Approximation of Optimal Control Problems in the Coefficient for the $p$-Laplace Equation. I. Convergence Result

Casas, Eduardo; Kogut, Peter I.; Leugering, Günter

doi:10.1137/15m1028108

Cited by 31 publications

(44 citation statements)

References 7 publications

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“…Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30].…”

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confidence: 99%

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Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

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Here, u ∈ V := H 1 0 (Ω) is the state and q ∈ Q := BV (Ω) is the control, where BV (Ω) denotes the space of functions of bounded variation (BV) on the interval Ω := (0, 1). The operator A is elliptic and α is a positive real number. The two finite element schemes that will be analyzed are identical in regard to the discretization of state and adjoint state, but they differ in the treatment of the control. In the variational discretization the control is not discretized, while in the second scheme the control is discretized by piecewise constant functions.The significance of the above control problem is given by the use of the BVseminorm q ′ M(Ω) in the objective. This favors piecewise constant controls with only a limited number of jumps, which makes this problem type interesting in many practical applications. The precise functional analytic setting will be provided in the next section.Optimal control problems with BV-controls defined in one space dimension are strongly related to control problems with measures as controls. Both BV optimal control problems and optimal control problems with measures have attracted significant research interest in the recent past, see, e.g. [8,13,14,17,22,23] for the former and [11,12,15,16,29,30] for the latter.Error estimates for PDE-constrained optimal control problems involving measures have been presented in [11,30,31,34,35]. For error estimates of further sparsity promoting optimal control problems with PDEs see for example [19,30]. The literature on error estimates for optimal control problems with controls in BV is rather limited. We are only aware of [14,18]. Error estimates and numerical analysis for inverse problems involving BV-functions are studied in [5,6]. Related discussion of ODE-constrained control problems involving discontinuous functions and their numerical analysis can be found in, e.g., [1,2,10,21,25,26,37,38].The main difficulty in deriving error estimates for the above problem is given by the fact that it lacks certain coercivity properties that are usually employed to obtain error estimates for the controls, for instance by suitably testing the first order necessary optimality conditions. Hence, only error estimates for the state and the adjoint state can be proven in a rather direct manner; these are, however, suboptimal. To obtain an error estimate for the control and also to improve the error estimates for state and adjoint state, we make use of a structural assumption on the Lagrange multiplierΦ arising from the convex subdifferential of the term q ′ M(Ω) . Specifically, we assume thatΦ, which is a C 2 function inΩ, has only finitely many global extreme points and that it exhibits quadratic growth near those points (i.e.,Φ ′′ = 0 near those points; see Assumption 4.4 and Assumption 4.5). Since the jump set of the optimal control is contained in the set of global extreme points ofΦ, see Corollary 1, this assumption implies that the optimal control admits only finitely many jumps, which is a rather typical situation in practice. In addition, it ensures ...

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Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Hafemeyer¹,

Mannel²,

Neitzel³

et al. 2019

Mathematical Control &Amp; Related Fields

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“…However, just a few papers are devoted to the control of quasilinear equations of p-Laplace type: [10], [14], [15], [16], [18], [26], [41]. However, control problems where the nonlinearity is in the state, not in the gradient, have not been extensively studied.…”

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Analysis and optimal control of some quasilinear parabolic equations

Casas¹,

Chrysafinos²

2018

Mathematical Control &Amp; Related Fields

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In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

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“…Optimal control problems with nonlinear PDE constraints have been studied for a long time in many works. In particular, employing the (regularized) p-Laplacian (see e.g., [29,22,34,46]) as a nonlinear constraint of an optimal control problem was considered for instance in [17].…”

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Multigoal-oriented optimal control problems with nonlinear PDE constraints

Endtmayer

Langer

Neitzel³

et al. 2020

Computers & Mathematics with Applications

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In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized p-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest.All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples. numerical example of a quasi-linear PDE constraint, and multiple goal-oriented a posteriori error estimation.In the following, we briefly refer to studies that treat parts of the three topics. Optimal control problems (specifically, a priori estimates and optimality conditions) with quasi-linear (as the p-Laplacian can be classified) elliptic PDE constraints were considered in [16,18,15]. More recently, the extension to optimal control with parabolic PDEs was discussed in [9] and [14].Optimal control problems with (single) goal functionals were investigated in [6,40,5,50,52,43]. The p-Laplacian and a posteriori error estimates were considered in [36,12,20,13], and, more specifically, for goal functional evaluations, we refer to [34,44,25]. To estimate goal functionals, we adopt the dual-weighted residual (DWR) method [7,8] in which an adjoint problem is solved to obtain (local) sensitivity measures that are used for mesh refinement. As is well-known, using a gradientbased approach for the numerical solution of optimal control problems, the same adjoint problem as for the DWR error estimator can be employed. For this reason, it is natural to combine gradient-based optimization with adjoint-based error estimation.We are specifically interested in an extended DWR version in which the discretization and (linear/nonlinear) iteration error are balanced [39,44,37]. As localization technique we employ integration by parts as done in [8] or, for residual based error estimates, in [49]. The extension of [44] to multiple goal functionals was recently undertaken in [25].Three major aims constitute the main contents of this paper: first, the design of a framework for goal-oriented error estimation for optimal control subject to a nonlinear PDE and balancing the discretization and nonlinear iteration error (Section 3). From the optimization point of view, we carefully revisit the important elements for the DWR estimator for optimization problems. The main result in this respect is the a posteriori error representation for the reduced optimal control system for an abstract problem formulation. The second aim is the extension to the simultaneous control of multiple goal functionals (Section 4). As a third goal, based on our theoretical developments, we carefully design an adaptive soluti...

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Approximation of Optimal Control Problems in the Coefficient for the $p$-Laplace Equation. I. Convergence Result

Cited by 31 publications

References 7 publications

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Analysis and optimal control of some quasilinear parabolic equations

Multigoal-oriented optimal control problems with nonlinear PDE constraints

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