On optimal control in a nonlinear interface problem described by hemivariational inequalities

The purpose of this paper is three-fold. Firstly we attack a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot analyzed in a reflexive Banach space setting. By boundary integral methods we obtain another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Secondly broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results, also establish a stability result with respect to the extended real-valued function as parameter. Thirdly based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous distributed-boundary control governed by the interface problem, as well as control of the obstacle driven by a related bilateral obstacle interface problem.


Introduction
Optimal control of partial differential equations (PDEs) is a vast field of applied mathematics.Here, we focus on the control of elliptic PDEs, governed by a hemivariational inequality (HVI) in a weak formulation.
The theory of HVIs was introduced and has been studied since the 1980s by Panagiotopoulos [44], as a generalization of variational inequalities with the aim to model many problems coming from mechanics when the energy functionals are nonconvex, but locally Lipschitz, so Clarke's generalized differentiation calculus [12] can be used [18,19,40].For more recent monographs on HVIs with application to contact problems, we refer to [38,55].
While optimal control in variational inequalities has already been treated for a longer time (see the monograph [5] and, e.g., the articles [1,13,15,30,37,47]), optimal control in HVIs has been more recently studied (see, e.g., [28,35,48,[52][53][54]).In particular, let us mention the very recent work on optimal control in HVIs and on  related inverse problems in HVIs presented in the articles [34] on optimal control for elliptic bilateral obstacle problems; [60] on well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, [59] on optimal control in nonlinear quasi-hemivariational inequalities; and [10] on inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems.Contrary to the work cited above, the underlying state problem of this article is not a boundary value problem on a bounded domain, but an interface problem involving a PDE on an unbounded domain.For the simplicity of presentation, we consider a scalar interface problem with a monotone PDE on the interior domain and the Laplacian on the exterior domain, connected by non-monotone set-valued transmission conditions as a novelty.This scalar problem models nonlinear contact problems with non-monotone friction in infinite elastic media that arise in various fields of science and technology; let us mention geophysics (see, e.g., [51]), soil mechanics, in particular soil-structure interaction problems (see, e.g., [16]), and civil engineering of underground structures (see, e.g., [57]).
It should be underlined that such interface problems involving a PDE on an unbounded domain are more difficult than standard boundary value problems on bounded domains, since a direct variational formulation of the former problems leads to a HVI, which lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting.Thanks to boundary integral methods (see the monograph [29]), we provide another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary.Let us note in passing that these integral methods lay the basis for the numerical treatment of such interface problems by the well-known coupling of boundary elements and finite elements (see [27,Chapter 12]).
A main novel ingredient of our analysis is a stability theorem that considerably improves a related result in the recent article [52] and extends it to more general extended real-valued HVIs augmented by convex extended real-valued functions.This stability theorem provides the key to a unified approach to the existence of optimal controls in various optimal control problems (OCPs): distributed control on the bounded domain, boundary control, simultaneous distributed-boundary control governed by the interface problem, as well as the control of the obstacle in a related bilateral obstacle interface problem.
The plan of this study is as follows.Section 2 provides preliminaries and consists of three parts: a collection of some basic tools of Clarke's generalized differential calculus for the analysis of the non-monotone transmission conditions, a description of the interface problem in strong form and in weak HVI formulation, and existence and uniqueness results for a class of abstract HVIs using an equilibrium approach.Section 3 establishes well-posedness results, in particular a stability theorem for a more general class of extended realvalued HVIs.Based on this stability theorem, Section 4 presents a unified approach to the existence of optimal controls in four OCPs: distributed, boundary, boundary-distributed, and obstacle control.Section 5 shortly summarizes our findings, gives some concluding remarks, and sketches some directions of further research.
2 Some preliminaries -Clarke's generalized differential calculus, the interface problem, and an equilibrium approach to HVIs 2.1 Some preliminaries from Clarke's generalized differential calculus From Clarke's generalized differential calculus [12], we need the concept of the generalized directional derivative of a locally Lipschitz function → ϕ X : on a real Banach space X at ∈ x X in the direction ∈ z X defined by: ) is finite, sublinear, hence convex, and continuous; furthermore, the function )is upper semicontinuous.The generalized gradient of the function ϕ at x, denoted by (simply) ∂ϕ x ( ), is the unique nonempty weak * compact convex subset of the dual space X *, whose support function is ϕ x; . 0( ).Thus, When X is finite dimensional, according to Rademacher's theorem, ϕ is differentiable almost everywhere, and the generalized gradient of ϕ at a point ∈ x n can be characterized by: where "co" denotes the convex hull.
In the interior part Ω, consider the nonlinear PDE where ) is a continuous function with ⋅ t p t ( ) being monotonously increasing with t.In the exterior part Ω c , consider the Laplace equation with the radiation condition at infinity for where a is a real constant for any u, but may vary with u.
| , the tractions on the coupling boundary Γ are given by the traces of and on Γ s analogously for the tractions: and the generally non-monotone, set-valued transmission condition: Here, the function ).Then, the set-valued transmission condi- tion (2.6) includes a transmission condition of Tresca's type analogous to Tresca's friction boundary condition (given friction model) (see [14,31]).Indeed, choose is monotone set-valued and with In this sense, (2.6) gives a simplified (scalar) model of an elastic transmission problem with frictional contact.
To arrive at a first variational formulation of the interface problem in form of a HVI, introduce some function spaces.For the bounded Lipschitz domain Ω, use the standard Sobolev space H Ω s ( ) and the Sobolev spaces on the bounded Lipschitz boundary By the trace theorem, Here the data ∈ ( ) ( ) enter the linear functional: Furthermore, in (2.10), the function g is given by p (see is Lipschitz continuous and strongly monotone in H Ω 1 ( ) with respect to the semi-norm: Analogously to [8,36], we first define and then, the affine, hence convex set of admissible functions: Then, it can be proved [26, Theorem 1] that the interface problem (2.1)-(2.6) is equivalent in the sense of distributions to the HVI problem However, since this HVI lives on the unbounded domain × Ω Ω c (as the original problem), this HVI cannot be treated in a reflexive Banach space setting and therefore provides only an intermediate step in the analysis.Therefore, employ boundary integral operator theory [27,29]  ( ) ( ) is a selfadjoint operator with the defining property: for solutions ∈ u 2 0 of the Laplace equation on Ω c .The operator S enjoys the important property that it can be expressed as: where ′ I V K K , , , , and W denote the identity, the single-layer boundary integral operator, the double-layer boundary integral operator, its formal adjoint, and the hypersingular integral operator, respectively (see [27,Section 12.2] for details).
Furthermore, S gives rise to the positive definite bilinear form ⋅ ⋅ S , ⟨ ⟩, i.e., there exists a constant > c 0 where Next, define the linear functional ∈ λ E* by: Using the representation formula of potential theory (see [17,36] for similar nonlinear interface problems), it can be proved [26,Theorem 2] that the intermediate HVI P Φ ( ) is equivalent to the following HVI problem P ( ): where

