Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.

(1.1) -We prove existence and uniqueness of solutions to (1.1) under a smallness assumption on the mapping Φ, see Sect. 3. -If, additionally, the functions A and Φ are differentiable and if K is polyhedric, we establish the directional differentiability of the solution mapping of (1.1), see Sect. 5. -For the associated optimal control problem, we derive necessary optimality conditions of strongly stationary type, see Sect. 6. In particular, our results are applicable if the Lipschitz constant of Φ is small. Let us put our work in perspective. In the following discussion, we will assume that A is μ A -strongly monotone and L A -Lipschitz and that Φ is L Φ -Lipschitz. We refer to Sect. 2 for the definitions. We further define the condition number of A via γ A := L A /μ A ≥ 1. An existence and uniqueness result for the general QVI (1.1) was given in [15,Theorem 9]. This result can be applied to the moving set case (1.2) via [14,Lemma 3.2]. One obtains the unique solvability of (1.1) under the condition In the work [14, Corollary 2] the requirement was relaxed to In this work, we shall show that is sufficient for existence and uniqueness under the condition that A is the derivative of a convex function. Note that A is indeed a derivative of a convex function in many important applications. Moreover, the conditions (1.4) and (1.5) are necessary for uniqueness in the following sense: Whenever the constants L Φ < 1, 0 < μ A ≤ L A violate (1.4) with γ A := L A /μ A , there exist bounded and linear operators A : V → V and Φ : V → V possessing these constants such that (1.1) does not have a unique solution for every f ∈ V . If even (1.5) is violated, A can chosen to be symmetric. We refer to Theorems 3.6 and 3.7 below for the precise formulation of this result. For a different approach to obtain uniqueness of solutions to (1.1), we refer to [10]. To our knowledge, [1] is the only contribution concerning differentiability of the solution mapping of (1.1). Their approach is based on monotonicity considerations and only the differentiability into non-negative directions is obtained. In what follows, we are able to relax the assumption required for the differentiability and we also obtain differentiability in all directions, see Theorem 5.5.
Finally, we are not aware of any contribution in which stationarity conditions for the optimal control of (1.1) are obtained.

Notation and preliminaries
Throughout this work, V will denote a Hilbert space. Its dual space is denoted by V . The radial cone, the tangent cone and the normal cone of a closed, convex set K ⊂ V at y ∈ K are given by respectively. The critical cone of K w.r.t. (y, λ) ∈ K × T K (y) • is given by The set K is called polyhedric at (y, λ) if K K (y, λ) = cl{R K (y) ∩ λ ⊥ }. We refer to [17] for a recent review of polyhedricity.
It is called μ-strongly monotone if μ > 0 satisfies If H is another Hilbert space, a mapping C : The monotonicity of an operator implies some weak lower semicontinuity.

Lemma 2.1 Let A : V → V be a monotone operator. Suppose that y n y in V and
Proof From the monotonicity of A we find A(y n ), y n ≥ A(y n ), y + A(y), y n − A(y), y .
The right-hand side converges towards A(y), y due to the weak convergences y n y and A(y n ) A(y). This implies the claim.
In the case that A is additionally bounded and linear, the above claim can be obtained from the observation that y → A(y), y is convex. This convexity does not hold in the nonlinear setting: consider A : R → R, y → max(− 1, min(1, y)).
In order to obtain unique solvability of (1.1) via contraction-type arguments, one typically requires an inequality like for some L Q ≥ 0, see, e.g., [14,Theorem 4.1]. Note that this inequality is not related to the Lipschitz continuity of the projection since the arguments of the projections in (2.1) coincide. By means of an example, we show that (2.1) does not hold for obstacle-type problems if Q is not of the moving-set type. We consider the setting Ω = (0, 1), It is easy to check that the projection of A −1 f onto the set K (h) is given by

Moving-set QVIs as VIs
In this section, we utilize the moving-set structure of Q(y) to recast the QVI (1.1) as an equivalent variational inequality (VI). This is a classical approach, see also [1,Section 2], [2, Section 5.1]. We start by defining the new solution variable z := y − Φ(y) ∈ K . In order to not lose any information, we require that the function I − Φ : V → V is a bijection. Hence, y = (I − Φ) −1 (z). Now, it is easy to check that (1.1) is equivalent to We shall see that this is also viable in the fully nonlinear case.
In order to analyze (3.1) we will frequently make use of the following assumption. Using the equivalence of (1. In the remainder of this section, we give some conditions implying Assumption 3.1.

