Homotopy trust-region method for phase-field approximations in perimeter-regularized binary optimal control

TL;DR

This work develops a phase-field homotopy trust-region method for binary optimal control with perimeter regularization, formulating the reduced problem as $\min_{w\in L^2(\Omega)} j(w) + \gamma C_0 P_\Omega(w^{-1}(\{1\}))$ subject to $w(x)\in\{0,1\}$ a.e. and $j(w)=J(S(w))$. It replaces the sharp perimeter by the Ginzburg--Landau energy $E_\varepsilon$ in subproblems and couples the interface parameter $\varepsilon$ to the trust-region radius, showing subsequential convergence to L-stationary points of the limit problem under regularity assumptions, with a Gamma-convergence type link between the diffuse and sharp problems. The algorithm leverages three subproblems (sharp, Ginzburg--Landau, and convexified variants) and integrates convex solves to accelerate computation, while proving boundedness of iterates and asymptotic L-stationarity of limit points. Numerical experiments in elliptic source control and acoustic-field scenarios illustrate the method’s ability to drive the design toward near-binary, sharp interfaces and demonstrate practical performance despite limitations in the regularity assumptions for PDEs with coefficient controls. The results highlight both the potential and the challenges of numerically solving perimeter-regularized binary optimal control via phase-field homotopy-trust-region strategies, suggesting future work on relaxing differentiability assumptions and extending to more general PDE settings.

Abstract

We consider optimal control problems that have binary-valued control input functions and a perimeter regularization. We develop and analyze a trust-region algorithm that solves a sequence of subproblems in which the regularization term and the binarity constraint are relaxed by a non-convex energy functional. We show how the parameter that controls the distinctiveness of the resulting phase field can be coupled to the trust-region radius updates and be driven to zero over the course of the iterations in order to obtain convergence to stationary points of the limit problem under suitable regularity assumptions. Finally, we highlight and discuss the assumptions and restrictions of our approach and provide the first computational results for a motivating application in the field of control of acoustic waves in dissipative media.

Figures (3)

• Figure 1: Positioning of $S$ (dark grey) with $s_a$ and $s_b$ inside $\Omega$ (white ellipse) along $e_1$, extended set $\hat{S}$ (light gray, extending to the exterior of $\Omega$), and smooth approximation of $\chi_S$.
• Figure 2: Global convergence of the iterates produced by \ref{['alg:trm']} indicated by the distance $\|w^n - \mathcal{I}w^*\|_{L^2(\Omega)}$ for the different values of $\varepsilon^0$. The locations of the marker symbols indicate the first iteration of an accepted iterate for a new (reduced) value of $\varepsilon^n$.
• Figure 3: First (a, c, e, g) and last accepted iterate (b, d, f, h) over the decreasing values of $\varepsilon^n$.

Theorems & Definitions (42)

• Proposition 2.1
• proof
• Proposition 2.3
• proof
• Definition 2.4: Definition 4.4 in manns2022on
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3