Discrete-time Riccati equations of stochastic control

Abstract

In this chapter a class of discrete-time backward nonlinear equations defined on the ordered Hilbert space S ^N _n is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time nonlinear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in Sections 5.3-5.6 provide sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations such as the maximal solution, the stabilizing solution, and the minimal positive semidefinite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities, LMIs. One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by positive operators on some ordered Hilbert spaces developed in Chapter 2. In Section 5.7 an iterative procedure is proposed for the computation of the maximal and stabilizing solution of the discrete-time backward nonlinear equations under consideration. In the last part of this chapter, one shows how the obtained results can be specialized to derive useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control in the stochastic framework.