Approximability of Sliding Modes and Regularization

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 minimisation problem in the context of relaxed controls. This allows us to give 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 non-approximable sliding mode control systems. In the flavour of regularization of ill-posed problems, we propose a method of selection of well-behaved approximating trajectories converging to a prescribed ideal sliding