This paper discusses the H∞ control problem for a class of nonlinear stochastic systems with both state-and disturbance-dependent noise. By means of Hamilton-Jacobi equations, both infinite and finite horizon nonlinear stochastic H∞ control designs are developed.Some results on nonlinear H∞ control of deterministic systems are generalized to a stochastic setting. We introduce some useful concepts such as "zero-state observability" and "zero-state detectability" which, together with the stochastic LaSalle invariance principle, yield some valuable consequences in infinite horizon nonlinear stochastic H∞ control. Introduction.In practice, when the exogenous disturbance enters the system, an H ∞ control design is often first considered, when a control law is sought to efficiently eliminate the effect of the disturbance; see [12], [13] and the references therein. Theoretically, study of H ∞ control first starts from the deterministic linear systems, and the derivation of the state-space formulation of the standard H ∞ control leads to a breakthrough; details can be found in the prize-winning paper [17]. From the viewpoint of the state space, the linear H ∞ control problem can be converted into the study of a game-theoretic Riccati equation, and the "completion of square methodology," similar to the linear quadratic (LQ) and linear quadratic Gaussian (LQG) theories, can be applied; see [15], [16], [20], [21], [25], and [30]. Soon after the appearance of [17], the nonlinear H ∞ control problem (on deterministic systems) was investigated by many authors; see [5], [14], [18], and [19]. From the time-domain perspective, an H ∞ norm of the transfer function is nothing else but the L 2 -induced norm of the input-output operator with initial state zero. This important feature makes it possible to develop nonlinear or stochastic H ∞ theory. Van der Schaft [5] made a contribution to the state feedback H ∞ control for nonlinear deterministic systems with infinite time horizon, where a relatively deep tool, i.e., the strict relation between Hamilton-Jacobi equations (HJEs) and invariant manifolds of Hamiltonian vector fields, was applied. Using this tool, he showed that a local solution to the primal nonlinear H ∞ control exists if its linearized H ∞ control problem is solvable. The authors of [18] and [19] dealt with the output feedback H ∞ control of nonlinear deterministic systems with incomplete state information, and a separation principle was obtained. In [5], [18], and [19], the differential geometric approaches

This paper discusses the stochastic 2 control problem with state-dependent noise. By means of the stabilization, exact observability and stochastic detectability of stochastic systems, the infinite horizon stochastic 2 control design is developed. For the finite horizon 2 control problem, our results generalize the corresponding deterministic ones to the stochastic models. Finally, the observer-based suboptimal stochastic 2 control is discussed in which the state variables cannot be measured directly, and a feasible design algorithm is proposed.Index Terms-Coupled Riccati equations, exact observability, state-dependent noise, stochastic detectability, stochastic 2 control.

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Matlab code and test data are available at http://www.ee.nthu.edu.tw/~bschen/quantitative/regulatory_network.htm

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In this paper, the finite horizon mixed H2/H∞ control problem is studied for the systems governed by Itôtype nonlinear stochastic differential equations with state, control and external disturbance-dependent noise. It is shown that the mixed H2/H∞ control under consideration is associated with the four cross-coupled Hamilton-Jacobi equations in nonlinear case. Some necessary and/or sufficient conditions for the existence of finite horizon stochastic H2/H∞ controllers are derived. In particular, some previous results on deterministic and stochastic H∞ (H2/H∞) can be viewed as simple corollaries of our main theorems.