Solutions of the Riccati equation for H Discrete Time Systems

Let H and Y be separable Hilbert spaces, and U finite dimensional. Let A ∈ L(H), B ∈ L(U,H), C ∈ L(H,Y ), D ∈ L(U, Y ), and suppose that the open loop transfer function D(z) := D+zC(I−zA)B ∈ H(D;L(U, Y )), where D is the open unit disk. We consider a subset of self adjoint solutions P of the discrete time algebraic operator Riccati equation (DARE)    APA− P + CJC = K PΛPKP , ΛP = D JD +BPB, ΛPKP = −DJC −BPA, where J = J ∈ L(Y ) is a cost operator, and Λ P ∈ L(U). Under further assumptions, we obtain the following results. To solutions of the DARE, we associate a coanalytic-analytic factorization of the Popov function D(z)JD(z). To each nonnegative solution of the DARE, we associate a (partial) inner-outer factorization of the transfer function D(z) (if J ≥ 0). We conclude that the natural partial ordering of the (adjoints of the) inner factors of DP (z) is consistent with the partial ordering of the solutions P , as self adjoint operators. We obtain a characterization of the critical solution as the maximal nonnegative solution (if J ≥ 0). Finally, generalizations of these results are indicated.