On the Kleinman iteration for periodic nonstabilizable systems

In this paper, the problem of finding periodic, Hermitian and nonnegative definite solutions of the differential periodic Riccati equation is considered. These solutions are the limit, P(t) = lim P_i(t), of a sequence of matrix functions obtained by solving a sequence of suitable differential periodic Lyapunov equations. The procedure parallels the well-known Kleinmann technique but it is applied here even for nonstabilizable systems. The convergence of the constructed sequence when the associated system is nonstabilizable is guaranteed by the minimality of P_i(·), in the set of Hermitian and nonnegative definite solutions of the Lyapunov equation which solves P_i(·).