Optimal control of a rate-independent evolution equation via viscous regularization
Abstract. We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.
Key words. rate-independent system, optimal control, necessary optimality conditions.
AMS Subject Classifications. 49K20, 35K87
Let a Lipschitz domain and be given and set . We study the optimal control of a non-smooth evolution problem given by the non-smooth dissipation
and the quadratic energy
which give rise to the differential inclusion
to be complemented by the initial condition . Here, has the role of the state variable, whereas is the control. The optimal control problem under consideration reads:
where denotes a suitable objective functional, see (6) below. The requirement arises as compatibility condition implying the stability of the initial state .
The aim of this article is to derive necessary optimality conditions. This turns out to be a quite demanding task, even in the basic setting of (3), for the dependence of the state on the control is non-smooth. This reflects the non-smoothness of the dissipation, which on the other hand is the trademark of rate-independent evolution. In this connection, we refer the reader to the recent monograph by Mielke and Roubíček , where a thorough discussion of the current state of the art on rate-independent systems is recorded.
Let us sketch the strategy of our method. Under rather mild assumptions, the optimal control problem (P) admits global solutions. By letting be locally optimal for the original optimal control problem, we find such that for all with and satisfying the constraints in (P). In order to prove necessary optimality conditions to be satisfied by we consider the regularized problem
subject to , , and the regularized problem
Here, is a smooth approximation of the modulus . The regularized state equation (5) is smooth. Hence, necessary optimality conditions for (P) can be derived by standard techniques. The main challenge is then to pass to the limit as in the optimality system.
As already mentioned above, the structure of the state equation (3) is inspired by the theory of rate-independent systems. These arise ubiquitously in applications, ranging from mechanics and electromagnetism to economics and life sciences, see Mielke and Roubíček  besides the classical monographs Visintin ; Brokate and Sprekels ; Krejčí . In particular, the presence of the elliptic operator (3) can be put in relation with the occurrence of exchange energy term in micromagnetics [DeSimone and James, 2002] or with gradient plasticity theories [Mühlhaus and Aifantis, 1991].
Our method is based on regularizing the equation by adding some viscosity. This relates with the classical vanishing-viscosity approach to rate-independent systems. Pioneered by Efendiev and Mielke , evolutions of this technology in the abstract setting are in a series of papers by Mielke et al. [2009, 2012]; Mielke and Zelik . See also Krejčí and Liero  for an existence theory for discontinuous loadings based on Kurzweil integration.
Vanishing viscosity has been applied in a number of mechanical contexts ranging from plasticity with softening [Dal Maso et al., 2008], generalized materials driven by nonconvex energies [Fiaschi, 2009], crack propagation [Cagnetti, 2008; Knees et al., 2008, 2010; Lazzaroni and Toader, 2011, 2013; Negri, 2010; Toader and Zanini, 2009], nonassociative plasticity of Cam-clay [Dal Maso et al., 2011], Armstrong-Frederick [Francfort and Stefanelli, 2013], cap type [Babadjian et al., 2012], and heterogeneous materials [Solombrino, 2014]. An application to adhesive contact is in Roubíček , and damage problems via vanishing viscosity are studied in Knees et al. [2013, 2015]. In all of these settings, the vanishing-viscosity approach has served as a tool to circumvent non-convexity of the energy toward existence of solutions. Our aim here is clearly different for the energy is convex. In particular, we exploit vanishing viscosity in order to regularize the control-to-state mapping and deriving optimality conditions.
Optimal control of finite-dimensional rate-independent processes has been considered in Brokate [1987, 1988]; Brokate and Krejčí  and we witness an increasing interest for the optimal control of sweeping processes, see Castaing et al. ; Colombo et al. [2012, 2015, 2016]. In the infinite-dimensional setting, the available results are scant. The existence of optimal controls, also in combination with approximations, was first studied by Rindler [2008, 2009] and subsequently applied in the context of shape memory materials by Eleuteri and Lussardi ; Eleuteri et al. ; Stefanelli . In these works, no optimality conditions were given.
To our knowledge, optimality conditions in the time-continuous, rate-independent, infinite-dimensional setting were firstly derived in Wachsmuth [2012, 2015, 2016] in the context of quasi-static plasticity, see also Herzog et al. . Let us however mention other works addressing optimality conditions for control problem for rate-independent systems in combination with time-discretizations, namely Kočvara and Outrata ; Herzog et al. [2012, 2013]; Adam et al. .
The plan of the paper is as follows. We firstly derive an optimality system for (P) by means of formal calculations in section 2. The argument is then made rigorous along the paper and brings to the proof of our main result, namely theorem 5.2. The existence of a solution of (P) is at the core of section 3, see lemma 3.5. In section 4, we address the regularization of (P) instead. We study the regularized state equation, and derive an optimality system for the regularized control problem by means of the regularized adjoint equations. Eventually, in section 5 we pass to the limit in the regularized control problem and rigorously obtain optimality conditions for (P) in theorem 5.2.
Recall that is a bounded Lipschitz domain, , and let and . We work with standard function spaces like and . The space is equipped with the norm and scalar product
respectively. Throughout the text, denotes the distributional Laplacian and denotes its inverse.
Moreover, we use the Bochner spaces , , and , where is a Hilbert space. By we denote the dual space of , where , . Since our state equation is equipped with homogeneous initial conditions, we also use
We will consider optimal control problems with an objective functional of the type
where the functions and are assumed to be continuously Fréchet differentiable and bounded from below.
