## 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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