightarrow\quad\liminf_{n\to\infty}\Psi(u_{n})\geq\Psi(u).$

An example for such a functional $\Psi$ will be given in Section 6 below.

######  Theorem 4.10 (Existence of optimal solutions).

There exists a globally optimal solution of Eq. P.

######  Proof 4.11.

Based on our above findings, the assertion immediately follows from the
standard direct method of calculus of variations. Nevertheless, let us shortly
sketch the arguments. First, we observe that the triple
$(u,\sigma,u_{D})\equiv(u_{0},\sigma_{0},u_{0})$ (constant in time) satisfies
the constraints in (P) so that the feasible set is nonempty. (At this point,
we need the additional regularity $u_{0}\in\mathbf{H}^{2}(\Omega)$.) Let
$(u_{n},\sigma_{n},u_{D,n})$ be a minimizing sequence. Then either
$(u_{0},u_{0})$ is already optimal or $J(u_{n},u_{D,n})\leq
J(u_{0},u_{0})<\infty$ for $n\in\mathbb{N}$ sufficiently large. Thus, since
$\Psi$ is bounded from below, $\\{u_{D,n}\\}$ is bounded in
$H^{1}(\mathbf{H}^{2}(\Omega))$. Via continuous and compact embedding, there
is thus a subsequence satisfying (17). Clearly, the associated limit satisfies
the conditions on the initial value in (P). Moreover, according to Proposition
4.8, a subsequence of weak solutions converges weakly in $\mathcal{U}\times
H^{1}(\mathbb{L}^{2}(\Omega))$ to a weak solution. Thus the weak limit is
feasible and the weak lower semicontinuity of norms and of $\Psi$ implies its
optimality.

######  Remark 4.12 (More general objectives).

The proof of existence readily transfers to slightly more general objectives
than the one in (P). For instance, one could add a term of the form
$\Phi(\sigma)$ with a function
$\Phi:H^{1}(\mathbb{L}^{2}(\Omega))\to\mathbb{R}$, which weakly lower
semicontinuous and bounded from below. Since objectives of this form have
already been discussed in the companion paper, we restrict ourselves to
objectives just depending on $u$ in order to keep the discussion concise.
Moreover, one could use other Tikhonov terms different from the
$H^{1}(\mathbf{H}^{2}(\Omega))$-norm to ensure the convergence properties in
(17) required for Proposition 4.8. For example, thanks to the Aubin-Lions
lemma, a Tikhonov term of the form

$\frac{\alpha}{2}\,\Big{(}\|u_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}^{2}+\|u_{D}\|_{L^{2}(X)}^{2}\Big{)}$

with any Banach space $X$ embedding compactly in $\mathbf{H}^{1}(\Omega)$
(such as e.g. $\mathbf{H}^{2}(\Omega)$) is sufficient to guarantee (17) for (a
subsequence of) a minimizing sequence. However, in order to shorten
presentation, we just consider the $H^{1}(\mathbf{H}^{2}(\Omega))$-norm.

##  5 Yosida Regularization and Reverse Approximation

As already mentioned above, the ultimate goal of our analysis is to establish
conditions that guarantee that optimal solutions to the optimization problem
(P) governed by perfect plasticity can be approximated via Yosida
regularization. The most crucial point in this respect is the so-called
_reverse approximation_ , which essentially means to construct a _recovery
sequence_ for a given perfect plastic solution. This is a rather challenging
task, as Example 3.11 illustrates: one easily verifies that every sequence of
regularized solutions tends to the linear solution $u(t,x)=2\,t\,x$ for
regularization parameter tending to zero, although there are infinitely many
other solutions. There is thus no hope that every perfect plastic solution can
be approximated via Yosida regularization! However, when it comes to
optimization, there is not only the state (i.e., the solution of the perfect
plasticity system), but also the _control_ variables, which can be used to
construct a recovery sequence. Unfortunately, the Dirichlet data $u_{D}$,
which serve as control variables in our case, are not sufficient for this
purpose. Instead we need a set of control variables that is rich enough to
generate a sufficiently large set of regularized solutions. For this purpose,
we introduce an _additional control variable in form of distributed loads_ and
end up with the following regularized version of the state equation:

(24a) $\displaystyle-\operatorname{div}\sigma_{\lambda}(t)$
$\displaystyle=\ell(t)$ $\displaystyle\text{in }\mathbf{H}^{-1}_{D}(\Omega),$
(24b) $\displaystyle\sigma_{\lambda}(t)$
$\displaystyle=\mathbb{C}(\nabla^{s}u_{\lambda}(t)-z_{\lambda}(t))$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (24c)
$\displaystyle\overset{\boldsymbol{.}}{z}_{\lambda}(t)$
$\displaystyle=\partial I_{\lambda}(\sigma_{\lambda}(t))$
$\displaystyle\text{in }\mathbb{L}^{2}(\Omega),$ (24d) $\displaystyle
u_{\lambda}(t)-u_{D}(t)$ $\displaystyle\in\mathbf{H}^{1}_{D}(\Omega),$ (24e)
$\displaystyle(u_{\lambda},\sigma_{\lambda})(0)$
$\displaystyle=(u_{0},\sigma_{0})$ $\displaystyle\text{in
}\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega).$

where $\lambda>0$ is the regularization parameter, $I_{\lambda}$ is the Yosida
regularization of the indicator functional, see (4), and $\ell\in
H^{1}(\mathbf{H}^{-1}_{D}(\Omega))$ is the mentioned load. Existence and
uniqueness of a solution to the regularized state equation Eq. 24 follows from
Banach’s fixed point theorem and can be proven by a reduction of the system to
an equation in the variable $z$ only, cf. e.g. [21, Proposition 3.15]. This
gives rise to the following

######  Lemma 5.1 (Existence of solutions to the regularized state system,
[21, Corollary 3.16]).

For every $\lambda>0$, $\ell\in H^{1}(\mathbf{H}^{-1}_{D}(\Omega))$, and
$u_{D}\in H^{1}(\mathbf{H}^{1}(\Omega))$ with $\ell(0)=0$ and
$u_{D}(0)|_{\Gamma_{D}}=u_{0}|_{\Gamma_{D}}$, there exists a unique solution
$(u_{\lambda},\sigma_{\lambda},z_{\lambda})\in
H^{1}(\mathbf{H}^{1}(\Omega))\times H^{1}(\mathbb{L}^{2}(\Omega))\times
H^{1}(\mathbb{L}^{2}(\Omega))$ of Eq. 24.

The associated solution operator is globally Lipschitz continuous with a
Lipschitz constant proportional to $\lambda^{-1}$.

The proof of existence is a direct consequence of the Lipschitz continuity of
$\partial I_{\lambda}$ and Banach’s contraction principle. In [21], the
external loads are set to zero, but it is straightforward to incorporate them
into the existence theory. The Lipschitz continuity of the solution mapping
directly follows from the Lipschitz estimate for the Yosida approximation, see
e.g. [3, Proposition 55.2(b)].

