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

Wachsmuth, Gerd

doi:10.1007/s00526-020-01743-3

Cited by 15 publications

(29 citation statements)

References 13 publications

(11 reference statements)

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“…For the sake of completeness, we give strong stationarity conditions for (2). After providing some background and context, we reduce this section to the essence of the statement of the result since a similar result has recently been obtained in [61] whilst this work was under preparation. Strong stationarity for the VI obstacle problem in the absence of constraints on the control was the focus of the classical works by Mignot [41,Theorem 5.2] and Mignot-Puel [42].…”

Section: Strong Stationaritymentioning

confidence: 91%

Directional differentiability for elliptic quasi-variational inequalities of obstacle type

2019

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The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments. arXiv:1802.03564v2 [math.OC] 27 Nov 2019 Remark 1.1. A space V under all of the previous assumptions except the second density assumption in (1) is referred to by Mignot in [49] as a 'Dirichlet space' -this is rather inconsistent with the modern literature [24] where Dirichlet spaces and Dirichlet forms are defined differently (see [24, §1.1]), for example in place of the T-monotonicity property the following Markov property should hold: if u ∈ V thenū := min(u + , 1) ∈ V and a(ū,ū) ≤ a(u, u). However, note that if the Markov property holds then so does T-monotonicity [48, Remark 1.4] [1, Proposition 5] and hence a Dirichlet form is also a positivity preserving form. Remark 1.2. It should be possible to generalise the above setting of positivity preserving spaces (V, a) to reflexive Banach spaces using for example the theory in [36] which generalises [49].Set H := L 2 (X; ξ) and let Φ : H → V be a possibly nonlinear map with Φ(0) ≥ 0 a.e., and suppose that it is increasing in the sense that u ≥ v a.e. (for u, v ∈ H) implies Φ(u) ≥ Φ(v) a.e. We define the set-valued mapping K :For fixed ϕ ∈ H, K(ϕ) is a closed, convex and non-empty set. Given f ∈ V * , consider the QVIIn general, there are multiple solutions of (P QVI ) (this and the existence theory for the QVI will be discussed later on) so we denote by Q : V * ⇒ V the set-valued mapping that takes a source term into the set of solutions of (P QVI ) with that right hand side source; hence (P QVI ) reads u ∈ Q(f ). When K is a constant mapping K(ϕ) ≡ K the problem (P QVI ) reduces to a standard variational inequality. In this work, we are interested in the differential sensitivity analysis of the map f → Q(f ); more precisely, we wish to show that a particular realisation of the multi-valued map Q is directionally differentiable. Such a result is of independent interest in itself but it is also a necessary step for deriving first order stationarity conditions for optimal control problems where the state is related to the control through a Remark 1.3. In fact, (A1) can be weakened significantly by requiring Hadamard differentiability of Φ only around the point u, i.e., locally, as in assumptions (A4) and (A5).Remark 1.4 (Compactness vs. complete continuity). Recall that a compact operator maps b...

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Section: Strong Stationaritymentioning

confidence: 91%

Directional differentiability for elliptic quasi-variational inequalities of obstacle type

2019

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“…Note that the differentiability results in Theorem 4.3 and Corollary 4.4 indeed do not require any conditions on the signs of the directions in the derivatives, on the smallness of Φ (or its differentiability), or on the existence of an underlying Dirichlet space structure, cf. [2,5,37]. As we will see in section 6, because of this, our theorems are in particular able to cover the elliptic and the parabolic setting simultaneously and to even yield Hadamard directional differentiability results in situations in which the solution set S(u) of (F) is a continuum and a characterization of derivatives via linearized auxiliary problems is provably impossible.…”

Section: • (Hadamard Directional Differentiability Of the Maximal Sol...mentioning

confidence: 79%

“…Due to the various processes in physics and economics that can be described by QVIs, there has been an increasing interest in the optimal control of this class of variational inequalities, cf. [1,6,19,37] and the references therein. Studying optimization problems with QVI-constraints, however, turns out to be a challenging task.…”

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confidence: 99%

“…Despite these difficulties, there have been several contributions in the recent years that have tried to establish stability and directional differentiability results for obstacle-type quasi-variational inequalities and, by doing so, to lay the foundation for the study of optimal control problems governed by QVIs. We mention exemplarily [2,4,5], which establish the directional differentiability of the solution maps of elliptic and parabolic obstacle-type quasi-variational inequalities in signed (i.e., nonnegative or nonpositive) directions by means of an approximation argument and classical results on ordinary variational inequalities; [6], which proves the continuity of the minimal and maximal solution mappings of elliptic obstacle-type QVIs in the L 2 -spaces; and [37], which establishes the directional differentiability of the solution operators of elliptic QVIs in all directions under a smallness assumption on the obstacle mapping and by means of the results of [17]. Unfortunately, all of the above papers have in common that they require very restrictive and, at times, even unrealistic assumptions on the involved operators and quantities.…”

mentioning

confidence: 99%

“…Unfortunately, all of the above papers have in common that they require very restrictive and, at times, even unrealistic assumptions on the involved operators and quantities. See, e.g., the conditions on the sign of the directions appearing in the derivatives in [2, Theorem 1], [4, Assumption 28], and [5,Theorems 3.6,4.4]; the assumptions on the image, the complete continuity, and the size of Φ and its derivatives in [2, Assumptions (A2), (A3), (A5)], [4, Assumptions 28,32,34], [5,Theorems 3.6,4.4], [6, Assumption 1], and [37,Assumption 3.1]; and the comments in [1, Remark 2], which emphasize that compactness assumptions on the obstacle map are a main bottleneck in the study of obstacle-type QVIs. We remark that all of these conditions in particular prevent the differentiability results of [2,5,37] from being applicable to the problem (1.1).…”

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confidence: 99%

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Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

Christof¹,

Wachsmuth²

2021

Preprint

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This paper is concerned with the sensitivity analysis of a class of parameterized fixedpoint problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that, if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal element of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that, if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. In contrast to prior results, our analysis requires neither a Dirichlet space structure, nor restrictive assumptions on the mapping behavior and regularity of the involved operators, nor sign conditions on the directions that are considered in the directional derivatives. Our approach further covers the elliptic and parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a partial differential equation.

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Vanishing‐viscosity solutions to a rate‐independent two‐field gradient damage model

Betz

2021

Z Angew Math Mech

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A rate-independent damage model which features two damage variables coupled through a penalty term in the stored energy is considered. Since the energy functional is nonconvex, solutions may be discontinuous in time. This calls for suitable notions of (weak) solutions which allow for jumps. We resort to a vanishing-viscosity approach based on an 𝐿 2 (Ω)−arclength parametrization, where parametrized solutions arise as a limit of the (reparametrized) graphs of the viscous solutions in the extended state space. This enables us to prove that vanishing-viscosity solutions exist and belong to the class of parametrized solutions. We show that the latter can be characterized in various different ways. These alternative formulations highlight the influence of the viscous effects at the jump points, while at the continuity points, the evolution displays a rateindependent behavior. As it turns out, the behavior of the system at each jump point is described by an ordinary differential equation in Banach space during which the physical time is constant.

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

Cited by 15 publications

References 13 publications

Directional differentiability for elliptic quasi-variational inequalities of obstacle type

Directional differentiability for elliptic quasi-variational inequalities of obstacle type

Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

Vanishing‐viscosity solutions to a rate‐independent two‐field gradient damage model

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