ON A CERTAIN ASYMPTOTIC-BEHAVIOR OF TIME-INVARIANT RICCATI MATRIX DIFFERENTIAL-EQUATIONS

The main result of this paper reads as follows: Given (real) (n x n)-matrices A, B, C, and C0 such that B, C, and C0 are symmetric, such that B and C0 are nonnegative definite, such that the pair (A, B) is (completely) controllable (i.e., rank [B, A, ..., A(n-1)B] = n), and such that the triple (A B, C0) is strongly observable (i.e., x = Ax + Bu, C0x = 0 on some nondegenerate interval implies x(t) = 0), then, for any t0 > 0 and any symmetric matrix Q0, the solution Q(t; lambda) of the Riccati matrix differential equation Q + A(T)Q + QA + QBQ - C + lambdaC0 = 0, Q(0) = Q0 exists on [0, t0] if lambda less-than-or-equal-to lambda0, and it satisfies lim(lambda --> -infinity) Q(t0; lambda) = infinity (i.e., all eigenvalues of the symmetric matrix Q tend to infinity). The result is that the new notion of strong observability is even necessary for the assertion. This result on Riccati matrix equations is motivated by the following application. It is shown that min {integral-t0/0 [x(T)Cx + u(T)Bu] dt/integral-t0/0 x(T)C0x dt, where x(t) not-equal 0, and x = Ax + Bu with u is-an-element-of C(s)[0, t0]} exists just under the same assumptions as above.