Abstract
We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our previous work which provided the first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
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Notes
We thank Jochen Glück for the idea of the proof.
In [69], the notation μ is used instead of σ.
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Acknowledgements
The authors extend their gratitude to the two referees for their careful reading and excellent comments which helped to greatly improve some of the results and presentation. AA and MH were partially supported by the DFG through the DFG SPP 1962 Priority Programme Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization within project 10. MH and CNR acknowledge the support of Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689) within project AA4-3. In addition, MH acknowledges the support of SFB-TRR154 within subproject B02, and CNR was supported by NSF grant DMS-2012391.
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Appendices
Appendix A: Technical Proofs
Proof
Footnote 1of Lemma 2.3 Take an arbitrary subsequence \(\{v_{n_{j}}\}\); this remains uniformly bounded hence we can extract a weakly convergent subsequence such that \(v_{n_{j_{k}}} \rightharpoonup v\) in V to some v.
Select an arbitrary \(f \in V^{*}_{+}\) and set ln := 〈f,vn〉 which is a monotonic sequence (since f is non-negative) and also bounded. Hence the monotone convergence theorem applies and we obtain the existence of l such that ln → l. Since also \(l_{n_{j_{k}}} \to l\), we conclude that l = 〈f,v〉.
Take another subsequence of {vn}, say \(\{v_{n_{m}}\}\), then by the above argument, we have \(v_{n_{m_{j}}} \rightharpoonup \hat {v}\) for some \(\hat {v}\) and \(l=\langle f, \hat {v} \rangle \). That is,
and from this, we can conclude via the weak-* density of \(V^{*}_{+} - V^{*}_{+}\) in V∗ (e.g., see [8, Lemma 2.7]) that \(\hat {v} = v\). The subsequence principle then yields the result. □
Proof Proof of Lemma 5.9
Define \(T_{n} = (\mathrm {I}-{\Phi }^{\prime }(z_{n}))\) and \(T=(\mathrm {I}-{\Phi }^{\prime }(z))\). Then
and we get \(T^{-1}(q_{n}-q) \rightharpoonup 0\) in V by continuity and linearity of T− 1. For the first term on the right-hand side above, we use the identity \(T_{n}^{-1}-T^{-1} = T_{n}^{-1}(T-T_{n})T^{-1}\) relating the inverses of operators to see that
with the convergence because we assumed that Φ is continuously Fréchet differentiable and hence the derivative is continuous. Therefore, \(T_{n}^{-1}q_{n} \rightharpoonup T^{-1}q\) in V. The strong convergence follows because if qn → q then T− 1(qn − q) → 0 in V.
For the final claim, we have
and the first term on the right-hand side tends to zero by the calculation above. Since by (??), AT− 1 is bounded and coercive (as well as being linear), we obtain
□
Appendix B: Sketch Proof of Theorem 2.16
Recall the notation αh which stands for the directional derivative in the direction h given through Theorem 3.2.
Lemma B.1
Denote by j: H → V∗ the inclusion map. Then 0 ∈ V∗ is a minimiser of the problem
Proof
Choosing the direction h = 0 in the inequality of Proposition 5.2 implies 0 ≤ (α0,y∗− yd) + ν(u∗, 0) = 0 with the equality because α0 = 0. Hence h = 0 is a minimiser of
As in Lemma 4.1 of [69], the feasible set can be enlarged (the continuity in V∗ of h↦αh assured by Proposition 3.12 is needed here) to obtain the desired result. □
The aim now is to rewrite (1) over the space
Using the characterisation of the critical cone from [69, Lemma 3.1], we see that \(\mathcal {K}^{y^{*}} \subset W\). Denote by i: W → V the inclusion map and define the closed convex set
which satisfies \(\mathcal {K}^{y^{*}} = i\mathcal {C}_{W}^{y^{*}}\). Now, note that, using (??), \((\mathrm {I}-{\Phi }^{\prime }(y^{*}))\colon V \to V\) is invertible.
Define
and observe that for any \(\tilde {d} \in W^{*}\) the inequality
has a unique solution by the Lions–Stampacchia theorem since AW is bounded and coercive due to the Lipschitz condition (??) (see [72, Lemma 3.3]). Now suppose that for d ∈ V∗, δ solves
Consider also
Then it is easy to see that that z = iδ.
Lemma B.2
Define the operator
by
Then (0,0) is a solution of
Proof
By defining \(\gamma _{h} := \alpha _{h} - {\Phi }^{\prime }(y^{*})(\alpha _{h}) =(\mathrm {I}-{\Phi }^{\prime }(y^{*}))\alpha _{h},\) the QVI (??) satisfied by αh can be written as
Now if βh satisfies
we have (as discussed above) γh = iβh, hence
Therefore, (1) can be restated and we get (using the continuity of \({\Phi }^{\prime }(y^{*})\)) that 0 is a solution of
this is well defined because u∗∈ W due to (??). Hence, similarly to Proposition 3.13, (0, 0, 0) is a solution of
Setting ξh = 0 leads to the result. □
We need to derive stationarity conditions for this problem and then transform the resulting system back to the original spaces and operators. Let us remark that under the assumptions of the theorem, we have that 𝜃 is linear and bounded.
Lemma B.3
Defining
there exists \((\tilde {p}, \tilde \lambda , \sigma ) \in \mathcal Y^{*} \cap C^{\circ }\) such that
Proof
In addition to the notation introduced above, let us also define the space \(\mathcal X := W \times W^{*}\).