Before we address the approximation properties of this regularization approach
and its convergence behavior for $\lambda$ tending to zero in Section 6 below,
see Proposition 6.3, we first lay the foundations for the construction of a
recovery sequence in the upcoming three lemmas. Unfortunately, as already
indicated in the introduction, the passage to the limit in the regularized
state equation in Proposition 6.3 below requires a rather high regularity of
the stress field, and the recovery sequence has to fulfill this regularity,
too, as it is a constraint in the regularized optimal control problem (Pλ).
The key issue for our reverse approximation argument is therefore to improve
the regularity of the stress field provided a displacement field with higher
regularity is given. To this end, we first need an auxiliary result on the
derivative of the Yosida regularization. Since the set of admissible stresses
admits a pointwise representation by the set $K$, the Fréchet-derivative of
the Yosida regularization does the same, i.e., given an arbitrary
$\tau\in\mathbb{L}^{2}(\Omega)$, it holds

(25) $\partial
I_{\lambda}(\tau)(x)=\frac{1}{\lambda}\big{[}\tau(x)-\pi_{K}(\tau(x))\big{]}\quad\text{f.a.a.\
}x\in\Omega,$

where $\pi_{K}:\mathbb{R}^{n\times n}_{\textup{sym}}\to\mathbb{R}^{n\times
n}_{\textup{sym}}$ is the projection on $K$. This pointwise representation
allows to derive the following

######  Lemma 5.2.

Let $\lambda>0$, $p>2$, and $\tau\in\mathbb{W}^{1,p}(\Omega)$ be arbitrary.
Then $\partial I_{\lambda}(\tau)\in\mathbb{W}^{1,p}(\Omega)$ and there holds

(26) $\displaystyle\|\partial I_{\lambda}(\tau)\|_{\mathbb{W}^{1,p}(\Omega)}$
$\displaystyle\leq\frac{1}{\lambda}\,\|\tau\|_{\mathbb{W}^{1,p}(\Omega)}$ and
(27) $\displaystyle\big{(}\partial_{i}(\partial
I_{\lambda}(\tau))\colon\partial_{i}\tau\big{)}(x)$ $\displaystyle\geq
0\quad\text{a.e.\ in }\Omega,\quad\forall\,i=1,...,n.$

######  Proof 5.3.

As a projection, $\pi_{k}:\mathbb{R}^{n\times
n}_{\textup{sym}}\to\mathbb{R}^{n\times n}_{\textup{sym}}$ is globally
Lipschitz continuous. Thus, the chain rule for Sobolev functions (see e.g.
[36, Thm 2.1.11]) implies that $\partial
I_{\lambda}(\tau)\in\mathbb{W}^{1,p}(\Omega)$ with

(28) $\frac{\partial}{\partial x_{m}}[\partial
I_{\lambda}(\tau)]_{ij}=\frac{1}{\lambda}\Big{(}\frac{\partial\tau_{ij}}{\partial
x_{m}}-\sum_{kl}\frac{\partial}{\partial\tau_{kl}}[\pi_{K}(\tau)]_{ij}\frac{\partial\tau_{kl}}{\partial
x_{m}}\Big{)}.$

Since the Lipschitz constant of the projection equals one, its directional
derivative clearly satisfies $|\pi_{K}^{\prime}(A;B)|_{F}\leq|B|_{F}$ for all
$A,B\in\mathbb{R}^{n\times n}_{\textup{sym}}$ and, consequently,

$\displaystyle\partial_{m}(\partial I_{\lambda}(\tau))\colon\partial_{m}\tau$
$\displaystyle=\frac{1}{\lambda}\big{(}|\partial_{m}\tau|_{F}^{2}-\pi_{K}^{\prime}(\tau;\partial_{m}\tau):\partial_{m}\tau\big{)}\geq
0,$

which is (27). It is moreover easily seen that
$\operatorname{Id}-\pi_{K}:\mathbb{R}^{n\times
n}_{\textup{sym}}\to\mathbb{R}^{n\times n}_{\textup{sym}}$ is globally
Lipschitz with Lipschitz constant 1, too. Thus, for every
$A,B\in\mathbb{R}^{n\times n}_{\textup{sym}}$, there holds
$|(\operatorname{Id}-\pi_{K})^{\prime}(A;B)|_{F}\leq|B|_{F}$. Since
$(\operatorname{Id}-\pi_{k})(0)=0$, the Lipschitz continuity moreover entails
$|(\operatorname{Id}-\pi_{k})(A)|_{F}\leq|A|_{F}$ for all
$A\in\mathbb{R}^{n\times n}_{\textup{sym}}$. In view of (28), this yields
(26).

The next lemma addresses the crucial regularity result for the stress field
$\sigma_{\lambda}$ as solution of

(29) $w-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{\lambda}=\partial
I_{\lambda}(\sigma_{\lambda}),\text{ \ \ \ \ }\sigma_{\lambda}(0)=\sigma_{0}.$

In the proof of our main result in Theorem 6.5, an optimal strain rate will
play the role of $w$ and the following regularity result will be essential for
the construction of a recovery sequence associated with that strain rate. The
required regularity of $w$ will carry over to this optimal strain rate and
represents the most restrictive assumption of our reverse approximation
approach.

######  Lemma 5.4 (Higher regularity of the stress field).

Let $\lambda>0$ be arbitrary and $w\in L^{2}(\mathbb{L}^{2}(\Omega))\cap
L^{1}(\mathbb{W}^{1,p}(\Omega))$ with $p\geq 2$ be given. Then (29) admits a
unique solution $\sigma_{\lambda}\in H^{1}(\mathbb{L}^{2}(\Omega))\cap
L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ and there holds

(30) $\|\sigma_{\lambda}\|_{L^{\infty}(\mathbb{W}^{1,p}(\Omega))}\leq
C_{p}\Big{(}\|w\|_{L^{1}(\mathbb{W}^{1,p}(\Omega))}+\|\sigma_{0}\|_{\mathbb{W}^{1,p}(\Omega)}^{p}\Big{)}$

with $C_{p}:=p\,\|\mathbb{A}\|^{p/2-1}$.

######  Proof 5.5.

Step 1. Existence of solutions in $H^{1}(\mathbb{L}^{2}(\Omega))$: First we
note that (29) is just an ODE in $\mathbb{L}^{2}(\Omega)$ and $\partial
I_{\lambda}$ is globally Lipschitz in $\mathbb{L}^{2}(\Omega)$. Thus, the
existence and uniqueness of solutions in $H^{1}(\mathbb{L}^{2}(\Omega))$
follows from the generalized Picard-Lindelöf theorem in Banach spaces.
However, a pointwise projection is in general not Lipschitz continuous in
Sobolev spaces. Therefore, we cannot apply this simple argument to show that
the solution is an element of $W^{1,1}(\mathbb{W}^{1,p}(\Omega))$.

Step 2. Higher regularity in case of smooth data: To prove this, we apply a
time discretization scheme, namely the explicit Euler method. At first we
consider the case $w\in C(\mathbb{W}^{1,p}(\Omega))$. For $N\in\mathbb{N}$ and
$n\in\\{0,...,N\\}$, we set $d_{t}^{N}:=\frac{T}{N}$ and
$t_{n}^{N}:=nd_{t}^{N}$ such that $0=t_{0}^{N}<t_{1}^{N}<...<t_{N}^{N}=T$. Now
define $\sigma_{0}^{N}:=\sigma_{0}\in\mathbb{W}^{1,p}(\Omega)$ and

$\sigma^{N}_{n}:=\sigma^{N}_{n-1}+d_{t}^{N}\mathbb{C}\big{(}w(t_{n-1}^{N})-\partial
I_{\lambda}(\sigma^{N}_{n-1})\big{)}\in\mathbb{W}^{1,p}(\Omega)\quad\text{(by
\lx@cref{creftype~refnum}{lem:yosida})}$

such that

(31) $\mathbb{A}\frac{\sigma^{N}_{n}-\sigma^{N}_{n-1}}{d_{t}^{N}}+\partial
I_{\lambda}(\sigma^{N}_{n-1})=w(t_{n-1}^{N})$

for all $N\in\mathbb{N}$ and $n\in\\{1,...,N\\}$. We define the piecewise
linear approximation $\sigma^{N}\in W^{1,\infty}(\mathbb{W}^{1,p}(\Omega))$ by

$\sigma^{N}(t):=\sigma_{n-1}^{N}+\frac{t-t_{n-1}^{N}}{d_{t}^{N}}(\sigma_{n}^{N}-\sigma_{n-1}^{N})$

and the piecewise constant approximation $\tilde{\sigma}^{N}\in
L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ by
$\tilde{\sigma}^{N}(t):=\sigma_{n-1}^{N}$ for $t\in[t_{n-1}^{N},t_{n}^{N})$.
Using (26), we deduce from (31) that

