Optimal stabilization in the critical case of a single zero root

Abstract

The problem of the optimal stabilization [1,2] of non-linear controlled systems in the critical case of a single zero root [3–5] is considered 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 the 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 established when the expansion of the integrand in the quality criterion in powers of the phase coordinates and the control forces begin with a positive 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 procedure for calculating the optimal control and the optimal Lyapunov function, which is based on results obtained previously [1, 2, 6, 7] and converges 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 constructed using perturbation methods [8] and estimates of the accuracy of the approximations are indicated.