The Feasible Control Lyapunov Function Of Nonlinear Control-affine Systems

In order to make the nonlinear control-affine systems become globally asymptotically stable, it is effective to design a kind of feedback controller by means of Lyapunov function. In addition, this way has drawn great attention in recent years.This paper is concerned with C 0-stabilizable of nonlinear control-affine systems. We introduce a limit condition instead of the small control property and prove that the Sontag formula is continuous with the accumulation point condition. From this, a result similar to Sontag's theorem is concluded.Three works are done in this paper. First, the sufficient and necessary condition about the lower semi-continuousness of feedback control mapping is introduced, which solves the problem on the existence of continuous feedback controller in the nonlinear control-affine systems. Second, the feasible control Lyapunov function is introduced, which restricts the state feedback under permissive scope and makes it more practical. Finally, the feasible control Lyapunov function is replaced by non-strict feasible control Lyapunov function in the nonlinear control-affine systems so that the scope of the control Lyapunov function is expanded.