Infinite horizon linear quadratic Pareto game of the stochastic singular systems

This paper is concerned with the linear quadratic (LQ) Pareto game of the stochastic singular systems in infinite horizon. Firstly, the optimal control problem of the weighted sum cost functional is discussed. Utilizing the equivalent transformation method, the weighted sum LQ optimal control problem is transformed into a stochastic LQ optimization problem. Based on the classical stochastic LQ optimal control theory, the necessary and sufficient condition for the solvability of the indefinite weighted sum LQ optimal control is put forward. Then, the LQ Pareto game of the stochastic singular systems is studied. By the discussion of the convexity of the cost functionals, a sufficient condition for the existence of the Pareto solutions is obtained via the solvability of the corresponding generalized algebraic Riccati equation (GARE). Moreover, we derive all Pareto solutions based on the solution of a Lyapunov equation. Finally, an example is given to show the effectiveness of the proposed results.