$\|\sigma^{N}_{n}\|_{\mathbb{W}^{1,p}(\Omega)}\leq\|\sigma_{0}\|_{\mathbb{W}^{1,p}(\Omega)}+d_{t}^{N}C\Big{(}\sum_{i=0}^{n-1}\|\sigma^{N}_{i}\|_{\mathbb{W}^{1,p}(\Omega)}\Big{)}+C\|w\|_{C(\mathbb{W}^{1,p}(\Omega)},$

which, together with the discrete Gronwall lemma (cf. [16, Lemma 5.1 and the
following remark]), shows that $\sigma^{N}$is bounded in
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ by a constant independent of
$d_{t}^{N}$. Thus, again owing to Eq. 31 and (26),
$\overset{\boldsymbol{.}}{\sigma}^{N}(t)=\frac{\sigma_{n}^{N}-\sigma_{n-1}^{N}}{d_{t}^{N}}$,
$t\in(t_{n-1}^{N},t_{n}^{N})$ is also bounded in
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$. Therefore, $\sigma^{N}$ is bounded in
$H^{1}(\mathbb{L}^{2}(\Omega))$ and consequently, there is a weakly converging
subsequence, for simplicity also denoted by $\sigma^{N}$, such that
$\sigma^{N}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$ and
$\sigma^{N}\rightharpoonup^{*}\sigma$ in
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ as $N\to\infty$. Note that, due to the
reflexivity of $\mathbb{W}^{1,p}(\Omega)$,
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ can be identified with the dual of
$L^{1}(\mathbb{W}^{1,p^{\prime}}(\Omega))$ so there is a weakly-$\ast$
converging subsequence. It remains to show that $\sigma$ solves (29). Since
$\sigma^{N}$ is bounded in $W^{1,\infty}(\mathbb{W}^{1,p}(\Omega))$ as seen
above, we have by compact embeddings that $\sigma^{N}\to\sigma$ in
$C(\mathbb{L}^{2}(\Omega))$. Thus, we find for the piecewise constant
interpolation that, for every $t\in[t_{n-1}^{N},t_{n}^{N})$,

$\|\tilde{\sigma}^{N}(t)-\sigma(t)\|_{\mathbb{L}^{2}(\Omega)}\leq\|\sigma^{N}(t_{n-1}^{N})-\sigma(t)\|_{\mathbb{L}}^{2}(\Omega)\to
0\quad\text{as }N\to\infty.$

Therefore, (31) and the Lipschitz continuity of $\partial I_{\lambda}$ in
$\mathbb{L}^{2}(\Omega)$ give

$\displaystyle\|\mathbb{A}\overset{\boldsymbol{.}}{\sigma}^{N}+\partial
I_{\lambda}(\sigma^{N})-w\|_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\leq\|\mathbb{A}\overset{\boldsymbol{.}}{\sigma}^{N}+\partial
I_{\lambda}(\tilde{\sigma}^{N})-\tilde{w}^{N}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}+\|\partial
I_{\lambda}(\sigma^{N})-\partial
I_{\lambda}(\tilde{\sigma}^{N})\|_{L^{2}(\mathbb{L}^{2}(\Omega))}+\|\tilde{w}^{N}-w\|_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle\leq\frac{1}{\lambda}\|\sigma^{N}-\tilde{\sigma}^{N}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}+\|\tilde{w}^{N}-w\|_{L^{2}(\mathbb{L}^{2}(\Omega))}\rightarrow
0\quad\text{as }N\to\infty,$

where $\tilde{w}^{N}$ denotes the piecewise constant interpolation of $w$,
which converges strongly in $C(\mathbb{L}^{2}(\Omega))$ to $w$ thanks to the
assumed regularity of $w$. Therefore, by the weak lower semicontinuity of the
$L^{2}(\mathbb{L}^{2}(\Omega))$-norm, we see that the limit satisfies (29).

Step 3. Higher regularity for nonsmooth data: Let now $w\in
L^{2}(\mathbb{L}^{2}(\Omega))\cap L^{1}(\mathbb{W}^{1,p}(\Omega))$ be
arbitrary and take a sequence $\\{w_{n}\\}\subset C(\mathbb{W}^{1,p}(\Omega))$
such that $w_{n}\rightarrow w$ in $L^{1}(\mathbb{W}^{1,p}(\Omega))$. Let
$\sigma_{\lambda}\in H^{1}(\mathbb{L}^{2}(\Omega))$ be the solution of (29)
and denote by $\sigma_{\lambda,n}\in H^{1}(\mathbb{L}^{2}(\Omega))\cap
L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ the solution of

(32) $\displaystyle
w_{n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{\lambda,n}=\partial
I_{\lambda}(\sigma_{\lambda,n}),\text{ \ \ \ \
}\sigma_{\lambda,n}(0)=\sigma_{0}.$

Since $\partial
I_{\lambda}:\mathbb{L}^{2}(\Omega)\rightarrow\mathbb{L}^{2}(\Omega)$ is
monotone, one obtains $\sigma_{\lambda,n}\rightarrow\sigma_{\lambda}$ in
$H^{1}(\mathbb{L}^{2}(\Omega))$ by standard arguments. Moreover, (32) holds
almost everywhere in time and space and so that, f.a.a. $t\in[0,T]$,

$\partial_{j}w_{n}(t)-\mathbb{A}\partial_{j}\overset{\boldsymbol{.}}{\sigma}_{\lambda,n}(t)=\partial_{j}\partial
I_{\lambda}(\sigma_{\lambda,n})(t)\quad\text{a.e.\ in }\Omega.$

follows. Testing this equation with
$((\mathbb{A}\partial_{j}\sigma_{\lambda,n}:\partial_{j}\sigma_{\lambda,n})^{p/2-1}\partial_{j}\sigma_{\lambda,n})(t)\in\mathbb{W}^{1,p^{\prime}}(\Omega)$
and using (27) leads to

(33)
$\displaystyle\frac{d}{dt}\int_{\Omega}(\mathbb{A}\partial_{j}\sigma_{\lambda,n}:\partial_{j}\sigma_{\lambda,n})^{p/2}\textup{d}x$
$\displaystyle\leq
p\int_{\Omega}(\mathbb{A}\partial_{j}\sigma_{\lambda,n}:\partial_{j}\sigma_{\lambda,n})^{p/2-1}\big{(}\mathbb{A}\partial_{j}\sigma_{\lambda,n}:\partial_{j}\overset{\boldsymbol{.}}{\sigma}_{\lambda,n}+\partial_{j}\partial
I_{\lambda}(\sigma_{\lambda,n}):\partial_{j}\sigma_{\lambda,n}\big{)}\textup{d}x$
$\displaystyle=p\int_{\Omega}(\mathbb{A}\partial_{j}\sigma_{\lambda,n}:\partial_{j}\sigma_{\lambda,n})^{p/2-1}\,\partial_{j}w_{n}:\partial\sigma_{n}\,\textup{d}x$
$\displaystyle\leq
C_{p}\,\int_{\Omega}|\partial_{j}w|_{F}\,|\partial_{j}\sigma_{\lambda,n}|_{F}^{p-1}\,\textup{d}x\leq
C_{p}\,\|w\|_{\mathbb{W}^{1,p}(\Omega)}\,\|\sigma_{\lambda,n}\|_{\mathbb{W}^{1,p}(\Omega)}^{p-1},$

with $C_{p}$ as defined in the statement of the lemma. Integrating this
inequality in time and taking the coercivity of $\mathbb{A}$ into account
gives

$\|\sigma_{\lambda,n}\|_{L^{\infty}(\mathbb{W}^{1,p}(\Omega))}\leq
C_{p}\Big{(}\|w_{n}\|_{L^{1}(\mathbb{W}^{1,p}(\Omega))}+\|\sigma_{0}\|_{\mathbb{W}^{1,p}(\Omega)}^{p}\Big{)}.$

Therefore, $\sigma_{\lambda,n}$ is bounded in
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ and we can select a weakly-$\ast$
converging subsequence. The uniqueness of the weak limit then gives $\sigma\in
L^{\infty}(\mathbb{W}^{1,p}(\Omega))$ as claimed. The estimate in (30) finally
follows from the above inequality and the lower semicontinuity of the norm
w.r.t. weak-$\ast$ convergence.

######  Remark 5.6.

We observe that (26), (29), and the proven regularity of $\sigma_{\lambda}$
even imply that $\sigma_{\lambda}\in W^{1,1}(\mathbb{W}^{1,p}(\Omega))$.
However, we do not obtain an estimate independent of $\lambda$ in this norm
(in contrast to (30)) and therefore, this additional regularity is not useful
for us.

