Optimal Control and Stability Analysis of Nonlinear Control-Affine Systems

Abstract

Abstract In this paper we will discuss the problems of optimal control and stabilization for nonlinear control-affine systems of the form x ˙ = A ( x ) + B ( x ) u where x = x ( t ) , x ∈ R n , u ∈ R , | u | ≤ 1 and vector functions A(x), B(x) are assumed to be smooth in the domain D ⊂ ℝ n , 0 ∈ D, A(0) = 0. We give an in-depth analysis of optimal control problems for nonlinear dynamical systems and then consider for the above systems the problem of synthesis of continuous control u = u(x), u(0) = 0, that stabilizes the system at the equilibrium point (x, u) = (0, 0). The solution to the problem is based on the transformation of the system to canonical form, and on nonlinear stabilization of the system.