Continuity of the Algebraic Riccati Equation for Stochastic Linear Systems

The algebraic Riccati equation plays the central role in the computation of the optimal control in the infinite time deterministic and stochastic linear regulator problem (Theorem 1). The problem of adaptive control of stochastic system combines the problems of identification and control of unknown system [3]. The most desirable solution of a stochastic adaptive control problem is to exhibit a family of estimators that is strongly consistent and to use these estimates to obtain a family of controls that suitably converge to the optimal control for the true system to obtain the self tuning and self optimizing properties. The continuity of the algebraic Riccati equation as a function of its coefficients is essential for the verification that a family of certainty equivalent controls based on a strongly consistent family of estimates of the unknown parameters provides the self tuning and self optimizing properties. In [3] an adaptive control problem for stochastic linear regulator is solved in this case when the unknown parameters appear only in the linear transformations of the state. This solution required the verification of the continuity of the solution of the algebraic Riccati equation as a function of its coefficients. This continuity is verified assuming that the unknown systems are reachable. However, it is desirable to replace this assumption by some more natural system theoretic assumption, such stabilizability. This is the purpose of this paper. In the paper [2] the asymptotic behaviour of the algebraic Riccati equation is consider when its coefficients are stabilizability and converge to the unstabilizability coefficients. Key-Words: Linear systems; optimal control; algebraic Riccati equation. 1 This research was supported by KBN Poland under Grant 7T11A 021 20 1. Preliminaries Consider stochastic control system (1) ( ) t t t t Cdw dt Bu Ax dx + + = , x x 0 ≡ , where x R t n ∈ , u R t m ∈ , A L R R n n ∈ ( , ) , B L R R m n ∈ ( , ), C L R R n n ∈ ( , ), w t t , ≥ 0 is a standard R n -valued Wiener process and x x R n 0 ≡ ∈ . The cost functional for finite time horizon is defined as (2) + = T T t T x Fx u x J , ) , ( 0 ( ) + + E Qx x Ru u ds s s s s T , , 0 , where Q L R R n n ∈ ( , ), R L R R m n ∈ ( , ) F L R R n n ∈ ( , ) are symmetric and Q, F are nonnegative, R positive definite and defined as (3) = ) , ( 0 t u x J ( ) = + →∞ lim , , T s s s s T