Optimal control of fractional semilinear PDEs

Antil, Harbir; Warma, Mahamadi

doi:10.1051/cocv/2019003

Cited by 33 publications

(30 citation statements)

References 43 publications

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“…The L ∞ (Ω) characterization of the solution of the above problem is expected in several settings. We state the following result that can be found in [5] (see also [3]).…”

Section: Boundedness Of the Solution To The Linear Problemmentioning

confidence: 67%

Fractional elliptic quasi-variational inequalities: Theory and numerics

Antil

Rautenberg

2018

Interfaces Free Bound.

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This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order s ∈ (0, 1), studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter τ tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.

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“…The L ∞ (Ω) characterization of the solution of the above problem is expected in several settings. We state the following result that can be found in [5] (see also [3]).…”

Section: Boundedness Of the Solution To The Linear Problemmentioning

confidence: 67%

Fractional elliptic quasi-variational inequalities: Theory and numerics

Antil

Rautenberg

2018

Interfaces Free Bound.

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“…In the connection of well-posedness and regularity results, we refer to [1,2] for the case of the fractional negative Laplacian with zero Dirichlet boundary conditions; general operators other than the negative Laplacian have apparently only been studied in [24,[33][34][35]. As of now, aspects of optimal control have been scarcely dealt with even for simpler linear evolutionary systems involving fractional operators; for such systems, some identification problems were addressed in the recent contributions [36,45], while for optimal control problems for such cases we refer to [6] (for the stationary (elliptic) case, see also [3][4][5][7][8][9]). However, to the authors' best knowledge, the present paper appears to be the first contribution that addresses optimal control problems for Cahn-Hilliard systems with general fractional order operators and potentials of double obstacle type.…”

Section: Introductionmentioning

confidence: 99%

Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials

Colli¹,

Gilardi²,

Sprekels³

2021

Discrete &Amp; Continuous Dynamical Systems - S

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In the recent paper "Well-posedness and regularity for a generalized fractional Cahn-Hilliard system", the same authors derived general well-posedness and regularity results for a rather general system of evolutionary operator equations having the structure of a Cahn-Hilliard system. The operators appearing in the system equations were fractional versions in the spectral sense of general linear operators A and B having compact resolvents and are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. The associated double-well potentials driving the phase separation process modeled by the Cahn-Hilliard system could be of a very general type that includes standard physically meaningful cases such as polynomial, logarithmic, and double obstacle nonlinearities. In the subsequent paper "Optimal distributed control of a generalized fractional Cahn-Hilliard system" (Appl. Math. Optim. (2018), https://doi.org/10.1007/s00245-018-9540-7) by the same authors, an analysis of distributed optimal control problems was performed for such evolutionary systems, where only the differentiable case of certain polynomial and logarithmic doublewell potentials could be admitted. Results concerning existence of optimizers and Colli -Gilardi -Sprekels first-order necessary optimality conditions were derived, where more restrictive conditions on the operators A and B had to be assumed in order to be able to show differentiability properties for the associated control-to-state operator. In the present paper, we complement these results by studying a distributed control problem for such evolutionary systems in the case of nondifferentiable nonlinearities of double obstacle type. For such nonlinearities, it is well known that the standard constraint qualifications cannot be applied to construct appropriate Lagrange multipliers. To overcome this difficulty, we follow here the so-called "deep quench" method. This technique, in which the nondifferentiable double obstacle nonlinearity is approximated by differentiable logarithmic nonlinearities, was first developed by P. Colli, M.H. Farshbaf-Shaker and J. Sprekels in the paper "A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles" (Appl. Math. Optim. 71 (2015), pp. 1-24) and has proved to be a powerful tool in a number of optimal control problems with double obstacle potentials in the framework of systems of Cahn-Hilliard type. We first give a general convergence analysis of the deep quench approximation that includes an error estimate and then demonstrate that its use leads in the double obstacle case to appropriate first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint state system.

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“…In this work, some regularity results, numerical schemes to aproximate the optimal solution and a priori error analysis are presented. We mention also [5], where the optimal control of fractional semilinear PDEs with both spectral and integral fractional Laplacians with distributed control is considered and [2]here linear PDEs and integral fractional Laplacian are studied. In these works, the necessary and sufficient optimality conditions for such problems are obtained.…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of this paper is study the existence of optimal solutions of problem (1)- (2). In view of a nonlinearity of f and f 0 , as well as, a general convexity assumption (H 3 ) a method of the proof of the main result differs from the method presented in [5]. Our study is based on the lower closure theorem for orientor fields ([17, Theorem 10.7.i]) and a measurable selection theorem of Filippov type ( [30,Theorem 2J]).…”

Section: Introductionmentioning

confidence: 99%