Optimal control of a rate-independent evolution equation via viscous regularization

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.


Introduction
Let a Lipschitz domain Ω ⊂ R d and T > 0 be given and set I := (0, T ). We study the optimal control of a non-smooth evolution problem given by the non-smooth dissipation D : H 1 0 (Ω) → R, D(ż) := Ω |ż| dx (1) and the quadratic energy which give rise to the differential inclusion 0 ∈ ∂|ż| − ∆z − g in H −1 (Ω), a.e. in I, to be complemented by the initial condition z(0) = 0. Here, z ∈ H 1 (I; H 1 0 (Ω)) has the role of the state variable, whereas g ∈ H 1 (I; L 2 (Ω)) is the control. The optimal control problem under consideration reads: Minimize J(z, g) subject to (3) and g(0) = 0, where J denotes a suitable objective functional, see (6) below. The requirement g(0) = 0 arises as compatibility condition implying the stability of the initial state z(0) = 0. 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 [2015], where a thorough discussion of the current state of the art on rate-independent systems is recorded.
Here, | · | ρ is a smooth approximation of the modulus | · |. The regularized state equation (5) is smooth. Hence, necessary optimality conditions for (P ρ ) can be derived by standard techniques. The main challenge is then to pass to the limit as ρ ց 0 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 [2015] besides the classical monographs Visintin [1994]; Brokate and Sprekels [1996]; Krejčí [1996]. 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 [2006], evolutions of this technology in the abstract setting are in a series of papers by Mielke et al. [2009Mielke et al. [ , 2012; Mielke and Zelik [2014]. See also Krejčí and Liero [2009] 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., 2008Knees et al., , 2010Toader, 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 [2013], and damage problems via vanishing viscosity are studied in Knees et al. [2013Knees et al. [ , 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 E is convex. In particular, we exploit vanishing viscosity in order to regularize the controlto-state mapping and deriving optimality conditions.
Optimal control of finite-dimensional rate-independent processes has been considered in Brokate [1987Brokate [ , 1988; Brokate and Krejčí [2013] and we witness an increasing interest for the optimal control of sweeping processes, see Castaing et al. [2014]; Colombo et al. [2012Colombo et al. [ , 2015Colombo et al. [ , 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 [2008Rindler [ , 2009 and subsequently applied in the context of shape memory materials by Eleuteri and Lussardi [2014]; Eleuteri et al. [2013]; Stefanelli [2012]. 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 [2012Wachsmuth [ , 2015Wachsmuth [ , 2016 in the context of quasi-static plasticity, see also Herzog et al. [2014]. 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 [2005]; Herzog et al. [2012Herzog et al. [ , 2013; Adam et al. [2015].
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.

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 Minimize J(z, g) such that (ż(t, x), g(t, x) + ∆z(t, x)) ∈ M ∀(t, x) ∈ (0, T ) × Ω. Here, The Lagrangian for this optimization problem is given by As (formal) optimality conditions, we would expect Here, N M is a normal-cone mapping associated with the closed set M ⊂ R 2 . Since M 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 relationṡ The above equations (7a)-(7b) for q, ξ could be written as Here, (9c) is equipped with the boundary conditions g(0) =ġ(T ) = 0. Hence, this formal derivation suggests that for each local solution (z, g) of (P), there exist functions q, ξ such that (8) and (9) are satisfied.

