[Paper Review] Optimal control of the coefficient for fractional and regional fractional {$p$}-{L}aplace equations: Approximation and convergence

Harbir Antil, Mahamadi Warma|arXiv (Cornell University)|Dec 24, 2016

Nonlinear Partial Differential Equations25 references4 citations

TL;DR

This paper establishes the existence and convergence of solutions to optimal control problems governed by fractional and regional fractional $p$-Laplace equations with coefficient control, using a regularization technique to overcome operator degeneracy. The key contribution is proving convergence of regularized optimal controls to a solution of the original problem, enabling first-order optimality system derivation in future work.

ABSTRACT

In this paper we study optimal control problems with either fractional or regional fractional $p$-Laplace equation, of order $s$ and $p\in [2,\infty)$, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the involved operator. To overcome the degeneracy of both fractional $p$-Laplacians, we introduce a regularization for both operators. We show existence and uniqueness of solution to the regularized state equations and existence of solution to the regularized optimal control problems. We also prove several auxiliary results for the regularized problems which are of independent interest. We conclude with the convergence of the regularized solutions.

Motivation & Objective

• To address the challenge of optimal control for quasilinear, possibly degenerate, nonlocal fractional $p$-Laplace equations with coefficient control.
• To overcome the degeneracy of regional and fractional $p$-Laplacian operators through a novel regularization approach.
• To establish existence and convergence of solutions to the regularized optimal control problem.
• To provide a foundation for deriving first-order optimality conditions in future work.

Proposed method

• Introduce a regularization of the fractional and regional $p$-Laplacian operators by adding a small parameter $\varepsilon$ and truncating the gradient term via $\mathcal{G}_n(u,s)$.
• Define a regularized optimal control problem (ROCP) with a Hilbert space-based $L^2$-based cost functional and box constraints on the coefficient $\kappa$.
• Use weak* convergence in $BV(\Omega)$ and strong convergence in $L^2(\Omega)$ to pass to the limit in the regularized system.
• Apply compactness and lower semicontinuity arguments to prove existence of solutions to the regularized and original problems.
• Leverage density results in fractional Sobolev spaces and equivalent norms in $W_0^{s,p}(\overline{\Omega})$ for the nonlocal case.
• Prove convergence of the regularized optimal controls $\kappa_{\varepsilon,n}^\star$ to a solution $\kappa_\star$ of the original optimal control problem.

Experimental results

Research questions

• RQ1 Can optimal control problems with coefficient control be well-posed for degenerate fractional $p$-Laplace equations?
• RQ2 How can regularization overcome the degeneracy of the regional and fractional $p$-Laplacian operators?
• RQ3 Does the solution of the regularized optimal control problem converge to a solution of the original problem?
• RQ4 What is the role of the $L^2$-based cost functional in enabling convergence analysis for nonlocal quasilinear problems?
• RQ5 Can the convergence framework be extended to the nonlocal fractional $p$-Laplacian with $0 < s < 1$?

Key findings

• Existence of a solution to the regularized optimal control problem is established via weak* compactness in $BV(\Omega)$ and strong convergence in $L^2(\Omega)$.
• The regularized state equation admits a unique solution due to the non-degeneracy introduced by the regularization.
• The optimal control $\kappa_{\varepsilon,n}^\star$ of the regularized problem converges to a solution $\kappa_\star$ of the original optimal control problem as $\varepsilon \to 0$ and $n \to \infty$.
• Convergence of the cost functional is verified through liminf and limsup estimates, showing $\mathbb{I}(\kappa_\star, u_\star) = \lim \mathbb{I}(\kappa_{\varepsilon,n}^\star, u_{\varepsilon,n}^\star)$.
• The results extend to the nonlocal fractional $p$-Laplacian $(-\Delta)_p^s$ for $0 < s < 1$ with minimal modifications to the framework.
• The convergence framework enables future derivation of first-order optimality systems for the original problem.