######  Lemma 5.7 ([20, Section 3]).

Let $w\in L^{2}(\mathbb{L}^{2}(\Omega))$ be given and $\lambda\searrow 0$.
Then $\sigma_{\lambda}\rightarrow\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$,
where $\sigma$ is the solution of

(34) $w-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}\in\partial
I_{\mathcal{K}(\Omega)}(\sigma),\text{ \ \ \ \ }\sigma(0)=\sigma_{0}.$

Moreover, there holds

(35)
$\|\sigma_{\lambda}-\sigma\|_{C(\mathbb{L}^{2}(\Omega))}^{2}\leq\lambda\,\frac{\|\mathbb{C}\|^{2}}{\gamma_{\mathbb{C}}}\,\|w-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}^{2},$

where $\gamma_{\mathbb{C}}>0$ is the coercivity constant of $\mathbb{C}$.

######  Proof 5.8.

The assertion is proven in [20], but, for convenience of the reader, we
shortly sketch the arguments. First, observe that $\sigma_{\lambda}\in
H^{1}(\mathbb{L}^{2}(\Omega))$ and $\sigma\in H^{1}(\mathbb{L}^{2}(\Omega))$
solve (29) and (34), respectively, if and only if
$z_{\lambda}:=W-\mathbb{A}\sigma_{\lambda}$ and $z:=W-\mathbb{A}\sigma$ with
$W(t):=\int_{0}^{t}w(s)\textup{d}s$ solve

(36) $\displaystyle\overset{\boldsymbol{.}}{z}_{\lambda}$
$\displaystyle=\partial I_{\lambda}(\mathbb{C}W-\mathbb{C}z_{\lambda}),$
$\displaystyle\quad z_{\lambda}(0)$
$\displaystyle=z_{0}:=-\mathbb{A}\sigma_{0}$ and (37)
$\displaystyle\overset{\boldsymbol{.}}{z}$ $\displaystyle\in\partial
I_{\mathcal{K}(\Omega)}(\mathbb{C}W-\mathbb{C}z),$ $\displaystyle\quad z(0)$
$\displaystyle=z_{0},$

respectively. These equations are exactly of the form studied in [20, Section
3] with the setting $A:=\partial I_{\mathcal{K}(\Omega)}$, $Q=R:=\mathbb{C}$,
and $\ell:=W$. The existence of $\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$
then follows from [20, Theorem 3.3], while the convergence
$\sigma_{\lambda}\rightarrow\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$ as well
as the estimate

$\|\mathbb{A}(\sigma_{\lambda}-\sigma)\|_{C(\mathbb{L}^{2}(\Omega))}^{2}\leq\frac{\lambda}{\gamma_{\mathbb{C}}}\|w-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}^{2}$

are consequences of [20, Proposition 3.5]. (Note that
$D(A)=\mathcal{K}(\Omega)$ is closed and $A^{0}\equiv 0$ in this case, hence,
the assumptions in [20, Section 2] are fulfilled). The inequality in (35) now
follows easily using
$\|\sigma_{\lambda}-\sigma\|_{\mathbb{L}^{2}(\Omega)}=\|\mathbb{C}\mathbb{A}(\sigma_{\lambda}-\sigma)\|_{\mathbb{L}^{2}(\Omega)}\leq\|\mathbb{C}\|\|\mathbb{A}(\sigma_{\lambda}-\sigma)\|_{\mathbb{L}^{2}(\Omega)}$.

######  Remark 5.9.

As a consequence of (30), the solution of (34) is an element of
$L^{\infty}(\mathbb{W}^{1,p}(\Omega))$, provided that $w\in
L^{1}(\mathbb{W}^{1,p}(\Omega))$. However, we do not need this regularity
result for the upcoming analysis.

As already mentioned, in the proof of our final convergence result in Theorem
6.5, $\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}$ will play the role of
the function $w$, where $\overline{u}$ is an optimal solution of (P). This
already indicates our most restrictive assumption, namely the existence of an
optimal solution providing the high regularity required for $w$. We will come
back to this point in Remark 6.7.

##  6 Convergence of Minimizers

We are now in the position to state the regularized optimal control problems.
Beside the additional control variable $\ell$ required for the reverse
approximation, they differ from (P) in an additional inequality constraint on
the stress field, which is needed to improve the regularity of the stress in
order to pass to the limit in the regularized state equation, see the proof of
Proposition 6.3 below. This additional regularity of the stresses is
unfortunately not enough to pass to the limit in the state system. We
additionally need to bound the displacement in $\mathcal{U}$, since this is
not guaranteed a priori by the regularized state system itself, unless the
loads fulfill a safe load condition. This however cannot be ensured for the
loads arising in the construction of the recovery sequence in the proof of our
main Theorem 6.5 (at least, we were not able to verify it). Therefore, we
directly enforce this boundedness by a special choice of the objective
functional as a tracking type objective of the following form:

(38)
$\Psi(u):=\int_{0}^{T}\|\nabla^{s}\overset{\boldsymbol{.}}{u}(t)-\mu(t)\|_{\mathfrak{M}(\Omega;\mathbb{R}^{n\times
n}_{\textup{sym}})}^{2}+\|\overset{\boldsymbol{.}}{u}(t)-v(t)\|_{\mathbf{L}^{1}(\Omega)}^{2}\,\textup{d}t$

with a given desired strain rate $\mu\in L^{2}(\mathbb{L}^{1}(\Omega))$ and a
desired displacement rate $v\in L^{2}(\mathbf{L}^{1}(\Omega))$. Note that this
objective trivially fulfills the lower semicontinuity assumption in (23). One
could even allow for less regular desired strain rates (in the space of
measures), but for convenience, we restrict to functions in
$L^{2}(\mathbb{L}^{1}(\Omega))$. The regularized counterpart of (P) now reads
as follows:

(Pλ) $\left\\{\;\begin{aligned}
\min\quad&J_{\lambda}(u,u_{D},\ell):=\|\nabla^{s}\overset{\boldsymbol{.}}{u}-\mu\|_{L^{2}(\mathbb{L}^{1}(\Omega))}^{2}+\|\overset{\boldsymbol{.}}{u}-v\|_{L^{2}(\mathbf{L}^{1}(\Omega))}^{2}\\\\[-2.15277pt]
&\quad\qquad\quad\quad+\frac{\alpha}{2}\,\|u_{D}\|_{H^{1}(\mathbf{H}^{2}(\Omega))}^{2}+\lambda^{-\theta}\|\ell\|_{L^{2}(\mathbf{H}^{-1}_{D}(\Omega))}^{2}+\|\overset{\boldsymbol{.}}{\ell}\|_{L^{2}(\mathbf{H}^{-1}_{D}(\Omega))}^{2}\\\
\text{s.t.}\quad&u_{D}\in H^{1}(\mathbf{H}^{2}(\Omega)),\quad\ell\in
H^{1}(\mathbf{H}^{-1}_{D}(\Omega)),\\\
&u_{D}(0)-u_{0}\in\mathbf{H}^{1}_{D}(\Omega),\quad\ell(0)=0,\\\
&(u,\sigma,z)\in\mathcal{U}\cap L^{2}(\mathbf{H}^{1}(\Omega))\times
L^{2}(\mathbb{L}^{2}(\Omega))\times H^{1}(\mathbb{L}^{2}(\Omega)),\\\
&(u,\sigma,z)\text{ is the solution of \eqref{eq:regularizedStateEquation}
w.r.t.\ $u_{D}$ and $\ell$,}\\\
&\|\overset{\boldsymbol{.}}{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}+\|\sigma\|_{L^{s}(\mathbb{W}^{1,p}(\Omega))}\leq
R\end{aligned}\right.$

with $0<\theta<1$ and

(39) $p>n\quad\text{and}\quad s>\max\Big{\\{}1,\frac{2np}{np+2(p-n)}\Big{\\}}$

and $R\geq\|\sigma_{0}\|_{\mathbb{W}^{1,p}(\Omega)}$ to be specified later,
see (43) below. With the exponents in (39), [19, Lemma 4.2(i)] is applicable
and tells us that $H^{1}(\mathbb{L}^{2}(\Omega))\cap
L^{s}(\mathbb{W}^{1,p}(\Omega))$ embeds compactly in
$L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}))$, which will be
useful at several places in the upcoming proofs. The term in the objective
associated with $\theta$ will be used to force the additional loads to zero in
the limit.

