Quantum stochastic linear quadratic control theory: Closed-loop solvability
Wang Penghui, Wang Shan, Zhao Shengkai
TL;DR
The paper addresses the closed-loop solvability of the quantum stochastic linear quadratic (QSLQ) control problem by deriving a Pontryagin maximum principle for infinite-dimensional quantum stochastic systems and establishing a precise equivalence between unique closed-loop solvability and the well-posedness of a quantum Riccati equation. The analysis shows that a unique weak solution $P(\cdot)$ to the Riccati equation, along with the feedback law $\Theta(\cdot) = -K(\cdot)^{-1} L(\cdot)$ (where $L = B^* P + D^* P C$ and $K = R + D^* P D$), fully characterizes the optimal closed-loop control and the value function via $V(t_0, \eta) = \tfrac{1}{2} \mathrm{Re}\langle P(t_0) \eta, \eta \rangle$. The authors navigate the challenges posed by infinite dimensions and Fermion Brownian motion by introducing a truncation approach and a relaxed transposition solution framework for adjoint QSDEs, proving existence and uniqueness of the Riccati solution. This work provides a rigorous theoretical foundation for designing optimal quantum controllers in infinite-dimensional noncommutative settings.
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 Lü and Zhang's approach [Q. Lü 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.