Hybrid Optimal Control Problems for a Class of Semilinear Parabolic Equations

TL;DR

This paper studies hybrid optimal control problems for semilinear parabolic equations, deriving optimality conditions and analyzing Hamiltonian properties, with a numerical example demonstrating the approach.

Contribution

It introduces a novel reformulation of hybrid control problems and derives first- and second-order optimality conditions for this class.

Findings

Hamiltonian is constant over time for autonomous systems.

Optimality conditions are established for switching times.

Numerical example illustrates the theoretical results.

Abstract

A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.