The Kleinman iteration for nonstabilizable systems (Q1402892)

A standard approach to obtaining a positive semidefinite solution (if it exists) of the algebraic matrix Riccati equation (ARE) \(A^*P+PA-PBB^*P+C^*C=0\) is by using the Kleinman iteration procedure. The procedure consists of solving consecutively a sequence of Lyapunov equations \(A^*_iP_{i+1} +P_{i+1}A_i+ C^*_iC_i=0\), \(i=0,1,\dots\), starting with a suitable initial matrix \(P_0\), and represents essentially an application of Newton's method to (ARE). So far in the literature, the stabilizability of \((A,B)\) was assumed to guarantee that the Kleinman iteration converges. In the reviewed paper, the stabilizability hypothesis is disposed of, at the expense of assuming a certain property of the initial matrix \(P_0\). As a result, a new criterion for the existence of positive semidefinite solutions of (ARE) is proved.