Convergence of the discrete-time Riccati equation to its maximal solution

If the initial and parameter matrices P₀, S, and Q are assumed to be positive (semi)definite, one pretty well knows necessary and sufficient conditions for the discrete-time matrix Riccati equation in the filtering form P_t+1=AP_tA^T-AP_tB^T (S+BP_tB^T)^-1BP_tA^T +Q to converge to its strong solution. In this paper we work out necessary and sufficient conditions for a more general Riccati equation to converge to its maximal solution, where we do not a priori assume the signature of the P₀, S, and Q. Since the Riccati equation is not genuinely quadratic for detectable systems, which in fact leads to the dimension reduction, we shall discuss the convergence under the condition that the pair (B, A) is observable. We shall also show how dimensions can be reduced for detectable systems. As auxiliary steps we discuss various solvability conditions for the corresponding algebraic Riccati equation