An equilibrium approach to a class of HVIsexistence and uniqueness results
Next, describe the functional analytic setting for the interface problem and provide the existence and uniqueness results using an equilibrium approach.To this end, let ≔ X L Γ s 2 ( ) and introduce the real-valued locally Lipschitz functional: Then, by Lebesgue's theorem of majorized convergence, where ⋅ ⋅ j s, ; 0 ( ) denotes the generalized directional derivative of ⋅ j s, ( ).As seen in the previous subsection, the weak formulation of problem (2.1)-(2.6)leads, in an abstract setting, to a HVI with a nonlinear operator and the nonsmooth functional J , namely: Here, ≠ ∅ is a closed convex subset of a real reflexive Banach space E, ≔ → γ γ E X is a linear continuous operator, the linear form λ belongs to the dual E*, and the nonlinear monotone operator → E E : * is Lipschitz continuous and strongly monotone with some monotonicity constant > c 0 , which results from the strong monotonicity of the nonlinear operator DG in H Ω 1 ( ) with respect to the semi-norm ⋅ = ∇⋅ and the positive definiteness of the Poincaré-Steklov operator S (see [8, Lemma 4.1]).
It is noteworthy that under the smallness condition (2.22) together with (2.21), fixed point arguments [7] or the theory of set-valued pseudomonotone operators [55] are not needed, but simpler monotonicity arguments are sufficient to conclude unique solvability.Moreover, the compactness of the linear operator γ is not needed either.In fact, (2.20) can be framed as a monotone equilibrium problem in the sense of Blum-Oettli [6]: Proposition 1. Suppose (2.21) and (2.22).Then, the bifunction × → φ : defined by: has the following properties: ( ) is convex and lower semicontinuous for all ∈ v ; • there exists some > μ 0 such that Proof.Obviously, φ vanishes on the diagonal and is convex and lower semicontinuous with respect to the second variable.To show strong monotonicity, estimate To show hemicontinuity, it is enough to consider the bifunction

