OPTIMAL STABILIZATION IN THE CRITICAL CASE OF A SINGLE ZERO ROOT

The problem of the optimal stabilization [1, 2] of non-linear controll ed systems in the critical case of a single zero root [3-5] is conside red when the right-hand sides of the equations of the perturbed motion and the integrand in the quality criterion are analytic with respect to the phase coordinates and the control forces. It is assumed that th e right-hand side of the critical equation is multiplied by a critical variable and its expansion begins with the terms of the second order. Sufficient conditions for the solvability of the problem are establis hed when the expansion of the integrand in the quality criterion in po wers of the phase coordinates and the control forces begin with a posi tive definite quadratic form, and it is shown that the optimal control is a non-smooth function of the critical variable and has the form of the permissible control proposed in [5] when constructing stabilizing forces in the critical case of a single zero root. An iterative proce dure for calculating the optimal control and the optimal Lyapunov func tion, which is based on results obtained previously [1, 2, 6, 7] and c onverges for sufficiently small initial perturbations with respect to the non-critical variables, is substantiated. An asymptotic expansion of the optimal result in powers of the critical variable is constructe d using perturbation methods [8] and estimates of the accuracy of the approximations are indicated. (C) 1998 Elsevier Science Ltd. All right s reserved.