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State feedback ${H}_{\infty }$ control for a class of nonlinear stochastic systems. (English) Zbl 1157.93019

This paper investigates an ${H}_{\infty }$ control problem for a class of nonlinear stochastic systems with state- and disturbance-dependent noise. This problem is discussed in both the finite and infinite horizon cases. In this regard, an approach involving Hamilton-Jacobi equations is used in order to develop infinite and finite horizon nonlinear stochastic ${H}_{\infty }$ control designs.

In the main, the authors generalize some results on nonlinear ${H}_{\infty }$ control for deterministic systems to a stochastic setting. In order to treat the infinite horizon nonlinear stochastic ${H}_{\infty }$ control problem, they introduce definitions for the concepts of “zero-state observability” and “zero-state detectability.” Another tool to solve the aforementioned problem is the stochastic LaSalle invariance principle.

Reviewer’s remark: The paper is well-written and of interest to experts in both ${H}_{\infty }$ control as well as nonlinear stochastic systems.

##### MSC:
 93C10 Nonlinear control systems 93D09 Robust stability of control systems 93E15 Stochastic stability