On the regularization of sliding modes

Approximability of sliding motions for control systems governed by nonlinear finite-dimensional differential equations is considered. This regularity property is shown to be equivalent to Tikhonov well-posedness of a related minimization problem in the context of relaxed controls. This allows the derivation of a general approximability result, which in the autonomous case has an easy to verify geometrical formulation. In the second part of the paper, we consider nonapproximable sliding mode control systems. In the flavor of regularization of ill-posed problems, we propose a method of selection of well-behaved approximating trajectories converging to a prescribed ideal sliding. © 2007 Society for Industrial and Applied Mathematics.