Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces

Antil, Harbir; Verma, Deepanshu; Warma, Mahamadi

doi:10.1007/s10957-020-01684-z

Cited by 20 publications

(24 citation statements)

References 23 publications

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“…In Sect. 3, we formulate and prove the main result of this paper, namely a theorem on the existence of optimal solutions for problem (1)- (2). We finish with an illustrative, theoretical example.…”

Section: Introductionmentioning

confidence: 93%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

Kamocki

2021

Appl Math Optim

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

Kamocki

2021

Appl Math Optim

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“…We will denote the space of Radon measures by M(Ω). For parabolic versions of these results, we refer to [20,21]. We further refer to [23] for semilinear problems, [22] for quasi-linear problems, [17,19] for variational and quasi-variational inequalities.…”

Section: Fractional Diffusion Equation: Analysis and Numerical Approx...mentioning

confidence: 99%

Optimal Control, Numerics, and Applications of Fractional PDEs

Antil¹,

Brown²,

Khatri³

et al. 2021

Preprint

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This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.

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confidence: 99%

Moreau-Yosida Regularization for Optimal Control of Fractional Elliptic Problems with State and Control Constraints

Antil,

Brown,

Verma

2019

Preprint

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Recently in [3], the authors have studied a state and control constrained optimal control problem with fractional elliptic PDE as constraints. The goal of this paper is to continue that program forward and introduce an algorithm to solve such optimal control problems. We shall employ the well-known Moreau-Yosida regularization to handle the state constraints. Similarly to the classical case, we establish the convergence (with rate) of the regularized control problem to the original one. We discretize the problem using a finite element method and establish convergence of our numerical scheme. We emphasize that due to the non-smooth nature of the fractional Laplacian, the proof for the classical case does not apply to the fractional case. The numerical experiments confirm all our theoretical findings.

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Optimal Control of Fractional Elliptic PDEs with State Constraints and Characterization of the Dual of Fractional-Order Sobolev Spaces

Cited by 20 publications

References 23 publications

Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

Optimal Control of a Nonlinear PDE Governed by Fractional Laplacian

Optimal Control, Numerics, and Applications of Fractional PDEs

Moreau-Yosida Regularization for Optimal Control of Fractional Elliptic Problems with State and Control Constraints

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