Analytic properties of matrix Riccati equation solutions

For matrix Riccati equations of platoontype systems and of systems arising from PDEs, assuming the coefficients are analytic functions in a suitable domain, the analyticity of the stabilizing solution is proved under various hypotheses. In addition, general results on the analytic behavior of stabilizing solutions are developed. I. NONSTANDARD RICCATI EQUATIONS In [10], [7] and elsewhere continuity and real analytic properties of solutions of standard Riccati equations depending on a parameter z have been considered: A(z)∗P (z) + P (z)A(z) +Q(z) = P (z)D(z)P (z) (I.1) In [1], in the context of spatially invariant systems, the question arose as to when the solution P (z) will have a complex-analytic extension to some open connected set Ω. Here we report on some of the results from [6], where a more general version of this problem was studied. The key mathematical problem concerns the existence of analytic solutions to the following nonstandard Riccati equation E(z)P (z) + P (z)A(z) +Q(z) = P (z)D(z)P (z), (I.2) where E(z), A(z), D(z), Q(z) are n×n matrices that are analytic in Ω, a connected open subset of the complex plane. The properties of solutions (I.2) are determined by the Hamiltonian H(z) = [ A(z) −D(z) −Q(z) −E(z) ] . (I.3) We call P (z) a stabilizing solution of (I.2) if the eigenvalues of A(z)−D(z)P (z) have negative real parts for all z ∈ Ω. We cite three main results from [6]. Ruth Curtain is with the Department of Mathematics and Computing Science, University of Groningen, 97400 AV Groningen, The Netherlands [email protected] Leiba Rodman is with the Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA [email protected] Theorem 1.1: Suppose that following two assumptions hold (a) H(z) has no eigenvalues on the imaginary axis, for every z ∈ Ω, (b) for some z0 ∈ Ω, H(z0) is similar to either −H(z0) or −H(z0), then either (A) or (B) holds: (A) There are no stabilizing solutions for any z ∈ Ω; (B) There is an n×n matrix function P (z) which is meromorphic on Ω and such that P (z0) is the unique stabilizing solution whenever z0 is not a pole of P (z). Moreover, there are no stabilizing solutions if z0 is a pole of P (z). 2 Theorem 1.2: If for every z ∈ Ω there is a unique stabilizing solution P (z), then P (z) is analytic in Ω.