Define the map \(g \colon \mathcal X \to \mathcal Y\) by g(β,h) := (AWβ − h,β,h) and observe that (2) can be compactly written as
We now proceed with checking the Zowe–Kurcyusz constraint qualification \(g^{\prime }((0,0))\mathcal X - \mathcal {R}_{C}(g(0,0)) = \mathcal Y\) to deduce the existence of Lagrange multipliers. First observe that D is a convex cone which in turn implies that C is a convex cone and then by [20, Example 2.62], \(\mathcal {R}_{C}((0,0,0)) = C\) and \(\mathcal {T}_{C}((0,0,0))^{\circ } = C^{\circ }\). Now, we see that g(0, 0) = (0, 0, 0) and \(\mathcal {R}_{C}(g(0,0)) = C\). We also have
Therefore, we are required to show that for every \((w_{1}^{*}, w_{2}, w_{3}^{*}) \in \mathcal {Y}\), there exist \((\gamma , d,v,h) \in \mathcal X \times \mathcal {C}_{W}^{y^{*}} \times D\) such that
The first equation written in terms of v and h reads \(A_{W}v - (w_{1}^{*} + w_{3}^{*}-A_{W}w_{2}) = h.\) In order to force solutions to belong to the desired sets, we consider the VI
associated to the above PDE.
As explained above, (5) has a solution and furthermore, the following complementarity system (which can be derived by the same arguments as before) is satisfied by any solution:
Using this, we see that \(h :=-\eta \in -(\mathcal {C}_{W}^{y^{*}})^{\circ }\). The manipulations in the paragraph after Lemma 5.1 of [69] show that \((i^{*}j\mathcal {T}_{U_{ad}}(y^{*}))^{\circ } \subset -\mathcal {C}_{W}^{y^{*}}\) which implies that \(-(\mathcal {C}_{W}^{y^{*}})^{\circ } \subset (i^{*}j \mathcal {T}_{U_{ad}}(y^{*}))^{\circ \circ } = D\), that is, g ∈ D. Then we simply define γ and d by (4). Thus the constraint qualification is met for (3).
Writing the objective functional in (3) as \(\hat J\), we obtain the existence of a Lagrange multipler \((\tilde {p}, \tilde \lambda , \sigma ) \in \mathcal Y^{*} \cap C^{\circ }\) such that
With x = (γ,d), we see that since 𝜃(0) = 0, the first term above is
where 𝜃∗: V∗→ W∗ is the adjoint of 𝜃: W → V (this exists due to the linearity assumption). We also have, by definition of the adjoint operator,
This implies the result. □
We now transform all quantities back to the space V.
Proof Conclusion of sketch proof of Theorem 5.16
Observe that under the assumptions, Proposition 5.2, Lemma B.3 and Theorem 3.2 are applicable. To start with, let us define
and
and for convenience, denote \(L := {\Phi }^{\prime }(y^{*}).\)
-
By definition of λ∗ and p∗, we get the first line in the system after etching away the inclusion map j.
-
We see from the definition of λ∗ and elementary manipulations to relate it to \(\tilde {\lambda } \in (C_{W})^{\circ }\) and the usage of the fact that \(iC_{W} = \mathcal {K}^{y^{*}}\) that \(\lambda ^{*} \in (\mathcal {K}^{y^{*}})^{\circ }\). This implies the final condition of the system thanks to [69, Lemma 3.1].
-
Since \(\tilde {p} \in W\), it vanishes q.e. on the strongly active set. As \(\tilde {p} = \nu u^{*} + \sigma \) and since σ ∈ D∘, Lemma 5.1 of [69] tells us that σ ≥ 0 q.e. on Ω ∖ Ua. Thus
$$ \sigma|_{\mathcal{B}(y^{*})} = \sigma|_{U_{a} \cap \mathcal{B}(y^{*})} + \sigma|_{({\Omega} \setminus U_{a}) \cap \mathcal{B}(y^{*})} \geq \sigma|_{U_{a} \cap \mathcal{B}(y^{*})} = 0 $$with the final equality because of (??). Note also that
$$ \begin{array}{@{}rcl@{}} u^{*}|_{\mathcal B(y^{*})} = u^{*}|_{\mathcal B(y^{*}) \cap U_{b}} + u^{*}|_{\mathcal B(y^{*}) \cap ({\Omega} \setminus U_{b})} \geq u^{*}|_{\mathcal B(y^{*}) \cap ({\Omega} \setminus U_{b})} \geq 0 \text{ q.e.,} \end{array} $$with the first inequality by (??) and the final inequality by the third sign condition on u∗ stated in Section ??. This implies the stated condition on p∗, which is equivalent to \(-p^{*} \in \mathcal {K}^{y^{*}}\) due to the characterisation of the critical cone in [69, Lemma 3.1].
-
We obtain \(\sigma \in \mathcal {N}_{U_{ad}}(u^{*})\) exactly as in the proof of Theorem 5.2 in [69]Footnote 2 (where \(\mathcal {N}_{U_{ad}}\) denotes the normal cone to Uad with respect to H), which is the polar cone of the tangent cone, see [20, §2.2.4]) and this is precisely the desired inequality constraint relating the control and the adjoint.
□
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Alphonse, A., Hintermüller, M. & Rautenberg, C.N. Optimal Control and Directional Differentiability for Elliptic Quasi-Variational Inequalities. Set-Valued Var. Anal 30, 873–922 (2022). https://doi.org/10.1007/s11228-021-00624-x
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DOI: https://doi.org/10.1007/s11228-021-00624-x
Keywords
- Quasi-variational inequality
- Obstacle problem
- Directional differentiability
- Sensitivity analysis
- Optimal control
- Stationarity conditions