Abstract
This paper deals with a certain type of matrix Riccati equation with periodic coefficients, that is, a slightly generalized equation with the addition of a linear positive operator Π (t, ·) to the linear terms of the standard Riccati equation.
Such an equation arises, for example, in the quasi-steady state optimal control problem of a linear periodic system with a state-dependent noise, which is a natural generalization of the steady state optimal control problem of a linear time-invariant system.
In this paper, we discuss the existence of a periodic solution of the Riccati equation of this type and the asymptotic behavior of the solutions in connection with the periodic solution.
As a result, it is verified that under certain conditions of the coefficients (i. e.“stabilizability” and“detectability”), there exists a periodic solution to the above equation, if the coefficients Di(t), which form the positive operator Π (t, ·), are not too large. Moreover, we can see that under almost the same conditions, any solution of the Riccati equation converges to the above periodic solution in a certain sense as t goes to minus infinity. Then it is made sure that those results which have been obtained by G. A. Hewer et al. can be generalized to this case.
In the proofs quasi-linearlization methods and succesive approximation procedures are used effectively, so that the discussion becomes rather similar to the one which was adopted by W. M. Wonham.
Finally, we present one concrete example and apply the derived results to it.