######  Proposition 6.1.

For every $\lambda>0$, there exists a globally optimal solution of (Pλ).

######  Proof 6.2.

The proof is almost standard, except for a lack of compactness with regard to
the control space. Let $(u_{n},\sigma_{n},z_{n},u_{D,n},\ell_{n})$ be a
minimizing sequence. As in the proof of Theorem 4.10,
$(u,\sigma,z,u_{D},\ell)\equiv(u_{0},\sigma_{0},\nabla^{s}u_{0}-\mathbb{A}\sigma_{0},u_{0},0)$
is feasible for (Pλ). Thus, $\\{u_{D,n},\ell_{n}\\}$ is bounded in
$H^{1}(\mathbf{H}^{2}(\Omega))\times H^{1}(\mathbf{H}^{-1}_{D}(\Omega))$.
Hence the Lipschitz continuity of the solution operator associated with (24)
implies that $\\{(u_{n},\sigma_{n},z_{n})\\}$ is bounded in
$H^{1}(\mathbf{H}^{1}(\Omega)\times\mathbb{L}^{2}(\Omega)\times\mathbb{L}^{2}(\Omega))$.
Therefore, there exist weakly convergent subsequences and we can pass to the
limit in (24) except for the nonlinearity in $\partial I_{\lambda}$. However,
the additional constraint on the stress implies that $\sigma_{n}$ also
converges weakly in $H^{1}(\mathbb{L}^{2}(\Omega))\cap
L^{s}(\mathbb{W}^{1,p}(\Omega))$, which is compactly embedded in
$L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}))$ as mentioned
above. Thus $\\{\sigma_{n}\\}$ converges strongly in
$L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}))$, which allows
to pass to the limit in $\partial I_{\lambda}(\sigma_{n})$ so that the weak
limit solves (24). Moreover, the inequality constraint on $\sigma_{n}$ is
clearly weakly closed so that the weak limit is indeed feasible for (Pλ).
Since the objective is convex and continuous and thus weakly lower
semicontinuous, the weak limit is also optimal.

######  Proposition 6.3 (Convergence of the Yosida regularization with varying
loads).

Let $\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ be a sequence converging to zero.
Suppose moreover that two sequences $\\{\ell_{n}\\}\subset
H^{1}(\mathbf{H}^{-1}_{D}(\Omega))$ and $\\{u_{D,n}\\}\subset
H^{1}(\mathbf{H}^{1}(\Omega))$ are given and denote the solution of (24)
associated with $\lambda_{n}$, $\ell_{n}$, and $u_{D,n}$ by
$(u_{n},\sigma_{n},z_{n})$. Furthermore, we assume that $\\{u_{D,n}\\}$
satisfies the convergence properties in (17), i.e.,

(40) $\begin{gathered}u_{D,n}\rightharpoonup u_{D}\;\text{ in
}H^{1}(\mathbf{H}^{1}(\Omega)),\quad u_{D,n}\rightarrow u_{D}\;\text{ in
}L^{2}(\mathbf{H}^{1}(\Omega)),\\\ u_{D,n}(T)\rightarrow u_{D}(T)\;\text{ in
}\mathbf{H}^{1}(\Omega),\end{gathered}$

and that

(41) $\displaystyle\ell_{n}$ $\displaystyle\rightharpoonup 0$
$\displaystyle\text{ in }L^{2}(\mathbf{H}^{-1}_{D}(\Omega)),\quad$
$\displaystyle u_{n}$ $\displaystyle\rightharpoonup u$ $\displaystyle\text{ in
}\mathcal{U},$ (42) $\displaystyle\sigma_{n}$
$\displaystyle\rightharpoonup\sigma$ $\displaystyle\text{ in
}H^{1}(\mathbb{L}^{2}(\Omega)),\quad$ $\displaystyle\sigma_{n}$
$\displaystyle\to\sigma$ $\displaystyle\text{ in
}L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}).$

Then $(u,\sigma)$ is a weak solution associated with $u_{D}$.

######  Proof 6.4.

The arguments are similar to the proof of Proposition 4.8. First, since
$\sigma_{n}\rightharpoonup\sigma$ in $H^{1}(\mathbb{L}^{2}(\Omega))$,
$\ell_{n}\rightharpoonup 0$ in $L^{2}(\mathbf{H}^{-1}_{D}(\Omega))$, and
$-\operatorname{div}\sigma_{n}=\ell_{n}$ for all $n\in\mathbb{N}$, it follows
that $\sigma(t)\in\mathcal{E}(\Omega)$ f.a.a. $t\in(0,T)$. Moreover, from
Lemma 3.20 and 3.21 in our compaion paper [21], we deduce that
$\sigma(t)\in\mathcal{K}(\Omega)$ a.e. in $(0,T)$, cf. also the first part of
the proof of [21, Theorem 3.22]. Moreover, due to
$H^{1}(\mathbf{L}^{\frac{n}{n-1}}(\Omega))\hookrightarrow
C(\mathbf{L}^{\frac{n}{n-1}}(\Omega))$ and
$H^{1}(\mathbb{L}^{2}(\Omega))\hookrightarrow C(\mathbb{L}^{2}(\Omega))$, the
weak limit satisfies the initial conditions.

To show the flow rule inequality, let $\tau\in L^{2}(\mathbb{L}^{2}(\Omega))$
with $\tau(t)\in\Sigma(\Omega)\cap\mathcal{K}(\Omega)$ f.a.a. $t\in(0,T)$ be
arbitrary. Then, (24b) and (24c) along with $I_{n}(a)=0$ for
$a\in\mathcal{K}(\Omega)$ and $I_{n}\geq 0$, imply

$\displaystyle 0$ $\displaystyle=\int_{0}^{T}I_{n}(\tau(t))dt$
$\displaystyle\geq\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau-\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}$
$\displaystyle=\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{n},\tau-\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}-\int_{0}^{T}\int_{\Omega}(\overset{\boldsymbol{.}}{u}_{n}-\overset{\boldsymbol{.}}{u}_{D,n})\operatorname{div}\tau\,\textup{d}x\textup{d}t$
$\displaystyle\qquad+\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}.$

Now one can pass to the limit with the first two terms on the right hand side
exactly as described at the end of the proof of Proposition 4.8, see (21) and
(22). Concerning the last term, we argue as follows: Since
$\operatorname{div}\sigma=0$ and $u_{n}$ satisfies the Dirichlet boundary
condition, i.e.,
$\overset{\boldsymbol{.}}{u}_{n}=\overset{\boldsymbol{.}}{u}_{D,n}$ on
$\Gamma_{D}$, we obtain

$\displaystyle\big{|}\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{n},\sigma_{n}\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\big{|}$
$\displaystyle\qquad=\big{|}\left(\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{n},\sigma_{n}-\sigma\right)_{L^{2}(\mathbb{L}^{2}(\Omega))}\big{|}$
$\displaystyle\qquad\leq\|\nabla^{s}\overset{\boldsymbol{.}}{u}_{D,n}-\nabla^{s}\overset{\boldsymbol{.}}{u}_{n}\|_{L^{2}(\mathbb{L}^{1}(\Omega))}\|\sigma_{n}-\sigma\|_{L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times
n}_{\textup{sym}})}\to 0,$

thanks to the boundedness of $u_{n}$ in $\mathcal{U}$ and the convergence of
$\sigma_{n}$.