Unregularized optimal control problem
In this section, we give some first results concerning the optimal control problem (P). We recall some known results for the state equation and prove the existence of solutions to (P).
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 g ∈ H 1 (I; H −1 (Ω)) with g(0) L ∞ (Ω) ≤ 1, there is a unique energetic solution S(g) := z ∈ H 1 ⋆ (I; H 1 0 (Ω)) and this is the unique solution to (3), see [Mielke and Roubíček, 2015, Section 1.6.4, Theorem 3.5.2].
The requirement g(0) L ∞ (Ω) ≤ 1 is needed as a compatibility condition. Indeed, it ensures that ∆z(0) + g(0) = g(0) is in the range of ∂żD = ∂ · L 1 (Ω) . 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 [1996].
Lemma 3.1. Let z ∈ H 1 ⋆ (I; H 1 0 (Ω)) and g ∈ H 1 (I; H −1 (Ω)) ∩ G 0 be given. Then, the state equation ( and toż Here, N Hilbert Proof. The assertion follows directly from standard results in convex analysis by using the definition of the dissipation (1) and of the energy (2).
The next lemma provides the energy equality (12), which will be crucial to prove the consistency of the regularization in H 1 ⋆ (I; H 1 0 (Ω)), cf. theorem 4.9.
Lemma 3.3. Let g ∈ H 1 (I; H −1 (Ω)) ∩ G 0 be given and set z = S(g). Then, we have Proof. Using (11) and ∆z(s) + g(s) ∈ K for all s ∈ I, we find for almost all t ∈ I and all h > 0 such that t ± h ∈ I. Using Lebesgue's differentiation theorem, see [Diestel and Uhl, 1977, Theorem II.2.9] for the version with Bochner integrals, we can pass to the limit h ց 0. This yields the claim, see also [Krejčí, 1996, (I.3.22 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 S. Recall, that H 1 (I; L 2 (Ω)) is not compactly embedded in H 1 (I; H −1 (Ω)), 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].
Proof. The assumptions imply that g n (0) ⇀ g(0) in H −1 (Ω). Hence, g(0) belongs to G 0 , which makes z = S(g) well-defined. Due to (13), the sequence {z n } n∈N is bounded in Adding these inequalities yields . Owing to (13), we have Due to the compact embedding H 1 (I; L 2 (Ω)) ֒→ L 2 (I; H −1 (Ω)), we can pass to the limit to obtain the convergence z n → z in C(Ī; Now, we are in the position to prove the existence of solutions of (P).
Lemma 3.5. There exists a (global) optimal control of (P).
The proof is standard, but included for the reader's convenience.
Proof. We denote by j the infimal value of the optimal control problem and by {(z n , g n )} n∈N a minimizing sequence. By the boundedness of {g n } n∈N in H 1 (I; L 2 (Ω)) we obtain the weak convergence of a subsequence (without relabeling) in H 1 (I; L 2 (Ω)) towardsḡ.

Regularized optimal control problem
In this section, we study the regularized optimal control problem.

Regularized dissipation
For given parameter ρ > 0, let us define the regularized dissipation by Note that the additional quadratic term in D ρ will add some viscosity to our state equation. In the regularization (14), |·| ρ is a regularized version of the modulus function |·| : R → R satisfying the following assumption: 3. |v| ρ = |v| for all v ∈ R with |v| ≥ ρ, and 4. |v| ′′ ρ ≤ 2 ρ for all v ∈ R.
Let us remark that Assumption 4.1 is satisfied, e.g., by

