Stochastic Linear-Quadratic Optimal Control With Input Delay and Quadratic Constraint

Although techniques in optimal control theory often address unconstrained problems, many applications involve constraints. Despite attempts to address constrained linear-quadratic (LQ) control problems, these efforts primarily focus on delay-free systems. For constrained LQ control problems with delays, there is little theoretical understanding due to the significant complexity added by the inclusion of time delays. This article explores the stochastic LQ optimal control problem, which includes control-dependent multiplicative noise and delays, along with quadratic constraint. Through the application of duality theory, we streamline the optimal control problem is reduced to a parameterized, unconstrained stochastic LQ control problem incorporating delays. By solving the Riccati-ZXL equation, the optimal control and cost function are explicitly formulated. Notably, the optimal parameter is determined by solving a semi-definite programming (SDP) problem. The main contribution is presenting the optimal control as a nonlinear function of both the initial state and the state’s conditional expectation. Numerical examples illustrate the efficacy of the derived results.