The Dissipation Inequality and Generalized Algebraic Riccati Equation for Linear Quadratic Control Problem of Descriptor System

We consider the infinite-horizon linear quadratic control problem for a descriptor system (DLQCP) based on the theory of dissipative system. First, we derive a dissipative inequality for the descriptor system satisfied by the optimal cost of DLQCP. This implies that the optimal cost is characterized by the solution of a linear matrix iniquality (LMI). Secondly, we show that the solution of LMI characterizing the optimal cost also satisfies a related generalized algebraic Riccati equation (GARE) if DLQCP is regular. Finally, we derive the optimal solution to DLQCP with fixed terminal condition.