Localization for nonlocal gradient-based optimal control problems

Technical Summary: Localization for Nonlocal Gradient-Based Optimal Control Problems

Problem Statement
This paper investigates optimal control problems formulated within a nonlocal function space framework characterized by two distinct parameters: a fractional parameter $s \in (0, 1)$ and a horizon parameter $\delta > 0$. The control variable $g$ belongs to an admissible set $Z_{ad}$, while the state variable $u$ resides in a nonlocal Sobolev space $H^{s,p,\delta}_0(\Omega-\delta; \mathbb{R}^n)$. The state $u$ is constrained to minimize a nonlocal energy density $W^{s,\delta}_g$, which depends on the control $g$. The objective is to minimize a cost functional $F(u, g)$ subject to these nonlocal constraints.

The study addresses two primary scenarios regarding the energy density $W$:

1. Convex Case: The energy is strictly convex, specifically modeled by a nonlocal $p$-Laplacian (or weighted $p$-Dirichlet energy).
2. Non-Convex Case: The energy is polyconvex or quasiconvex, satisfying standard $p$-growth bounds. In this setting, minimizers exist but are not necessarily unique.

The central mathematical challenge is to analyze the "localization" of these problems, demonstrating how solutions converge to classical partial differential equation (PDE)-based optimal control problems as the nonlocality vanishes. This is achieved through two distinct limiting processes:

• Taking the fractional parameter $s \to 1^-$ (with $\delta$ fixed).
• Taking the horizon parameter $\delta \to 0^+$ (with $s$ fixed).

Methodology and Framework
The authors utilize a nonlocal gradient operator $D^s_\delta$, defined via a truncated kernel $\rho^s_\delta$ that generalizes the Riesz fractional gradient. This operator acts on functions defined over an extended domain $\Omega_\delta$ but enforces zero boundary conditions on the "collar" regions $\Omega_\delta \setminus \Omega$ and $\Omega \setminus \Omega_{-\delta}$.

The analysis relies on several functional analytic tools:

• Nonlocal Sobolev Spaces: The spaces $H^{s,p,\delta}$ are defined as the closure of smooth functions under a norm involving the nonlocal gradient.
• Compactness and Embeddings: The authors employ nonlocal Poincaré inequalities (Proposition 2.5) and compact embedding results (Proposition 2.6) to establish the boundedness of admissible states and the existence of converging subsequences.
• $\Gamma$-Convergence: To handle the asymptotic analysis, the paper utilizes $\Gamma$-convergence of the energy functionals. This ensures that the limit of minimizers of the nonlocal energies corresponds to a minimizer of the limiting local energy.
• Direct Method of Calculus of Variations: For the non-convex case, existence is proven using the direct method, leveraging the weak lower semi-continuity of polyconvex and quasiconvex functionals.

Key Results

1. Well-posedness in the Convex Case:
   For the nonlocal $p$-Laplacian energy, the set of admissible pairs $(u, g)$ is convex. The authors prove the existence and uniqueness of a minimizer for the optimal control problem (Theorem 3.7). This relies on the compactness of the solution operator mapping the control $g$ to the state $u$.

2. Existence in the Non-Convex Case:
   When the energy is polyconvex or quasiconvex, the admissible set is generally non-convex, and uniqueness of the state $u$ for a fixed $g$ cannot be guaranteed. Nevertheless, the authors establish the existence of at least one optimal pair $(u, g)$ (Theorem 3.12) by proving the boundedness of the set of admissible states and the weak closedness of the constraint set.

3. Localization as $s \to 1^-$:
   With $\delta$ fixed, the paper demonstrates that as $s \to 1^-$, the nonlocal gradients converge strongly to the classical gradient.

   • Convex Case: The minimizers of the nonlocal control problems converge strongly (for states and gradients) and weakly (for controls) to the unique minimizer of the corresponding local $p$-Laplacian control problem (Theorem 4.10). Furthermore, the cost functional values converge.
   • Non-Convex Case: Due to the lack of uniqueness in the state equation, a full convergence of minimizers cannot be guaranteed. Instead, the authors prove a "relative minimization" result (Corollary 4.11), showing that any limit of a sequence of nonlocal minimizers is a minimizer relative to the set of locally admissible pairs that can be approximated by nonlocal solutions.
   • Strong Convergence of Controls: A significant result is that the control variables $g_s$ converge strongly in $L^r$ for any $r \in [1, \infty)$, not just weakly (Theorem 4.12). This is derived from the specific structure of the cost functional and the regularization term.

4. Localization as $\delta \to 0^+$:
   With $s$ fixed, the paper analyzes the limit as the horizon $\delta \to 0^+$. The results mirror those for $s \to 1^-$. The nonlocal gradients converge to the classical gradient, and the nonlocal optimal control problems converge to their local counterparts (Theorem 4.19). Similar to the $s \to 1$ case, strong convergence of controls is established (Theorem 4.21).

Significance and Claims
The paper claims to be the first work to consider optimal control problems where the state constraints involve minimizing polyconvex or quasiconvex energies, even within the local setting. By extending these results to the nonlocal framework, the authors bridge the gap between modern nonlocal modeling (such as peridynamics and fractional calculus) and classical optimal control theory.

The significance of the work lies in:

• Generalization: Moving beyond strictly convex energies to include materials and models described by quasiconvex or polyconvex energies, which are essential for nonlinear elasticity.
• Dual Localization: Providing a rigorous analysis of convergence via two distinct parameters ($s$ and $\delta$), offering flexibility in how nonlocal models are approximated by local PDEs.
• Strong Convergence of Controls: Establishing strong convergence for control variables, which is a stronger result than the typical weak convergence found in many $\Gamma$-convergence applications.

The authors modestly note that while $\Gamma$-convergence guarantees the convergence of minimizers, it does not provide rates of convergence. They suggest that future work could utilize regularity theory to recover convergence rates. Additionally, they identify the study of control problems with non-convex cost functionals (requiring second-order optimality conditions) and simultaneous limits ($s \to 1$ and $\delta \to 0$) as potential directions for future research.