Construction of a Set of Restricted Inertial Controls for C(1)-Smooth Affine Systems with Multidimensional Control

Abstract

Affine control systems of the class C(1) with multidimensional control are considered. For those systems which can be reduced to linear systems, the inertial synthesis problem is solved, that is, the problem of finding feedback controls satisfying preassigned constraints on a control and its derivatives up to a given order l. The problem of stabilization by use of inertial controls is also solved. The controllability function method is the basis of the investigation. It is shown that each collection f of r nonnegative non-increasing functions which have no less than n1,?,nr points of decrease ( n1+?+nr=n), respectively, and satisfy certain conditions generates a family of controllability functions {?f,?(x)} and a family of controls {uf,?(x)} which transfer an arbitrary point x0 from a certain neighborhood of the origin to the origin in some finite time Tf,?(x0) and satisfy given constraints if ??2l+1. We estimate the time of motion from below and from above. In the limiting case ?=?, the function ?f(x) is a Lyapunov function and uf(x) solves the stabilization problem and satisfies given constraints.