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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 systems in control theory
93D09 Robust stability
93E15 Stochastic stability in control theory
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