The last step of the above proof illustrates, where the high regularity of the
stress field enforced by the additional inequality constraint in (Pλ) comes
into play: we need the strong convergence of the stress in
$L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}})$ in order to pass
to the limit in the flow rule inequality. Unfortunately, the recovery sequence
needs to be feasible for (Pλ) and thus has to fulfill this inequality
constraint, too. Using our results from Section 5, this can be guaranteed,
provided that there is at least one optimal solution, whose strain rate admits
higher regularity. This is the most severe restriction for our main result:

######  Theorem 6.5 (Approximation of global minimizers).

Let the objective in (P) be of the form (38). Assume moreover that there
exists a global minimizer $(\overline{u},\overline{\sigma},\overline{u}_{D})$
of Eq. P such that $\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}\in
L^{2}(\mathbb{L}^{2}(\Omega))\cap L^{1}(\mathbb{W}^{1,p}(\Omega))$ and
$\overline{u}-\overline{u}_{D}\in\mathbf{H}^{1}_{D}(\Omega)$ for all
$t\in(0,T)$. Suppose in addition that $R$ in (Pλ) is chosen so large that

(43)
$R\geq\frac{1}{\gamma_{\mathbb{A}}}\,\|\overline{u}_{D}\|_{H^{1}(\mathbf{H}^{1}(\Omega))}+p\,\|\mathbb{A}\|^{p/2-1}\Big{(}\|\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}\|_{L^{1}(\mathbb{W}^{1,p}(\Omega))}+\|\sigma_{0}\|_{\mathbb{W}^{1,p}(\Omega)}^{p}\Big{)}.$

Furthermore, let
$\\{\overline{u}_{\lambda},\overline{\sigma}_{\lambda},\overline{z}_{\lambda},\overline{u}_{D,\lambda},\overline{\ell}_{\lambda}\\}_{\lambda>0}$
be a sequence of global minimizers of (Pλ) for $\lambda\searrow 0$.

Then there exists an accumulation point of
$\\{\overline{u}_{\lambda},\overline{\sigma}_{\lambda},\overline{u}_{D,\lambda}\\}_{\lambda>0}$
w.r.t. weak convergence in $\mathcal{U}\times
H^{1}(\mathbb{L}^{2}(\Omega))\cap L^{s}(\mathbb{W}^{1,p}(\Omega))\times
H^{1}(\mathbf{H}^{2}(\Omega))$. Moreover, every such accumulation point is a
global minimizer of Eq. P.

Furthermore, if $(\tilde{u},\tilde{\sigma},\tilde{u}_{D})$ is such an
accumulation point and
$\\{\overline{u}_{\lambda},\overline{\sigma}_{\lambda},\overline{u}_{D,\lambda}\\}_{\lambda>0}$
the associated sequence converging weakly to it, then

(44) $\displaystyle\overline{u}$ $\displaystyle\to\tilde{u}$
$\displaystyle\text{in }H^{1}(L^{1}(\Omega;\mathbb{R}^{n})),\quad$
$\displaystyle\overline{u}_{D,\lambda}$ $\displaystyle\to\tilde{u}_{D}$
$\displaystyle\text{in }H^{1}(\mathbf{H}^{2}(\Omega)),$ (45)
$\displaystyle\overline{\sigma}_{\lambda}$ $\displaystyle\to\tilde{\sigma}$
$\displaystyle\text{in }L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times
n}_{\textup{sym}})),\quad$ $\displaystyle\overline{\ell}_{\lambda}$
$\displaystyle\to 0$ $\displaystyle\text{in
}H^{1}(\mathbf{H}^{-1}_{D}(\Omega)).$

######  Proof 6.6.

_Step 1. Existence of an accumulation point._ Since
$\\{\overline{u}_{\lambda},\overline{\sigma}_{\lambda}\overline{z}_{\lambda},\overline{u}_{D,\lambda},\overline{\ell}_{\lambda}\\}_{\lambda>0}$
is a global solution of (Pλ) and the constant tuple
$(u,\sigma,z,u_{D},\ell)\equiv(u_{0},\sigma_{0},\nabla^{s}u_{0}-\mathbb{A}\sigma_{0},u_{0},0)$
is feasible for (Pλ), we obtain

(46)
$J_{\lambda}(\overline{u}_{\lambda},\overline{u}_{D,\lambda},\overline{\ell}_{\lambda})\leq
J_{\lambda}(u_{0},u_{0},0)=\frac{\alpha}{2}\|u_{0}\|_{L^{2}(\mathbf{H}^{2}(\Omega))}^{2}=:C<\infty.$

Since all $\overline{u}_{\lambda}$ share the same initial value and due to the
special structure of the objective in (38), this implies that
$\overline{u}_{\lambda}$ satisfies the boundedness assumption in (18) such
that Lemma 4.6 yields the existence of a subsequence converging weakly in
$\mathcal{U}$. Moreover, the inequality constraint on the stress and the
$H^{1}(H^{2})$-norm in the objective immediately yield the boundedness of
$\overline{\sigma}_{\lambda}$ and $\overline{u}_{D,\lambda}$ in their
respective spaces, and the reflexivity of the latter imply the existence of a
weakly convergent subsequence.

_Step 2. Feasibility of an accumulation point._ Let us now assume that a given
subsequence of
$\\{\overline{u}_{\lambda},\overline{\sigma}_{\lambda},\overline{u}_{D,\lambda}\\}_{\lambda>0}$,
denoted by the same symbol for simplicity, converges weakly to
$(\tilde{u},\tilde{\sigma},\tilde{u}_{D})$ in $\mathcal{U}\times
H^{1}(\mathbb{L}^{2}(\Omega))\cap L^{s}(\mathbb{W}^{1,p}(\Omega))\times
H^{1}(\mathbf{H}^{2}(\Omega))$. By the compact embedding of
$H^{1}(\mathbf{H}^{2}(\Omega))$ in $C(\mathbf{H}^{1}(\Omega))$, this ensures
the convergence properties required in (40) and in addition
$\tilde{u}_{D}(0)-u_{0}\in\mathbf{H}^{1}_{D}(\Omega)$. Moreover, the
assumptions on $p$ and $s$ in (39) guarantee that
$H^{1}(\mathbb{L}^{2}(\Omega))\cap L^{s}(\mathbb{W}^{1,p}(\Omega))$ embeds
compactly in $L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}))$,
as already mentioned above, so that (42) is valid. Furthermore, considering
again Eq. 46, we see that
$\lambda^{-\theta}\|\overline{\ell}_{\lambda}\|_{L^{2}(\mathbf{H}^{-1}_{D}(\Omega))}$
is bounded, hence, $\overline{\ell}_{\lambda}\rightarrow\ell=0$ in
$L^{2}(\mathbf{H}^{-1}_{D}(\Omega))$ (even with strong convergence).
Altogether, we observe that the convergence properties in (40)–(42) are
fulfilled such that Proposition 6.3 yields that the weak accumulation point
$(\tilde{u},\tilde{\sigma})$ is a weak solution associated with
$\tilde{u}_{D}$ and therefore feasible for the original optimization problem
(P).

_Step 3. Construction of a recovery sequence._ First, observe that, since
$\overline{u}$ is assumed to be in $H^{1}(\mathbf{H}^{1}(\Omega))$ and to
satisfy the Dirichlet boundary conditions, Corollary 3.5 gives that
$(\overline{\sigma},\overline{u})$ is a strong solution associated with
$\overline{u}_{D}$.

The recovery sequence for $(\overline{u},\overline{\sigma},\overline{u}_{D})$
is constructed based on our findings in Section 5. To be more precise, we
apply Lemma 5.4 and Lemma 5.7 with
$w=\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}$. According to these
lemmas, $\sigma_{\lambda}\in H^{1}(\mathbb{L}^{2}(\Omega))$ defined as unique
solution of

$\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}_{\lambda}=\partial
I_{\lambda}(\sigma_{\lambda}),\text{ \ \ \ \ }\sigma_{\lambda}(0)=\sigma_{0},$

satisfies the bound in (30) and converges strongly in
$H^{1}(\mathbb{L}^{2}(\Omega))$ to $\sigma$, which is the solution to

$\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}-\mathbb{A}\overset{\boldsymbol{.}}{\sigma}\in\partial
I_{\mathcal{K}(\Omega)}(\sigma),\text{ \ \ \ \ }\sigma(0)=\sigma_{0}.$

This equation is just the strong form of the flow rule in (8c). The
monotonicity of $\partial I_{\mathcal{K}(\Omega)}$ immediately gives that (8c)
is uniquely solvable. Therefore, the limit $\sigma$ coincides with
$\overline{\sigma}$, i.e., the stress associated with $\overline{u}_{D}$. If
we now define

$z_{\lambda}:=\nabla^{s}\overline{u}-\mathbb{A}\sigma_{\lambda}\in
H^{1}(\mathbb{L}^{2}(\Omega))\quad\text{and}\quad\ell_{\lambda}:=-\operatorname{div}\sigma_{\lambda}\in
H^{1}(\mathbf{H}^{-1}_{D}(\Omega)),$

then we observe that $(\overline{u},\sigma_{\lambda},z_{\lambda})$ is the
solution of the regularized plasticity system in (24) w.r.t.
$\overline{u}_{D}$ and $\ell_{\lambda}$. In addition, we have
$\ell_{\lambda}(0)=-\operatorname{div}\sigma_{0}=0$ and
$\overline{u}_{D}(0)-u_{0}=\overline{u}_{D}(0)-\overline{u}(0)\in\mathbf{H}^{1}_{D}(\Omega)$.
Therefore, since $\sigma_{\lambda}$ satisfies the bounds in (30) and (16) (by
Lemma 4.1),
$(\overline{u},\sigma_{\lambda},z_{\lambda},\overline{u}_{D},\ell_{\lambda})$
satisfies all constraints in (Pλ).