y t z y z y t z y t J y t z y z y
and thus, hemicontinuity follows from upper semicontinuity of J 0 : Since strong monotonicity implies coercivity and uniqueness, the fundamental existence result [6, Theorem 1] applies to the HVI (2.20) to conclude the following.Theorem 1. Suppose (2.21) and (2.22).Then, the HVI (2.20) is uniquely solvable.
Thus, under the smallness condition, unique solvability holds for P ( ).
3 Extended real-valued HVIsexistence, uniqueness, and stability In view of the subsequent study of OCPs in Section 4 governed by the interface problem which we have described in the previous section, we broaden the scope of analysis and consider the extended real-valued HVIs: where V is a real reflexive Banach space, the nonlinear operator → V V : * is a monotone operator, ≔ → γ γ V X with X a real Hilbert space (in the interface problem we have = X L Γ s 2 ( )) denotes a linear continuous operator, J 0 stands for the generalized directional derivative of a real-valued locally Lipschitz functional J , and now in addition, → ∪ +∞ F V : { } is a convex lower semicontinuous function that is supposed to be proper (i.e., ≢ ∞ F on V ).This means that the effective domain of F in the sense of convex analysis [49], is nonempty, closed, and convex.To resume the HVI (2.20) of Section 2.2, let is the indicator function on in the sense of convex analysis [49].
Next, similar to (2.23) in Section 2.2, define and apply Proposition 1.Thus, under Assumptions (2.21) and (2.22), the aforementioned HVI (3.1) falls into the framework of an extended real-valued equilibrium problem of monotone type in the sense of [23].Clearly, strong monotonicity implies uniqueness.Note by the separation theorem it can be shown that any convex proper lower semicontinuous function → ∪ +∞ ϕ V : { } is conically minorized, i.e., it enjoys the estimate with some > c 0 ϕ .Hence, strong monotonicity implies the asymptotic coercivity condition in [23], too.Thus the existence result [23,Theorem 5.9] applies to the HVI (3.1) to conclude the following.

By this solvability result, we can introduce the solution map by
, the solution of (3.1).Next, we investigate the stability of the solution map with respect to the extended real-valued function F .Here, we follow the concept of epi-convergence in the sense of Mosco [3,39] In view of our later applications, it is not hard to require that the functions F n are uniformly conically minorized, i.e., there holds the estimate with some ≥ d 0 0 . Moreover, similar to [52], in addition to the one-sided Lipschitz continuity (2.21), we assume that the locally Lipschitz function J satisfies the following growth condition: for some > d 0 , which is immediate from the growth condition (2.7) for the integrand j.Now, we are in the position to state the main result of this section, which extends the stability result of [21] for monotone variational inequalities to extended real-valued HVIs with an unperturbed bifunction φ in the coercive situation.
Theorem 3. Suppose that the operator is continuous and strongly monotone with monotonicity constant > c 0 , the linear operator γ is compact, the generalized directional derivative J 0 satisfies the one-sided Lipschitz condition (2.21) and the growth condition (3.4).Moreover, suppose the smallness condition (2.22).
) be convex lower semicontinuous proper functions that satisfy the lower estimate Proof.We divide the proof into three parts.We first show that the = u F ˆn n ( ) are bounded, before we can establish the convergence result.In the following, c c , ,… .
) and use the strong monotonicity of the operator and the estimate (3.3) to obtain On the other hand, write Hence, by the one-sided Lipschitz condition (2.21), Furthermore, by (3.4), By the convergences (3.6),

9) result in
A nonlinear interface problem and its optimal control  9 Hence, by the smallness condition (2.22), a contradiction argument proves the claimed boundedness of u ˆn To prove this claim, we employ a "Minty trick" similar to the proof of [23,Prop. 3.2] using the monotonicity of the operator .
Take ∈ v V arbitrarily.By M 2 ( ), there exist We test (3.5) with v n , use the monotonicity of the operator , and obtain On the other hand, by the previous step, there exists a subsequence { } ∈ u ˆn k k that converges weakly to some . Thus, the continuity of , the upper semicontinuity of ∈ × ↦ y z X X J y z , ; ) inserted above, the positive homogeneity of ⋅ J γu ˜; 0 ( ) and the convexity of F imply after division by the factor − > s 1 0 ( ) , This shows by uniqueness that = u F ˜ ( ) and the entire sequence u ˆn On the other hand, an inspection of the above proof of Theorem 3 shows that it is enough to demand ), the generalized directional derivative of the real-valued locally Lipschitz functional  ( ).Concerning the monotone operator → V V : * , we only require its norm continuity, not needing Lipschitz continuity.More importantly, we can also dispense with the condition [52, (4.1)]: It seems that this condition forces an elliptic operator, which stems from an elliptic PDE on the domain, to be linear.

