Turnpike property of linear quadratic control problems with unbounded control operators

TL;DR

This paper proves the turnpike property for linear quadratic control problems with unbounded control operators, extending existing results to more general and realistic systems using approximation and convergence techniques.

Contribution

It introduces a novel approximation approach for unbounded control operators and establishes the turnpike property under broad conditions, filling a gap in the existing literature.

Findings

Established convergence of approximate bounded control problems to the original unbounded problem.

Extended the theory of linear quadratic control to systems with unbounded control operators.

Provided a framework for analyzing optimal control problems with more general control operators.

Abstract

We establish the turnpike property for linear quadratic control problems for which the control operator is admissible and may be unbounded, under quite general and natural assumptions. The turnpike property has been well studied for bounded control operators, based on the theory of differential and algebraic Riccati equations. For unbounded control operators, there are only few results, limited to some special cases of hyperbolic systems in dimension one or to analytic semigroups. Our analysis is inspired by the pioneering work of Porretta and Zuazua \cite{PZ13}. We start by approximating the admissible control operator with a sequence of bounded ones. We then prove the convergence of the approximate problems to the initial one in a suitable sense. Establishing this convergence is the core of the paper. It requires to revisit in some sense the linear quadratic optimal control theory with admissible control operators, in which the roles of energy and adjoint states, and the connection between infinite-horizon and finite-horizon optimal control problems with an appropriate final cost are investigated.