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This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a gen- eral class of the second order distributed systems with arbitrary damping terms. It is well understood that the feedback control law of the linear quadratic optimal control problem can be determined by the solution of a proper Riccati equation. The studies of Riccati equations for parabolic and hyperbolic control systems has been developed in full extent (cf. (5), (8), (19), (20)). In this paper we consider the second order damped evolution equation system described by

RICCATI EQUATION IN QUADRATIC OPTIMAL CONTROL PROBLEM OF DAMPED SECOND ORDER SYSTEM