Lemma 3.3
We assume that A is μ A -strongly monotone and L A -Lipschitz and that Φ is L Φ -Lipschitz. We further assume that In particular, Assumption 3.1 is satisfied and (1.1) has a unique solution for every f ∈ V .
Proof First we remark that we have L Φ < γ −1 A ≤ 1. Thus, Banach's fixed point theorem implies that I − Φ is a bijection. We claim that (1 − L Φ ) −1 is a Lipschitz constant of (I − Φ) −1 . Indeed, let y 1 , y 2 ∈ V be arbitrary. We define Then, x i − y i = Φ(x i ), i = 1, 2 and this yields This shows the claim concerning a Lipschitz constant of (I − Φ) −1 . Moreover, this directly shows that L B is a Lipschitz constant of B.
For arbitrary y 1 , y 2 ∈ V we again use The estimate yields the assertion concerning the strong monotonicity of B. The final claim follows from Theorem 3.2.
We recall that the condition (3.2) was used in [14, Corollary 2] to obtain existence and uniqueness for solutions of (3.1). The above analysis shows, that this condition even implies Assumption 3.1.
Next, we show that the estimate (3.2) can be significantly relaxed if A is the derivative of a convex function. To this end, we need to recall an important inequality for convex functions. This inequality is well-known in the finite-dimensional case, see, e.g., [13,Theorem 2.1.12] or [6,Lemma 3.10], and the proof carries over to arbitrary Hilbert spaces. We are, however, not aware of a reference in the infinite-dimensional case.

Lemma 3.5 We assume that A is μ A -strongly monotone and L A -Lipschitz and that
We further assume that there exists a Fréchet differentiable convex function g : V → R such that A = g and In particular, Assumption 3.1 is satisfied and (1.1) has a unique solution for every f ∈ V .
Proof By arguing as in the proof of Lemma 3.3, we obtain that I − Φ is invertible and the value of the Lipschitz constant L B follows. Now, let y 1 , y 2 ∈ V be arbitrary and we set Then, we apply Lemma 3.4 to obtain Next, we employ Young's inequality From the proof of Lemma 3.3 we find  Theorem 3.6 Let the constants 0 < μ A < L A be given. We define L Φ := μ A /L A < 1, i.e., (3.2) is violated. Then, there exist linear operators A and Φ on V = R 2 (equipped with the Euclidean inner product), such that A is μ A -strongly monotone and

and the operators
Since A is the combination of a rotation and a scaling by L A , it is easy to check that z A z = μ A z 2 and A z = L A z hold for all z ∈ R 2 . Moreover, the Lipschitz constant of Φ is L Φ . However, The next result shows that (3.3) is sharp.
i.e., ( Proof We define It is clear that the operator A is μ A -strongly monotone and L A -Lipschitz. We further set It can be checked that Φ is L Φ -Lipschitz and x = 1. However, Moreover, if we set K = span{y} it is clear that (3.1) cannot be uniquely solvable for all f ∈ V = R 2 .
Since I −Φ is a bijection, this implies that (1.1) is not uniquely solvable for all f ∈ V = R 2 .
We further mention that it is also possible to obtain Assumption 3.1 in situations in which Φ is "not small", e.g., if Φ = λ I with some λ < 1, Assumption 3.1 follows automatically if A is strongly monotone and Lipschitz since (I − Φ) −1 = (1 − λ) −1 I in this case. Moreover, it is possible to analyze the situation in which A is a small perturbation of the derivative of a convex function by combining the ideas of Lemmas 3.3 and 3.5.
The combination of Theorem 3.2 and 3.3 yields a well-known result: under the assumption (3.2), the QVI (1.1) has a unique solution. Such a result is typically shown via contractiontype arguments, see, e.g., [14] or [2, Section 3.1.1]. Thus, the approach of this section is able to reproduce this classical result. However, the combination of Theorem 3.2 and Lemma 3.5 yields a new result in case that A has a convex potential in which the condition (3.2) on the Lipschitz constant L Φ of Φ is relaxed to (3.3).

Localization of the smallness assumption
We localize the assumptions concerning the Lipschitz constant of Φ. Assumption 4. 1 We assume that A : V → V is (globally) μ A -strongly monotone and L A -Lipschitz. Further, letf ∈ V be given and letȳ be a solution of (1.1). We suppose that there is a closed, convex neighborhood Y ⊂ V ofȳ such that Φ is L Φ -Lipschitz continuous on Y . Finally,

Theorem 4.2 Suppose that Assumption 4.1 is satisfied. There is a neighborhood F ⊂ V of f such that (1.1) has exactly one solution in Y for all f ∈ F. Moreover, this solution depends
Lipschitz-continuously on f .
Note that we do not claim that (1.1) is uniquely solvable for all f ∈ F and (1.1) might have further solutions in V \Y .