Next we show the convergence of the objective functional. As
$\overline{\sigma}$ fulfills the equilibrium condition, i.e.,
$\overline{\sigma}\in\mathcal{E}(\Omega)$, the convergence of
$\sigma_{\lambda}$ by Lemma 5.7 implies

$\ell_{\lambda}=-\operatorname{div}\sigma_{\lambda}\to-\operatorname{div}\overline{\sigma}=0\quad\text{in
}H^{1}(\mathbf{H}^{-1}_{D}(\Omega)).$

Furthermore, (35) gives

$\displaystyle\lambda^{-\theta}\|\ell_{\lambda}\|_{L^{2}(\mathbf{H}^{-1}_{D}(\Omega))}^{2}$
$\displaystyle=\lambda^{-\theta}\,\|\operatorname{div}\sigma_{\lambda}-\operatorname{div}\overline{\sigma}\|_{L^{2}(\mathbf{H}^{-1}_{D}(\Omega))}^{2}$
$\displaystyle\leq
C\,\lambda^{-\theta}\,\|\sigma_{\lambda}-\overline{\sigma}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}^{2}$
$\displaystyle\leq
C\,\lambda^{1-\theta}\,\|\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}-\mathbb{A}\overset{\boldsymbol{.}}{\overline{\sigma}}\|_{L^{2}(\mathbb{L}^{2}(\Omega))}^{2}\to
0\quad\text{as }\lambda\searrow 0.$

To summarize, we found that
$(\overline{u},\sigma_{\lambda},z_{\lambda},\overline{u}_{D},\ell_{\lambda})$
is feasible for (Pλ) and fulfills

(47) $J_{\lambda}(\overline{u},\overline{u}_{D},\ell_{\lambda})\rightarrow
J(\overline{u},\overline{u}_{D}).$

_Step 4. Strong convergence and global minimizer._ The feasibility and the
convergence of the recovery sequence and the optimality of
$(\overline{u}_{\lambda},\overline{u}_{D,\lambda},\overline{\ell}_{\lambda})$
give

(48) $\displaystyle J(\tilde{u},\tilde{u}_{D})$
$\displaystyle\leq\liminf_{\lambda\searrow
0}J(\overline{u}_{\lambda},\overline{u}_{D,\lambda})$
$\displaystyle\leq\limsup_{\lambda\searrow
0}J(\overline{u}_{\lambda},\overline{u}_{D,\lambda})$
$\displaystyle\leq\limsup_{\lambda\searrow
0}J_{\lambda}(\overline{u}_{\lambda},\overline{u}_{D,\lambda},\overline{\ell}_{\lambda})\leq\limsup_{\lambda\searrow
0}J_{\lambda}(\overline{u},\overline{u}_{D},\ell_{\lambda})=J(\overline{u},\overline{u}_{D}),$

which, together with the feasibility of
$(\tilde{u},\tilde{\sigma},\tilde{u}_{D})$ for (P) shown in step 2, implies
that $(\tilde{u},\tilde{\sigma},\tilde{u}_{D})$ is a global minimizer of Eq.
P.

To show the strong convergence in (44) and (45), we first observe that (48)
yields $J(\overline{u}_{\lambda},\overline{u}_{D,\lambda})\rightarrow
J(\tilde{u},\tilde{u}_{D})$, from which we deduce the convergence of the norms
$\|\overset{\boldsymbol{.}}{\overline{u}}_{\lambda}\|_{L^{2}(\mathbf{L}^{1}(\Omega))}$
and $\|\overline{u}_{D,\lambda}\|_{H^{1}(\mathbf{H}^{2}(\Omega))}$ to
$\|\overset{\boldsymbol{.}}{\tilde{u}}\|_{L^{2}(\mathbf{L}^{1}(\Omega))}$ and
$\|\tilde{u}_{D}\|_{H^{1}(\mathbf{H}^{2}(\Omega))}$, respectively. Since both
norms are Kadec norms and we already have weak convergence in the respective
spaces, this implies (44). Similarly, (48) yields
$\|\overline{\ell}_{\lambda}\|_{H^{1}(\mathbf{H}^{-1}_{D}(\Omega))}\to 0$.
Finally, the strong convergence of the stresses follows from the compact
embedding of $H^{1}(\mathbb{L}^{2}(\Omega))\cap
L^{s}(\mathbb{W}^{1,p}(\Omega))$ in $L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times
n}_{\textup{sym}}))$, already used above.

Some comments concerning our approximation result are in order:

######  Remark 6.7 (Crucial regularity assumption).

The assumption of existence of a global minimizer
$(\overline{u},\overline{\sigma},\overline{u}_{D})$ with the properties listed
in Theorem 6.5 is admittedly very restrictive. Notice in particular that the
regularity assumptions on $\overline{u}$ imply that
$(\overline{u},\overline{\sigma})$ is a _strong solution_ w.r.t.
$\overline{u}_{D}$, whose existence can in general not be guaranteed. The
regularity assumption however seems to be indispensable, as the above proof
demonstrates: In order to pass to the limit in the flow rule inequality to
show the feasibility of an accumulation point in step 2 of the proof, we need
the additional regularity of the stress ensured by the inequality constraint
in (Pλ). The generic regularity of the stress, which is
$H^{1}(\mathbb{L}^{2}(\Omega))$ (see Lemma 4.1), is by far not sufficient for
this passage to the limit. It therefore appears to be unavoidable to enforce
the required regularity by additional inequality constraints in (Pλ). The
elements of the recovery sequence however have to be feasible for (Pλ) and
thus have to fulfill this inequality constraint, too. As the generic
regularity of the stress is $H^{1}(\mathbb{L}^{2}(\Omega))$, it is not
possible to guarantee this constraint to be fulfilled without further
hypotheses on the recovery sequence and its limit, respectively. At the end,
this leads to the regularity assumption on
$\nabla^{s}\overset{\boldsymbol{.}}{\overline{u}}$.

We however emphasize that we do not require the existence of a strong solution
with the addition regularity of the strain rate for every Dirichlet
displacement $u_{D}\in H^{1}(\mathbf{H}^{2}(\Omega))$ (which would really be
unrealistic), but only for one optimal $\overline{u}_{D}$. (Of course, there
might be many optimal solutions, since (P) is a non-convex problem). Whether
an optimal solution fulfilling these regularity assumptions exists or not,
clearly depends on the data, especially on the smoothness of the desired
strain rate $\mu$ in (38).

