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

TL;DR

The paper introduces a PDE-constrained high-index saddle dynamics (PCHiSD) framework to systematically map the non-convex control landscape of elliptic optimal control problems. By coupling HiSD with the PDE state and adjoint equations, it defines a reduced gradient and develops downward/upward search strategies to locate multiple saddle points and minima without relying on favorable initial guesses. It also extends the method to integral-constrained controls via a Riemannian-manifold projection, enabling constrained landscape exploration. Numerical experiments in one and two dimensions reveal rich solution structures, including nonintuitive transitions where higher-index saddles can have lower objective values. The work provides a robust, globally informed approach for identifying both local and global optima in non-convex PDE-constrained control problems and suggests broad applicability beyond elliptic systems.

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.

Figures (5)

• Figure 1: Control landscapes of one-dimensional problem with different $\lambda$. Each plot shows either a distribution of the control $u$ (blue curve) corresponding to a saddle point of the reduced cost functional $\hat{J}$, or the corresponding state $y$ (red curve). The optimal control and the corresponding optimal state are denoted as $u^*$ and $y^*$, respectively.
• Figure 2: Domains of convergence of the gradient method and the Newton's method. Each colored segment represents an interval in which the constant initial value of $u$ could be selected such that the algorithm converges to the corresponding minima.
• Figure 3: Control landscapes of symmetric and unconstrained case 1 with $\lambda = 0.01,0.005, 0.002$. Each colored plot shows the distribution of the control $u$, and each three-dimensional surface depicts the corresponding state $y$.
• Figure 4: Control landscapes of asymmetric and unconstrained case 2 with $\lambda = 0.01,0.005,0.0025$, $0.002$.
• Figure 5: Control landscapes of asymmetric and constrained case 3 with $\lambda = 0.002, 0.0015$.

Theorems & Definitions (11)

• Remark 2.1
• Lemma 1
• Proof 1
• Theorem 2
• Proof 2
• Theorem 3
• Proof 3
• Theorem 4
• Proof 4
• Theorem 1