Some OCPs governed by the interface problem
In this section, we rely heavily on the stability result of Theorem 3 and present a unified approach to existence results for various OCPs governed by the interface problem, which was described in Section 2. For convenience, let us recall the boundary/domain HVI formulation P ( ) of the interface problem: find Here } on the bounded domain Ω and the boundary part Γ s .The operator is given for all extends the L 2 duality on Γ.
A nonlinear interface problem and its optimal control  11 Now, for simplicity, we set ≔ u 0 0 and impose for the data f and q that ∈ f L Ω 2 ( ) and ∈ q L Γ 2 ( ).Thus, we can write

(
) as the simplest case of a cost functional and regularize this functional by the norm of the control with a given regularization parameter > ρ 0. The subsequent analysis can extended to cover more general cost functionals under appropriate lower semicontinuity and coerciveness assumptions (see, e.g., [52, sec. 5]), see also [56, sec. 4.4]), and also to a more general setting of regularization (see, e.g., [33,II Sect. 7.5 (7.51)], [24, (26)]).

Distributed OCP governed by the interface problem
Here, we control by ∈ f L Ω 2 ( ) distributed on the domain Ω.Thus, in the abstract setting of Section 5, we choose the convex functional F as the linear functional: By the abstract existence and uniqueness result of Theorem 2, we have the control-to-state map ∈ ), the solution of (4.1).Thus, we can pose 1 ( ) : for which we can prove the following existence result.
Theorem 4. Suppose that the generalized directional derivative J 0 satisfies the one-sided Lipschitz condition (2.21) and the growth condition (3.4).Moreover, suppose the smallness condition (2.22) with the monotonicity constant c of the operator .Then, there exists an optimal control to 1 ( ) .
Proof.The proof follows the standard pattern of existence proofs in optimal control.Since the cost function I 1 is bounded below: ( ) be a minimizing sequence of Now, we control by ∈ q L Γ 2 ( ) on the boundary Γ.Thus, in the abstract setting of Section 5, we now choose the convex functional F as the linear functional: F u v q τu ιv τ q ι q u v F τ q ι q E , , * , * , , * , * *.
By the abstract existence and uniqueness result of Theorem 2, we have the control-to-state map ∈ ↦ q L Γ 2 ( ) ), the solution of (4.1).Thus, we can pose 2 ( ) : for which we can prove the following existence result.
Theorem 5. Suppose that the generalized directional derivative J 0 satisfies the one-sided Lipschitz condition (2.21) and the growth condition (3.4).Moreover, suppose the smallness condition (2.22) with the monotonicity constant c of the operator .Then, there exists an optimal control to 2 ( ) .
Proof.The proof follows from arguments similar to those that were given in the proof of Theorem 4. So the are omitted.□ Let us remark that we can also treat the simultaneous distributed-boundary 3 ( ) as in [9], here driven by the interface problem: where now we have the control-to-state map , for the unbounded domain = Ω \Ω c d , introduce the Frechet space (see, e.g., [29, Section 4.1, (4.1.43)]):

, M 1 (
), and (3.10) entail together with (3.11): to reformulate the interface problem (2.1)-(2.6) in the weak sense as a boundary-domain variational inequality on × Γ Ω.From now on, concentrate on the analysis to the case of dimension = differential operator in the bounded domain Ω, the Poincaré-Steklov operator on the bounded boundary Γ, and a nonsmooth functional on the boundary part Γ s .To this end, recall the Poincaré-Steklov operator for the exterior problem: → [8], since as already the distinction in the radiation condition(2.3)indicates, in the case = d 2, some peculiarities of boundary integral methods for exterior problems come up that need extra attention (see, e.g.,[8], [27, Sec.12.2]).As a result, arrive at an equivalent hemivariational formulation of the original interface problem (2.1)-(2.6)that lives on × Ω Γ and consists of a weak formulation of the nonlinear Boundary OCP and a simultaneous distributed-boundary OCP governed by the interface problem