Quantum stochastic linear quadratic control theory: Closed-loop solvability

TL;DR

This paper develops a theoretical framework for the closed-loop control of quantum stochastic systems, establishing conditions for optimal control and solving associated infinite-dimensional Riccati equations.

Contribution

It introduces a novel approach to quantum stochastic LQ control, proving existence and uniqueness of solutions to quantum Riccati equations and linking them to closed-loop solvability.

Findings

Established the equivalence between closed-loop solvability and Riccati equation well-posedness.

Proved existence and uniqueness of solutions to infinite-dimensional quantum Riccati equations.

Provided a theoretical basis for optimal quantum control design.

Abstract

In this paper, we investigate the closed-loop solvability of the quantum stochastic linear quadratic optimal control problem. We derive the Pontryagin maximum principle for the linear quadratic control problem of infinite-dimensional quantum stochastic systems. The equivalence between unique closed-loop solvability for quantum stochastic linear quadratic optimal control problems and the well-posedness of the corresponding quantum Riccati equations is established. Notably, although the quantum Riccati equation is an infinite-dimensional deterministic operator-valued ordinary differential equation, classical methods are not applicable. Inspired by L\"{u} and Zhang's approach [Q. L\"{u} and X. Zhang, Probability Theory and Stochastic Modelling, 101. Springer, Cham, (2021) \& Mem. Amer. Math. Soc. 294 (2024)] to stochastic Riccati equations, we prove the existence and uniqueness of its solutions. The results provide a theoretical foundation for the optimal design of quantum control.