The behaviour of extremals in the neighbourhood of singular regimes and non-smooth Liapunov functions in optimal control problems

We consider a wide class of multi-dimensional control systems with bounded scalar control. This class consists of systems which are perturbations of the canonical control system. Our main assumption is that these perturbations are small with respect to the action of Fuller group. We prove the existence of an optimal solution of the perturbed multi-dimensional Fuller problem. We show that the optimal trajectory attains equilibrium point within finite time. For some class of perturbations we prove that the optimal control switches infinitely in a finite time interval when the solution approaches a singular trajectory. As a mechanical application of our theory we consider problems of controlling robots. In these examples we find singular regimes of high order and optimal chattering trajectories. We use Liapunov's direct method for solving the stabilisation problem for nonlinear control systems. We propose a new method (the so-called “cut-off” method) for constructing non-smooth Liapunov functions. In the neighbourhood of the origin the obtained Liapunov functions are of the same order as the time-optimal function for the canonical system. By using this method we show that there exists a local synthesis which steers some neighbourhood of the equilibrium point to the equilibrium point. This synthesis is the same for all perturbed systems that belong to the class of systems under consideration. The time to reach the equilibrium point for the perturbed system has the same order as for the non-perturbed system.