Backstepping Controller Design for Nonlinear Stochastic Systems Under a Risk-Sensitive Cost Criterion

This paper develops a methodology for recursive construction of optimal and near-optimal controllers for strict-feedback stochastic nonlinear systems under a risk-sensitive cost function criterion. The design procedure follows the integrator backstepping methodology, and the controllers obtained guarantee any desired achievable level of long-term average cost for a given risk-sensitivity parameter $\theta$. Furthermore, they lead to closed-loop system trajectories that are bounded in probability, and in some cases asymptotically stable in the large. These results also generalize to nonlinear systems with strongly stabilizable zero dynamics. A numerical example included in the paper illustrates the analytical results.