2 Formal derivation of an optimality system
In this section, we formally derive an optimality system. It is clear that the resulting system may not be a necessary optimality condition. However, this derivation sheds some light on the situation and we get an idea what relations can be expected as necessary conditions.
We start by (formally) restating the optimal control problem by
The Lagrangian for this optimization problem is given by
As (formal) optimality conditions, we would expect \cref@addtoresetequationparentequation
Here, is a normal-cone mapping associated with the closed set . Since is not convex, the different normal cones of variational analysis, namely Fréchet, Clarke, Mordukhovich, do not coincide. In particular, by using the Fréchet normal cone, which is the smallest among these, we would expect the relations \cref@addtoresetequationparentequation
3 Unregularized optimal control problem
A concept tailored to rate-independent systems is the notion of energetic solutions, see [Mielke and Roubíček, 2015, Section 1.6]. Since the energy (2) is convex, our situation is much more comfortable and we can use the formulation (3), which is strong in time. Indeed, for every with , there is a unique energetic solution and this is the unique solution to (3), see [Mielke and Roubíček, 2015, Section 1.6.4, Theorem 3.5.2].
The requirement is needed as a compatibility condition. Indeed, it ensures that is in the range of . Hence, we define
Due to the quadratic nature of the energy, it is possible to recast the state equation as an evolution variational inequality in the sense of Krejčí .
Let and be given. Then, the state equation (3) in is equivalent to
is the (Hilbert space) normal cone of the set
is the normal cone of the set
The control-to-state map is continuous from to and Lipschitz continuous from to .
Let be given and set . Then, we have
We note that the a-priori energy estimate
follows immediately from (12).
In order to prove the existence of solutions of the optimal control problem (P), we need to show a weak continuity result for . Recall, that is not compactly embedded in , hence, the following result is not a simple consequence of lemma 3.2. Similarly, it does not directly follow from Helly’s selection theorem, which would only give pointwise weak convergence of the state variable. We note that a similar argument was used in [Wachsmuth, 2012, Theorem 2.3, Section 2.3].
Let be given such that in . Then, in and in .
The assumptions imply that in . Hence, belongs to , which makes well-defined. Due to (13), the sequence is bounded in .
Now, we are in the position to prove the existence of solutions of (P).
There exists a (global) optimal control of (P).
The proof is standard, but included for the reader’s convenience.
4 Regularized optimal control problem
In this section, we study the regularized optimal control problem.
4.1 Regularized dissipation
For given parameter , let us define the regularized dissipation by
Note that the additional quadratic term in will add some viscosity to our state equation. In the regularization (14), is a regularized version of the modulus function satisfying the following assumption:
The family satisfies
is and convex,
for all ,
for all with , and
for all .
Note that this assumption implies
by convexity of .
Let satisfy creftype 4.1. Then it holds
Let us remark that creftype 4.1 is satisfied, e.g., by
4.2 Regularized state equation
Let us now discuss the regularized state equation. In particular, we will prove the differentiability of the solution map and show a-priori stability results.
We recall the regularized problem (5)
By using the differentiability of , we obtain the equivalent formulation
This equation can be written as the system \cref@addtoresetequationparentequation
equipped with the initial condition . In order to discuss the solvability of (17), we first analyze the semilinear equation
Due to the monotonicity of , this equation has a unique weak solution for all . Moreover, the solution depends Lipschitz continuously on the right-hand side. Let us denote by the associated solution mapping, which is globally Lipschitz continuous from to for fixed, positive .
Using this mapping, equation (17) can be written as
which is an ODE in . Due to the global Lipschitz continuity of , we have the following classical result.
Let be given. For each , there exists a unique solution of the regularized state equation (5). The mapping , which maps to , is continuous with respect to these spaces.
The result follows directly from [Gajewski et al., 1974, Satz 1.3, p. 166]. ∎
In the next step, we will investigate the differentiability of . Due to the properties of , the operator is Fréchet differentiable from to . Let be given with . By standard arguments it can be proven that is given as the unique weak solution of the equation
Moreover due to , we can bound the norm of uniformly with respect to by
Hence, the linearized ODE
with the initial condition is uniquely solvable provided , , and , see again Gajewski et al. . Summarizing these arguments leads to the following differentiability result.
Let be given. The regularized control-to-state map is Fréchet differentiable from to . The directional derivative satisfies the system \cref@addtoresetequationparentequation
where is given by .
Now, we show a regularized counterpart to the Lipschitz continuity of , cf. lemma 3.2.
Let and be given. Then it holds
with solely depending on .
By testing the state equations (16) for and by , integrating over , and taking the difference, we get
Using the monotonicity of and , we get for all
Taking the supremum on the left-hand side, we obtain
which shows the assertion. ∎
As last result in this section, we provide some a-priori estimates and, in particular, provide the boundedness of in independent of .
Let and be given, and let . Then it holds . In addition, there is a constant independent of (and ) such that
We start by showing . Since is globally Lipschitz continuous, we have
with a -dependent constant . Since both and are in , one can prove with the help of finite differences that it holds .
Moreover, we obtain by continuity. Testing the associated semilinear elliptic equation by and using yields
By using the second inequality in (15) for as well as the assumption we obtain
Now, let us differentiate (16) w.r.t. to obtain
Testing with and integrating, we find
Let us introduce the function
This construction implies