Constructing control landscape for non-convex optimal control of elliptic equation by PDE-constrained high-index saddle dynamics

TL;DR

This paper introduces a PDE-constrained high-index saddle dynamics method to systematically explore the control landscape of non-convex elliptic optimal control problems, enabling the computation of multiple solutions without relying on initial guesses.

Contribution

The paper develops a novel PDE-constrained high-index saddle dynamics approach for non-convex optimal control, allowing systematic exploration of control landscapes and computation of multiple solutions.

Findings

The method effectively constructs the control landscape for elliptic equations.

Numerical experiments demonstrate the method's ability to find multiple solutions.

Unintuitive phenomena observed support the need for high-index solution computation.

Abstract

Non-convex optimal control arises from various applications but may contain multiple stationary points. Classical solvers usually perform a ``local'' search near a saddle point or a local minimum, thus rely on good initial guess to reach the (quasi-)optimal control. We introduce a novel solution strategy for the non-convex optimal control of an elliptic equation. We develop a PDE-constrained high-index saddle dynamics (PCHiSD) to construct the control landscape. This method depicts the macroscopic configuration of control and state spaces such that the local and global minima could be systematically computed along the transition pathways in control landscape without requiring good initial conditions. We establish the well-posedness of the state equation and the existence of an optimal control, and then implement the PCHiSD and control landscape algorithms for numerical experiments and comparisons. Numerical results not only indicate the effectiveness of the proposed method, but reveal unintuitive phenomena that supports the necessity of computing multiple solutions of high indices.