From Assumption 4.1, Lemmas 3.3 and 3.5, and Theorem 3.2 it follows that (4.1) has a unique solution y =S( f ) for every f ∈ F and the solution operator
Since Q(y) =Q(y) for all y ∈ Y , it is clear that y ∈ Y is a solution of (1.1) if and only if y ∈ Y solves (4.1). Hence, (1.1) has a unique solution in Y for all f ∈ F.

Differential stability
In this section, we consider the situation of Assumption 4.1. However, we do not need Assumption 4.1 directly, but the assertion of Theorem 4.2 is enough.

Assumption 5.1
We suppose that the following assumptions are satisfied.
(i) We assume the existence of sets F ⊂ V , Y ⊂ V such that for every f ∈ F, (1.1) has a unique solution y in Y and the solution map S : F → Y , f → y is Lipschitz continuous. For fixedf ∈ F, we setȳ := S(f ). The sets F, Y are assumed to be neighborhoods off ,ȳ, respectively.

Due to (1.2), the last assumption is equivalent to the polyhedricity of Q(y) at (ȳ,f − A(ȳ)).
First, we show that Assumption 5.1 follows from Assumption 4.1 and from the differentiability of Φ and A. Since L Φ < 1, Banach's fixed point theorem implies that I −Φ is bijective with a Lipschitz continuous inverse. The invertibility of I − Φ (ȳ) follows from the Neumann series since Φ (ȳ) ≤ L Φ < 1.
If A is Fréchet differentiable atȳ, Assumption 4.1 implies that A (ȳ) is μ A -strongly monotone and L A -Lipschitz. In case that (3.2) is satisfied, we can invoke Lemma 3.3 to obtain Assumption 5.1 (iii). Otherwise, A is the Fréchet derivative of a convex function. Hence, A (ȳ) is symmetric since it is a second Fréchet derivative, see [7, Theorem 5.1.1]. Thus, A (ȳ) is the derivative of the convex function v → A (ȳ) v, v /2. Therefore, we can invoke Lemma 3.5 to obtain Assumption 5.1 (iii).

Lemma 5.3 Let us assume that Assumption
Using the Fréchet differentiability of Φ atȳ implies Finally, using the fact that (I − Φ) −1 is Lipschitz implies This shows the claim.

Lemma 5.4 Let us assume that Assumption 5.1 (i)-(iii) is satisfied. The operator B := A • (I − Φ) −1 is Fréchet differentiable atz and its Fréchet derivative is given by B
Proof Follows from Lemma 5.3 together with a chain rule.

Theorem 5.5 Let us assume that Assumption 5.1 is satisfied. Then, the solution map S is directionally differentiable atf and the directional derivative x := S (f ; h) in direction h ∈ V is given by the unique solution of the QVI
where the set-valued mapping Qȳ : V ⇒ V is given by Note that we have due to (1.2).
Proof Let h ∈ V be given. There exists T > 0 such thatf + t h ∈ F for all t ∈ [0, T ). For t ∈ (0, T ) we define Since S is assumed to be Lipschitz continuous on F, the difference quotients {x t t ∈ (0, T )} are bounded in V . The Lipschitz continuity of Φ implies the boundedness of Since z t solves the VI (3.1), i.e.,  Thus, the application of [9, Theorem 2.13] yields that all accumulation points w of w t for t 0 are solutions of the linearized VI Since B (z) is coercive, this linearized VI has a unique solution. Hence, the last part of [9, Theorem 2.13] implies w t → w as t 0. It remains to prove the convergence of x t towards the solution of (5.2). Using the differentiability of (I − Φ) −1 , we find The change of variables w = (I − Φ (ȳ)) x shows the equivalence of (5.2) and (5.3). Thus, x is the unique solution of (5.2). Some remarks concerning Theorem 5.5 are in order. Remark 5.6 (i) The polyhedricity assumption Assumption 5.1 (iv) can be replaced by the strong twice epi-differentiability of the indicator function δ K in the sense of [9, Definition 2.9]. Under this generalized assumption, the second epi-derivative of δ K appears as a curvature term in the linearized inequalities (5.2) and (5.3). Note that the indicator function of the critical cone K K (z,f − A(ȳ)), which appears implicitly in (5.2) and (5.3), is just the second epi-derivative of δ K in the case of K being polyhedric. (ii) We have derived the differentiability result under the assumption that Φ is Fréchet differentiable atȳ. In the notation of [9], this translates to linearity of the operator A x . However, the inspection of the proof of [9, Theorem 2.13] entails that it is possible to replace the Fréchet differentiability of Φ by the following set of assumptions: (a) Φ is Bouligand differentiable atȳ, i.e., there exists Φ (ȳ; ·) : V → V such that (b) For every sequence w n w in V , we assume ) is invertible and Lemmas 3.3 and 3.5 can be used to obtain the strong monotonicity of A (ȳ) (I − Φ (ȳ; ·)) −1 . Property (5.4a) can be verified by assuming, e.g., weak continuity of Φ (ȳ; ·). Indeed, the sequence z n := (I − Φ (ȳ; ·)) −1 (w n ) is bounded, hence, z n z along a subsequence. Now, weak continuity implies w n = z n − Φ (ȳ; z n ) z − Φ (ȳ; z) and w n w implies z = (I − Φ (ȳ; ·)) −1 (w), i.e. z n (I − Φ (ȳ; ·)) −1 (w) along a subsequence. The uniqueness of the limit point implies the convergence of the entire sequence. Finally, (5.4b) can be obtained via (5.4a) and Lemma 2.1.
In the next remark, we compare our differentiability result with [1, Theorem 1].

