Stochastic nonlinear stabilization

The fact that many physical systems are nonlinear and subject to disturbances motivates the study of stochastic nonlinear control, with stabilization as the most basic question. Because of a fundamental technical difficulty in the Lyapunov analysis, research attention since the 1960's has been moved from stabilization to optimization. The development of differential geometric nonlinear control theory in the 1980's and a recent discovery of a simple constructive formula for Lyapunov stabilization for deterministic nonlinear systems has inspired a large amount of research in adaptive, robust, and optimal nonlinear control. These achievements naturally led to re-examining the stochastic nonlinear stabilization problem, which is the topic of this dissertation.; In this dissertation, we first introduce a new set of stability definitions in class $K$ formalism, in order to connect the results in stochastic nonlinear control with the results in modern deterministic nonlinear control literature, and to introduce the new concept of Noise-to-State Lyapunov function. Then we rigorously prove a stochastic version of the convergence result of LaSalle and Yoshizawa. When we consider systems with unknown noise covariance, we extend Sontag's concept of input-to-state stability (ISS) to stochastic systems by introducing the concept Noise-to-State Lyapunov functions (ns-lf). For systems with a control input as well as stochastic noise (affine in both), we extend the ISS-control Lyapunov functions to stochastic systems, introduce ns-control Lyapunov functions (ns-clf), and prove that there exists a feedback law continuous away from the origin that guarantees the controlled system has an ns-lf if there exists an ns-clf. After addressing the stabilization issue, we pursue inverse optimality since direct optimal control problem involves a formidable computational task—solving HJB PDE. We show that if we know an ns-clf, then the inverse optimal control problem is solvable for general stochastic systems affine in control.; Based on these theorems, we proceed with controller design for the stochastic strict-feedback systems—both state-feedback and output-feedback. Using back-stepping design method, we design closed form stabilizing and inverse optimal controllers when the noise has unity intensity. For the system with unknown covariance noise, we develop adaptive stabilization and disturbance attenuation schemes without a priori knowledge of a bound on the covariance.