Discrete-time Indefinite Stochastic Linear Quadratic Optimal Control: Inequality Constraint Case

It is known that the Karush-Kuhn-Tucker (KKT) theorem gives necessary conditions for the existence of optimal solutions to constrained optimization problems. It is shown that a class of discrete-time indefinite stochastic linear quadratic (LQ) optimal control problems with an inequality constraint on terminal state, can be transformed into a mathematical programming problem with equality and inequality constrains. In this paper, the KKT condition for the existence of optimal linear state feedback controllers is given. More importantly, the previous results on discrete-time stochastic LQ optimal control without constraints or with equality constraints, can be viewed as corollaries of the main theorems of this paper.