Finite-time Linear-Quadratic Optimal Control of Partial Differential-Algebraic Equations

TL;DR

This paper extends finite-time linear-quadratic optimal control theory from PDEs to a specific class of PDAEs, providing existence, uniqueness, and a feedback solution, along with numerical validation.

Contribution

It introduces a novel approach to solve finite-time LQ control problems for radial PDAEs with index zero, generalizing PDE results and avoiding explicit PDAE projection.

Findings

Existence and uniqueness of optimal control established.

Derived Riccati-like and algebraic equations for feedback control.

Numerical simulations demonstrate practical applicability.

Abstract

Finite-time linear-quadratic control of partial differential-algebraic equations (PDAEs) is considered. The discussion is restricted to those that are radial with index $0$; this corresponds to a nilpotency degree of 1. We establish the existence of a unique minimizing optimal control. A projection is used to derive a system of differential Riccati-like equation coupled with an algebraic equation, yielding the solution of the optimization problem in a feedback form. This generalizes the well-known result for PDEs to this class of PDAEs. These equations and hence the optimal control can be calculated without construction of the projected PDAE. Finally, we provide numerical simulations to illustrate application of the theoretical results.