Non-convex optimal control governed by nonlinear elliptic partial differential equation: Analysis, functional-based algorithm and control landscape

Abstract

Non-convex optimal control arises from various applications but may contain multiple stationary points. Classical solvers usually perform a local search and therefore rely on good initial guesses to reach appropriate local optimal controls. In this work we introduce a novel solution strategy for the non-convex optimal control of an elliptic equation, the main idea of which is to construct the control landscape based on the functional high-index saddle dynamics (FHiSD) method. This method reduces the dependence on prescribed initial guesses by using the information of high-index saddle points, and depicts the macroscopic configuration of the space of the control variable. Then various minima could be systematically computed along transition pathways and the control strategy can then be selected among them. We prove the existence of the optimal control, and then propose and analyze the FHiSD in locating saddle points. Subsequently, the applicability of FHiSD in non-convex optimal control is rigorously justified. Numerical results not only indicate the effectiveness of the proposed method, but reveal unintuitive phenomena (e.g. the non-monotonicity of the values of the cost functional with respect to the Morse indices) that support the necessity of computing multiple solutions of high indices.