Regularized state equation
Let us now discuss the regularized state equation. In particular, we will prove the differentiability of the solution map S ρ and show a-priori stability results. We recall the regularized problem (5) a.e. on I, z(0) = 0.
By using the differentiability of |·| ρ , we obtain the equivalent formulation This equation can be written as the systeṁ −ρ ∆w + |w| ′ ρ = ∆z + g in H −1 (Ω), a.e. on I, equipped with the initial condition z(0) = 0. In order to discuss the solvability of (17), we first analyze the semilinear equation Due to the monotonicity of |·| ′ ρ , this equation has a unique weak solution w ∈ H 1 0 (Ω) for all v ∈ H −1 (Ω). Moreover, the solution depends Lipschitz continuously on the right-hand side. Let us denote by T ρ := (−ρ ∆ + |·| ′ ρ ) −1 the associated solution mapping, which is globally Lipschitz continuous from H −1 (Ω) to H 1 0 (Ω) for fixed, positive ρ. Using this mapping, equation (17) can be written aṡ which is an ODE in H 1 0 (Ω). Due to the global Lipschitz continuity of T ρ , we have the following classical result.
Theorem 4.3. Let ρ > 0 be given. For each g ∈ L 2 (I; H −1 (Ω)), there exists a unique solution z ∈ H 1 ⋆ (I; H 1 0 (Ω)) of the regularized state equation (5). The mapping S ρ , which maps g to z, is continuous with respect to these spaces.
In the next step, we will investigate the differentiability of S ρ . Due to the properties of |·| ′ ρ , the operator T ρ is Fréchet differentiable from H −1 (Ω) to H 1 0 (Ω). Let v, h ∈ H −1 (Ω) be given with w = T ρ (v). By standard arguments it can be proven that y = T ′ ρ (v) h is given as the unique weak solution of the equation Moreover due to |w| ′′ ρ ≥ 0, we can bound the norm of T ′ ρ (v) uniformly with respect to v by with the initial condition ζ(0) = 0 is uniquely solvable provided g ∈ L 2 (I; H −1 (Ω)), z = S ρ (g), and h ∈ L 2 (I; H −1 (Ω)), see again Gajewski et al. [1974]. Summarizing these arguments leads to the following differentiability result.
Now, we show a regularized counterpart to the Lipschitz continuity of S, cf. lemma 3.2.
As last result in this section, we provide some a-priori estimates and, in particular, provide the boundedness of z = S ρ (g) in H 1 (I; H 1 0 (Ω)) independent of ρ.
Testing withż and integrating, we find Let us introduce the function f ρ (r) = r 0 |s| ′′ ρ s ds.

This construction implies
where we used in addition the estimate (22) ofż(0). Due to the assumptions on |·| ρ , the auxiliary function f ρ is bounded, and it holds 0 ≤ f ρ (s) ≤ ρ. Hence, we obtain Using Young's inequality, we finally obtain This shows the claim.
We emphasize that the compatibility condition g ∈ G 0 , i.e., g L ∞ (Ω) ≤ 1, is crucial for the validity of lemma 4.6.
Adding both inequalities and integrating on (0, t) for t ∈Ī yields With lemma 4.2, we can estimate the first integral. Applying Young's inequality we obtain which proves the convergence claim due to g n → g in L 2 (I; H −1 (Ω)), see the end of the proof of lemma 3.4, as well as estimate (24).
Note thatḡ is a feasible control for this problem. With similar arguments as in the proof of lemma 3.5 we can show the existence of global solutions of (P ρ ).
Due to special construction of (P ρ ), we can prove convergence of global minimizers to the local solution (z,ḡ).
Remark 4.12. In view of the assumptions of theorem 4.9, we could relax the constraint in (P) and (P ρ ) on g(0) to g(0) L ∞ (Ω) ≤ 1.

Regularized optimality system
Let us now turn to the first-order optimality system of (P ρ ). At first, we study the regularity of solutions of the adjoint system to (17). For given ρ > 0 and z ρ ∈ H 1 (I; H 1 0 (Ω)) it reads With the help of the adjoint variables we will express derivatives of the reduced objective functional. Let us first prove existence and uniqueness of solutions.
As a consequence, we can derive first-order optimality conditions for (P ρ ).
Let us now derive bounds on ξ ρ and q ρ that are explicit with respect to ρ.
It remains to get an estimate for ξ ρ .

Passing to the limit
In this final section we investigate the limit ρ ց 0 and prove our main result, namely theorem 5.2.
In particular, lemma 5.1 shows that (8a) and (8e) hold in a distributional sense. We are now in the position to formulate and prove the main result of this article.

Conclusions and outlook
In this work, we have derived optimality conditions for the optimal control of a rateindependent process. The full set of conditions has been formally derived and we have succeeded in presenting rigorous arguments for the validity of a specific subset of those. The verification of the remaining optimality conditions as well as their possible validity in a stronger regularity setting will be the object of further research. A time-discretization or a decoupling of the smoothing of |·| and the additional viscosity might offer the chance of deriving the complementarity (8c) as well. Note however that this task is challenging by the low regularity of the adjoint variables.