On convergence of solutions to nonlocal optimal control problems with quasi-minimization constraints

Technical Summary: Convergence of Solutions to Nonlocal Optimal Control Problems with Quasi-Minimization Constraints

Problem Statement
This paper investigates a class of ill-posed, nonlocal optimal control problems where the state constraints are defined not by global minimizers, but by $\epsilon$-quasi-minimizers of a quasiconvex energy functional. The problem is parameterized by a fractional differentiability order $s \in (0, 1)$ and a horizon parameter $\delta > 0$. The state $u$ and control $g$ are sought to minimize a cost functional $F(u, g)$ subject to the constraint that the state $u$ is an $\epsilon$-quasi-minimizer of a nonlocal energy $W^{\delta,s}_g$.

The energy density $W$ is assumed to be quasiconvex (or polyconvex), a setting common in nonlinear elasticity where minimizers may not be unique. The nonlocal gradient operator $D^\delta_s$ incorporates long-range interactions via an integral kernel, distinguishing this framework from classical PDE-constrained control. The primary challenge addressed is the lack of uniqueness in the constraint's minimization problem, which previously hindered the proof of convergence for nonlocal control problems to their local counterparts (as seen in prior work [20] which utilized global minimization constraints).

Methodology
The analysis is conducted within the framework of nonlocal function spaces $H^{s,p,\delta}_0(\Omega-\delta; \mathbb{R}^n)$, which are closures of smooth compactly supported functions under a norm involving the nonlocal gradient. The methodology relies on three main pillars:

1. Existence Theory: The authors establish the existence of solutions to the nonlocal control problem using the direct method of the calculus of variations. Key to this is proving the weak closedness of the set of admissible pairs $T^\epsilon_{\delta,s}$ and the lower semi-continuity of the cost functional. Unlike standard control problems, a solution mapping cannot be defined due to the potential non-uniqueness of the energy minimizers; instead, the set of $\epsilon$-quasi-minimizers is shown to be bounded and weakly closed.
2. $\Gamma$-Convergence: To analyze the asymptotic behavior, the paper employs $\Gamma$-convergence theory. The authors prove that the family of nonlocal energy functionals $\{W^{\delta,s}_g\}$ $\Gamma$-converges to the corresponding local energy functional $W^{\text{loc}}_g$ in the strong $L^p$ topology under two distinct limits:
   • $s \to 1^-$ (fractional order approaching classical differentiability).
   • $\delta \to 0^+$ (vanishing horizon).
3. Recovery Sequence Construction: A critical technical contribution is the construction of recovery sequences for the convergence of minimizers. Because the constraint involves an inequality ($W(u) - \min W \le \epsilon$) rather than an equality, the authors must carefully handle cases where the energy difference is strictly less than $\epsilon$ versus when it equals $\epsilon$. In the latter case, they construct recovery sequences using convex combinations of the candidate limit state and local minimizers to ensure the $\epsilon$-constraint is satisfied in the limit.

Key Results
The paper establishes the following main theorems:

• Existence (Theorem 3.4): For fixed parameters $\delta, s, \epsilon$, there exists a pair $(u_{\delta,s}, g_{\delta,s})$ solving the nonlocal optimal control problem with an $\epsilon$-quasi-minimization constraint.
• Convergence as $s \to 1^-$ (Theorem 4.10): Solutions to the nonlocal problem converge (up to a subsequence) to a solution of the corresponding local optimal control problem (where the constraint depends on the classical gradient $\nabla u$). The convergence includes strong convergence of the state in $L^p$ and weak convergence of the control in $L^{p'}$.
• Convergence as $\delta \to 0^+$ (Theorem 4.20): Analogous convergence results hold when the horizon parameter vanishes, provided the kernel satisfies specific scaling assumptions.
• Strong Convergence of Controls (Corollaries 4.13 & 4.21): Under the established convergence of the cost functionals, the control variables $g_{\delta,s}$ converge strongly in $L^r$ spaces.

Significance and Claims
The paper claims that the introduction of the $\epsilon$-quasi-minimization constraint resolves a specific limitation found in previous literature (specifically [20]), where the lack of uniqueness in global minimizers prevented the proof of convergence for nonlocal control problems to local ones. By relaxing the constraint to allow for approximate minimizers, the authors can:

1. Prove the existence of solutions for a broader class of quasiconvex energies where uniqueness is not guaranteed.
2. Establish rigorous convergence of solutions from nonlocal to local settings for general cost functionals that depend on the nonlocal gradient.
3. Provide a novel analytic technique for constructing recovery sequences in parameterized optimal control problems, leveraging the tolerance $\epsilon$ to manage the oscillation of energy differences.

The authors note that their results are not heavily dependent on the specific function space framework chosen and suggest that the techniques are applicable to other families of parameterized optimal control problems, particularly those that are ill-posed but admit at least one solution. The paper explicitly avoids claiming applications to specific physical systems beyond the general modeling of solid mechanics and diffusion, and does not propose new numerical schemes, though it references recent work on Discontinuous Galerkin methods for context.