Two Types Of Stochastic Nonlinear Systems Control Problems

State-feedback stabilization for a class of high-order stochastic nonlinear systems, and output-feedback stabilization for a class of stochastic nonlinear systems with linearly growth condition of unmeasurable states are considered in the paper, which is composed of the following two parts.1. The problem of state-feedback stabilization for a class of high-order stochastic nonlinear systems.Consider the following high-order stochastic nonlinear systems described by.where and are the measurable state and control input, respectively;ω∈R^r is independent standard Wiener process vector; p≥1 is an odd integer; is smooth function which satisfies the following Assumption 1, andφ_i(0) = 0.Assumption 1: For , there exists a known nonnegative smooth functionρi(·), such that:The objective of this part is to design a smooth state-feedback controller under Assumption 1, such that the closed-loop system is globally asymptotically stable in probability.2. The problem of output-feedback stabilization for a class of stochastic nonlinear systems with linearly growth condition of unmeasurable states. Consider the following stochastic nonlinear systems:where x = [x₁,…,x_n]^T∈Rⁿ, u∈R and y∈R are unmeasurable state vector, control input and measurable output, respectively;ω∈R^m is independent standard Wiener process vector;φi(x) : Rⁿ�'R^m is continuous function which satisfies the following assumption condition, andφi(0) = 0.Assumption 2: Forφi(x), i = 1,…, n, there exists a constant c≥0 satisfyingThe objective is to design an observer and a output-feedback controller:under Assumption 2, such that the closed-loop system is globally asymptotically stable in probability.