Bilinear optimal control for the fractional Laplacian: analysis and discretization

TL;DR

This work addresses the optimal control of a nonlocal diffusion model governed by the integral fractional Laplacian $(-\Delta)^s$ with a coefficient control $q$ on Lipschitz domains. It adopts the integral definition of the operator and develops a rigorous analysis for existence, first- and second-order optimality conditions, and regularity of the optimal variables, together with two finite element discretization strategies: a fully discrete scheme with piecewise-constant controls and a semidiscrete variational discretization. The authors establish convergence of discrete global solutions to global solutions, show that continuous strict local solutions can be approximated by discrete local minima, and derive a priori error estimates for both discretization approaches. The numerical experiments illustrate the theoretical rates and reveal how regularity and the choice of control bounds influence the observed convergence, providing a practical framework for reliable simulation and potential coefficient identification in nonlocal models.

Abstract

We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.

Figures (5)

• Figure 1: Experimental rates of convergence for $\| \bar{u} - \bar{u}_h\|_s$ and $\| \bar{p} - \bar{p}_h \|_s$ considering the fully discrete (A.1)--(A.2) and semidiscrete schemes (B.1)--(B.2) for $s \in \{0.1,0.2,...,0.9\}$.
• Figure 2: Experimental rates of convergence for $\| \bar{u} - \bar{u}_h \|_{L^2(\Omega)}$, $\| \bar{p} - \bar{p}_h \|_{L^2(\Omega)}$, and $\| \bar{q} - \bar{q}_h \|_{L^2(\Omega)}$ considering the fully discrete (C.1)--(C.3) and semidiscrete schemes (D.1)--(D.3) for $s \in \{0.1,0.2,...,0.5\}$.
• Figure 3: Experimental rates of convergence for $\| \bar{u} - \bar{u}_h \|_{L^2(\Omega)}$, $\| \bar{p} - \bar{p}_h \|_{L^2(\Omega)}$, and $\| \bar{q} - \bar{q}_h \|_{L^2(\Omega)}$ considering the fully (E.1)--(E.3) and semidiscrete scheme (F.1)--(F.3) for $s \in \{0.6,0.7,0.8,0.9\}$.
• Figure 4: Experimental rates of convergence for $\| \bar{q} - \bar{q}_h\|_{L^{2}(\Omega)}$ considering the fully discrete (G.1) and semidiscrete schemes (G.2) for $a = 0.001\|\bar{u}\bar{p}\|_{L^{\infty}(\Omega)}$ and $s \in \{0.1,0.2,..., 0.9\}$.
• Figure 5: Experimental rates of convergence for $\| \bar{q} - \bar{q}_h\|_{L^{2}(\Omega)}$ considering the fully discrete (H.1) and semidiscrete schemes (H.2) for $a = 0.95\|\bar{u}\bar{p}\|_{L^{\infty}(\Omega)}$ and $s \in \{0.1,0.2,..., 0.9\}$.

Theorems & Definitions (50)

• Lemma 2.1: continuity of the product
• Proof 1
• Lemma 2.2: embedding results
• Proof 2
• Theorem 3.1: Sobolev regularity
• Proof 3
• Theorem 3.2: $L^{\infty}(\Omega)$--regularity
• Proof 4
• Theorem 3.3: error estimates
• Proof 5