Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Abstract

We consider the numerical solution of the rational algebraic Riccati equations in R n , arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n 3 ) computational complexity and O(n 2 ) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided. Introduction.In this paper, we consider the numerical solution of rational Riccati equations (RREs) arising in stochastic optimal control. Linear time-invariant (LTI) systems [2,46,53] with stochastic components will be considered. These can be considered to be local approximations to more general nonlinear and time-variant systems. The theory of LTI systems has been vital in control system theory and has contributed successfully to many real-life applications. In other words, the solution of the difficult RREs, which leads to the optimal control of the corresponding LTI systems with stochastic components, is of some importance to control engineers in many applications.

Stochastic algebraic Riccati equations are almost as easy as deterministic ones

“…As we can see, the stochastic AREs are still algebraic, and it is quite natural to ask whether algebraic methods could be developed to solve them. However, limited by lack of clear algebraic structures, to the best of the authors' knowledge, nearly all of the existing algorithms are based on the differentiability or continuity of the equations, such as Newton's method [9,8], modified Newton's method [15,21,7], Lyapunov/Stein iterations [12,22,26], comparison theorem based method [13,14], LMI's (linear matrix inequality) method [25,19], and homotopy method [28].…”