Time-Inconsistent singular stochastic Linear-Quadratic control for stochastic evolution equations

This paper addresses general discounted stochastic linear-quadratic (LQ) optimal control problems, where the control cost weighting operator may be singular, and the system is governed by controlled stochastic evolution equations in a Hilbert space. In the time-consistent setting, we establish conditions on the problem coefficients that guarantee the well-posedness of the generalized Riccati equation, despite the singularity of the control weighting operator. For the time-inconsistent case, a necessary and sufficient condition for the existence of a closed-loop equilibrium operator is established in terms of the solvability of the equilibrium Riccati evolution equation with certain regularity. In the last part of the paper, the solvability of the equilibrium Riccati evolution equation is established through the application of the multi-person differential game method. These results extend the recent work of Lü and Ma [Science China Mathematics, 67 (2024), 211–236], which addressed this problem for controlled stochastic differential equations.