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

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.