Homogenization of an optimal control problem for nonlocal semilinear elasticity with soft inclusions

Technical Summary: Homogenization of an Optimal Control Problem for Nonlocal Semilinear Elasticity with Soft Inclusions

1. Problem Formulation
This paper investigates the asymptotic analysis of an Optimal Control Problem (OCP) defined on a high-contrast elastic medium containing soft periodic inclusions. The physical domain $\Omega \subset \mathbb{R}^N$ is decomposed into a connected stiff matrix phase ($\Omega^2_\varepsilon$) and a soft inclusion phase ($\Omega^1_\varepsilon$). The system is governed by a semilinear elasticity equation featuring a nonlocal term, modeling materials where the mechanical response depends on a global measure of deformation amplitude (e.g., due to solvent redistribution or mean-field feedback).

The governing state equation for the displacement $u_{\varepsilon\delta}$ is:
$$-\nabla \cdot \sigma_{\varepsilon\delta}(u_{\varepsilon\delta}) + \alpha \|u_{\varepsilon\delta}\|_{L^p(\Omega)}^p u_{\varepsilon\delta} = f(x) + \mathbb{1}_{\Omega^1_\varepsilon} \theta_\varepsilon \quad \text{in } \Omega,$$
subject to Dirichlet and Neumann boundary conditions. Here:

• $\varepsilon > 0$ represents the periodicity scale.
• $\delta > 0$ represents the contrast parameter, where the stiffness in the soft phase scales as $\delta^2$ relative to the stiff phase (order 1).
• $\theta_\varepsilon \in L^2(\Omega^1_\varepsilon)^N$ is the distributed control acting exclusively within the soft inclusions.
• The nonlocal term $\alpha \|u\|_{L^p}^p u$ (with $p \ge 2$) couples all spatial points through the global $L^2$-amplitude of the displacement.

The objective is to minimize a reduced cost functional $j_{\varepsilon\delta}(\theta_\varepsilon)$, which balances the $L^2$-tracking error of the displacement against a desired profile $u_d$ and a regularization term for the control energy:
$$j_{\varepsilon\delta}(\theta_\varepsilon) = \frac{1}{2} \int_\Omega |u_{\varepsilon\delta}(\theta_\varepsilon) - u_d|^2 dx + \frac{\gamma}{2} \int_{\Omega^1_\varepsilon} |\theta_\varepsilon|^2 dx.$$

The analysis focuses on the limit as $(\varepsilon, \delta) \to (0, 0)$ under the scaling regime where the ratio $\kappa = \lim \delta/\varepsilon$ belongs to $(0, +\infty]$.

2. Methodology
The authors employ a combination of the periodic unfolding operator and $\Gamma$-convergence arguments to derive the limit systems.

• Unfolding Operator: The periodic unfolding operator $T_\varepsilon$ is utilized to handle the oscillating domains and coefficients. This technique allows for the separation of macroscopic and microscopic scales, transforming integrals over $\Omega$ into integrals over $\Omega \times Y$ (where $Y$ is the unit periodic cell).
• Energy Decomposition and A Priori Estimates: Due to the high-contrast nature (stiffness degeneracy in the soft phase), standard $H^1$-bounds fail. The authors introduce a specific decomposition of the displacement field into a macroscopic part and a microscopic corrector part (Lemmas 3.2–3.3). This allows for the derivation of uniform estimates necessary for compactness.
• Strong Convergence via Nonlocal Terms: A critical methodological challenge is passing to the limit in the nonlocal semilinear term $\alpha \|u\|_{L^p}^p u$. Weak convergence is insufficient for this term. The authors establish strong $L^2$ convergence of the unfolded state variables by leveraging the convergence of the total energy (via $\Gamma$-convergence) and the strict convexity of the energy functional.
• Adjoint System Analysis: To characterize the optimal control, the authors derive the adjoint state equation. They prove the existence of a unique adjoint state and establish its asymptotic behavior, which is essential for deriving the limit optimality conditions.

3. Key Contributions and Results

• Derivation of the Homogenized State System:
  The paper derives the limit (homogenized) state system associated with the scaling $\delta/\varepsilon \to \kappa$. The limit system involves a two-scale structure:

  • A macroscopic displacement $u_0$.
  • Microscopic correctors $\tilde{W}$ (associated with the soft phase) and $\hat{U}$ (associated with the stiff phase).
  • The limit energy functional $T_\kappa$ depends explicitly on the parameter $\kappa$.
  • In the specific case $\kappa = +\infty$, the system decouples into a fully scale-separated homogenized system.
  • The limit problem is characterized by a variational inequality involving the homogenized stiffness tensor $A^{hom}$ and the nonlocal term acting on the combined limit field $u_0 + \frac{1}{\kappa}\tilde{W}$.

• Homogenization of the Optimal Control Problem:
  The authors prove that the sequence of microscopic optimal controls $\Theta_\varepsilon$ converges to an optimal control $\hat{\Theta}$ for the limit problem.

  • Existence and Uniqueness: Existence of optimal controls for both the microscopic and limit problems is established. Uniqueness is guaranteed for the limit problem under specific conditions, though the non-convexity of the reduced cost functional (due to the nonlinearity) implies that global uniqueness for the microscopic problem is not guaranteed in general.
  • Convergence: Using $\Gamma$-convergence of the reduced cost functionals and the strong convergence of the state and adjoint variables, it is proven that any limit point of a sequence of microscopic optimal controls is an optimal control for the limit OCP.
  • Limit Optimality Conditions: The limit optimal control is explicitly characterized via the limit adjoint state: $\hat{\Theta} = -\frac{1}{\gamma} (v_0 + \frac{1}{\kappa}\hat{Z})$, where $v_0$ and $\hat{Z}$ are components of the limit adjoint state.

• Handling High-Contrast and Nonlocality:
  The work addresses the simultaneous difficulty of high-contrast elasticity (where energy degenerates in soft phases) and nonlocal semilinearity. The authors demonstrate that the nonlocal term requires strong convergence, which is achieved through energy convergence arguments, distinguishing this analysis from standard local semilinear homogenization where weak convergence often suffices.

4. Significance and Novelty
The paper claims significance primarily in the structure of the nonlocal semilinearity and the high-contrast setting.

• Nonlocal Operator Structure: Unlike previous works on OCPs with semilinear equations (e.g., those using local terms $S(u)$), this model employs a Hilbert-space operator $u \mapsto \alpha \|u\|_{L^p}^p u$. This couples all spatial points globally. The authors note that this structure necessitates strong $L^2$ convergence (convergence of norms) rather than just weak convergence, a requirement that is non-trivial in high-contrast settings where uniform $H^1$ bounds typically fail.
• High-Contrast Elasticity: The analysis extends homogenization theory to optimal control problems in elasticity with soft inclusions, a regime where the elastic energy degenerates. The paper provides a rigorous framework for deriving limit systems in this regime, showing how the parameter $\kappa$ influences the effective model.
• Methodological Rigor: The work combines the indirect method (optimality systems) with direct $\Gamma$-convergence arguments to handle the convergence of the control-to-state map and the cost functional, ensuring that the limit of the optimal controls is indeed optimal for the limit problem.

The paper concludes by organizing the results into existence proofs for the state and control, a priori estimates, the derivation of the two-scale limit systems, and the final convergence of the OCP, providing a complete mathematical framework for shape/response optimization in such complex metamaterials.