A Singular Linear Quadratic Time-Inconsistent Optimal Control Problem

Yong J [Acta Math. Appl. Sin. Engl. Ser. 28 (2012), 1-30] [Math. Control Relat. Fields 1 (2011), 83-118] studied a standard linear quadratic time-inconsistent optimal control problem via a cooperative and non-cooperative approach, respectively. The authors extend his results to a singular case. To handle the singularity, the authors prove the solvability of a generalized Riccati equation, and introduce a notion of MP-convergence of matrix. It is shown that the authors can obtain a family of parameter equilibrium controls in both cases. Another interesting outcome is that a new type of parameter forward-backward Volterra integral equations is derived.