######  Remark 6.8 (Extensions and modifications of the approximation result).

  1. (i)

One essential drawback of the approximation result is that the bound $R$ given
in (43) depends on the unknown solution $(\overline{u},\overline{u}_{D})$ and
is therefore in general unknown, too. One could replace the inequality
constraints on the stress involving this bound in (Pλ) by an additional
tracking term in the objective of the form
$\|\sigma-\sigma_{d}\|_{H^{1}(\mathbb{L}^{2}(\Omega))}^{2}+\|\sigma-\sigma_{d}\|_{X}$
with a given desired stress distribution $\sigma_{d}$ and a reflexive Banach
space $X$ with the following properties: On the one hand,
$H^{1}(\mathbb{L}^{2}(\Omega))\cap X$ should compactly embed in
$L^{2}(C(\bar{\Omega};\mathbb{R}^{n\times n}_{\textup{sym}}))$. On the other
hand, $H^{1}(\mathbb{L}^{2}(\Omega))\cap L^{\infty}(\mathbb{W}^{1,p}(\Omega))$
should compactly be embedded in $X$. Provided these embeddings hold, the steps
2 and 3 of the previous proof can easily be adapted. At this point, one
benefits from the strong convergence of the recovery sequence in
$H^{1}(\mathbb{L}^{2}(\Omega))$ by Lemma 5.7.

  2. (ii)

The above analysis is restricted to objectives of the type (38) or other types
of objectives ensuring the boundedness of $\\{\overline{u}_{\lambda}\\}$ in
$\mathcal{U}$. This bound cannot be deduced from the regularized plasticity
system in (24) unless the loads fulfill a _safe load condition_ , see [30].
One could thus allow for more general objectives, if a safe load condition
would be included in the set of constraints in (Pλ). We were however not able
to find a safe load condition that is satisfied by the loads associated with
the recovery sequence. This is due to several reasons, among these a lack of
regularity of the recovery sequence. This issue is subject to future research.

  3. (iii)

By contrast, it is well possible to consider objectives, which give the
boundedness of the displacement in more regular spaces such as
$H^{1}(\mathbf{H}^{1}(\Omega))$. In this case, the inequality constraints on
the stress in (Pλ) can be weakened or even be completely left out, since the
higher regularity of the displacement enables the passage to the limit at the
end of the proof of Lemma 5.7. Such a setting is treated in [35].

  4. (iv)

We have chosen the space $H^{1}(\mathbf{H}^{2}(\Omega))$ as the control space
for the Dirichlet displacement in order to guarantee the compact embeddings in
step 2 of the above proof and in the proof of Theorem 4.10. Of course, one
might want to avoid the $\mathbf{H}^{2}(\Omega)$-norm in the objective, which
could be achieved by an additional (pseudo-)force-to-Dirichlet-map, for
example by solving an additional linear elasticity system. This strategy was
employed in [21, Subsection 6.1].

######  Remark 6.9 (Numerical treatment of (Pλ)).

Although they are still nonsmooth optimization problems, the regularized
problems in (Pλ) offer ample possibilities for a numerical treatment. A
popular strategy is to further regularize the problem by smoothing the Yosida
approximation $\partial I_{\lambda}$. This has been used for the numerical
computations in the companion paper [21]. Moreover, the non-smooth objective
in (Pλ) calls for an additional regularization of the $L^{1}$-norms for
instance in terms of a Huber-regularization. In this way, one ends up with a
smooth optimal control problem, which can be treated by the classical adjoint
approach. Our convergence result in Theorem 6.5 implies that, under the
certainly restrictive assumptions of this theorem, there is an optimal
solution of the original optimization problem governed by the perfect
plasticity system that can be approximated by this procedure.

## Acknowledgments

We are very grateful to Hannes Meinlschmidt (RICAM Linz) for various fruitful
discussions on regularity issues associated with the construction of the
recovery sequence.

##  Appendix A Auxiliary results

######  Lemma A.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain and $X$ be a Banach
space and $A\subset X$ be a convex and closed set with $0\in A^{\circ}$. Set
$\mathcal{A}(\Omega):=\\{v\in L^{2}(\Omega;X):v\in A\text{ a.e.~{}in
}\Omega\\}$. Then $C^{\infty}_{c}(\Omega;X)\cap\mathcal{A}(\Omega)$ is dense
in $\mathcal{A}(\Omega)$.

######  Proof A.2.

Let $v\in\mathcal{A}(\Omega)$ and $\varepsilon\in(0,1)$ be arbitrary. By
assumption there exists $\delta>0$ such that
$\overline{B_{X}(0,\delta)}\subset A$. We set $\overline{v}:=(1-\varepsilon)v$
and select a sequence $\\{v_{n}\\}_{n\in\mathbb{N}}\subset
C^{\infty}_{c}(\Omega;X)$ such that

(49)
$\|v_{n}-\overline{v}\|_{L^{2}(\Omega;X)}^{2}\leq\frac{\delta^{2}\varepsilon^{3}}{4n}\quad\forall\,n\in\mathbb{N}.$

We moreover define

$\displaystyle S_{n}^{c}:=\\{x\in\Omega:v_{n}(x)\in X\setminus
A^{\circ}\\},\quad S_{n}^{o}:=\\{x\in\Omega:v_{n}(x)\in
X\setminus(1-\tfrac{\varepsilon}{2})A\\}.$

Hence, $S_{n}^{c}\subset S_{n}^{o}$ and, by continuity and compact support of
$v_{n}$, $S_{n}^{c}$ is compact, while $S_{n}^{o}$ is open. Thus, for every
$n\in\mathbb{N}$, there is a function $\varphi_{n}\in
C^{\infty}(\mathbb{R}^{n};[0,1])$ with $\varphi_{n}\equiv 1$ in
$\mathbb{R}^{n}\setminus S_{n}^{o}$ and $\varphi_{n}\equiv 0$ in $S_{n}^{c}$.
Furthermore, if
$\|v_{n}(x)-\overline{v}(x)\|_{X}\leq\frac{\varepsilon}{2}\,\delta$, then the
convexity of $A$ and $\overline{B_{X}(0,\delta)}\subset A$ imply

$\displaystyle\frac{v_{n}(x)}{1-\frac{\varepsilon}{2}}=\frac{1-\varepsilon}{1-\frac{\varepsilon}{2}}\,v(x)+\Big{(}1-\frac{1-\varepsilon}{1-\frac{\varepsilon}{2}}\Big{)}\,\frac{2}{\varepsilon}\,\big{(}v_{n}(x)-\overline{v}(x)\big{)}\in
A$

Therefore, we obtain by contraposition that

$\displaystyle\|v_{n}-\overline{v}\|_{L^{2}(\Omega;X)}^{2}\geq\int_{S_{n}^{o}}\|v_{n}-\overline{v}\|_{X}^{2}\textup{d}x\geq\frac{\varepsilon^{2}}{4}\,\delta^{2}\,|S_{n}^{o}|$

so that (49) yields $|S_{n}^{c}|\leq|S_{n}^{o}|\leq\varepsilon/n$. Thus, due
to Lebesgue’s dominated convergence theorem, there exists
$N=N(\varepsilon)\in\mathbb{N}$ such that

(50)
$\|\overline{v}\|_{L^{2}(S_{N}^{o};X)}\leq\|v\|_{L^{2}(S_{N}^{o};X)}\leq\varepsilon.$

Now we define $v_{s}:=\varphi_{N}\,v_{N}$. Then, by construction
$v_{s}\in\mathcal{A}(\Omega)\cap C^{\infty}_{c}(\Omega;X)$ and, in addition,
(49) and (50) imply

$\displaystyle\|v-v_{s}\|_{L^{2}(\Omega;X)}$
$\displaystyle\leq\|v-\overline{v}\|_{L^{2}(\Omega;X)}+\|\overline{v}-v_{N}\|_{L^{2}(\Omega;X)}+\|v_{N}-v_{s}\|_{L^{2}(\Omega;X)}$
$\displaystyle\leq\varepsilon\|v\|_{L^{2}(\Omega;X)}+\|\overline{v}-v_{N}\|_{L^{2}(\Omega;X)}+\|v_{N}\|_{L^{2}(S_{N}^{o};X)}$
$\displaystyle\leq\varepsilon\|v\|_{L^{2}(\Omega;X)}+2\,\|\overline{v}-v_{N}\|_{L^{2}(\Omega;X)}+\|\overline{v}\|_{L^{2}(S_{N}^{o};X)}$
$\displaystyle\leq\varepsilon\Big{(}\|v\|_{L^{2}(\Omega;X)}+\frac{\delta\sqrt{\varepsilon}}{\sqrt{N}}+1\Big{)}.$

Since $\varepsilon$ was arbitrary, this finishes the proof.

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