Remark 5.7
In [1, Theorem 1] a similar differentiability result is obtained in a more restrictive setting: (i) Therein, the leading operator A has to be linear and T -monotone (w.r.t. a vector space order on V ). Our approach also allows for non-linear operators and we do not need any order structure on V . Similarly, we do not need any monotonicity assumptions on Φ. (ii) They require the complete continuity of Φ (ȳ), which is not needed in Theorem 5.5. (iii) One of their most restrictive assumptions is the assumption (A5). Via [7,Theorem 3.1.2], this assumption is equivalent to Φ being L Φ -Lipschitz with This inequality is much stronger than (3.2). Thus, their assumption (A5) implies that the solutions to the QVI (1.1) are unique.
Moreover, they obtained the differentiability only for non-negative directions whereas our approach is applicable to arbitrary perturbations of the right-hand side.

Optimal control
In this section, we consider the optimal control problem Minimize J (y, u) Here, f ∈ V is fixed, U is a Hilbert space and the bounded, linear operator B : U → V is assumed to have a dense range. Moreover, the objective J : The main goal of this section is the derivation of stationarity conditions for local solutions of (6.1). Since the constraints of (6.1) contain a QVI, this is a delicate issue. Using the (local) solution map S of the QVI, one can consider the reduced problem Under the assumptions of Theorem 5.5, this reduced objective function is directionally differentiable and we obtain the stationarity condition In the literature, such an inequality is called B-stationarity. In some situations, it is possible to introduce dual variables to obtain a so-called system of strong stationarity which is equivalent to B-stationarity. The next theorem shows that this is indeed possible for (6.1).
Proof We use classical arguments dating back to [12,Proposition 4.1], see also [17,Theorem 5.3]. Due to Assumption 5.1 we can invoke Theorem 5.5 to obtain the directional differentiability of the control-to-state map. Combined with the local optimality of (ȳ,ū), this implies Due to the Lipschitz estimate S (Bū Hence, there is p ∈ V ∼ = V (by defining it as in the next line on the dense subspace image(B) ⊂ V and extending it by continuity on the whole space V ) such that This yields (6.2b) and Using the density of image(B) in V we get In what follows, we set K := K K (z,λ) for convenience. We recall that S (Bū + f ; h) is the unique solution of where the set-valued mapping Qȳ : V ⇒ V is given by We choose h ∈ K • in (*). We check that (**) implies S (ū + f ; h) = 0. Indeed, 0 ∈ Qȳ(0) = K and i.e., p ∈ −K, which shows (6.2c). Now, we choose w ∈ (I − Φ (ȳ)) −1 K and set h = A (ȳ) w. It can be checked that (**) implies S (Bū + f ; h) = w. Indeed, w ∈ Qȳ(w) = K + Φ (ȳ) w and due to the definition of h. With this choice, (*) implies J y (·) + A (ȳ) p, w V ,V ≥ 0 ∀w ∈ (I − Φ (ȳ)) −1 K.
Since I − Φ (ȳ) is a bijection, this is equivalent to (6.2d). The uniqueness of p and μ follows from the injectivity of B and the bijectivity of (I − Φ (ȳ)) .
The approach of [8, Section 6.1] can be used to provide strong stationarity systems under less restrictive assumptions on K , i.e., the polyhedricity assumption can be replaced by the twice epi-differentiability of the indicator function δ K .

Lemma 6.2 Let
Proof For an arbitrary h ∈ U we define x := S (Bū + f ; B h). Then From the linearized QVI (5.2) and the strong stationarity system (6.2), we have where we used (6.3). Thus, (6.4) follows.