Journal of Dynamical and Control Systems

, Volume 21, Issue 4, pp 513–538

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

Article

## 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 n 1,…,n r points of decrease ( n 1+…+n r =n), respectively, and satisfy certain conditions generates a family of controllability functions {Θ f,α (x)} and a family of controls {u f,α (x)} which transfer an arbitrary point x 0 from a certain neighborhood of the origin to the origin in some finite time T f,α (x 0) and satisfy given constraints if α≥2l+1. We estimate the time of motion from below and from above. In the limiting case α=, the function Θ f (x) is a Lyapunov function and u f (x) solves the stabilization problem and satisfies given constraints.

## Keywords

C(1)-smooth nonlinear control systems Synthesis problem Controllability function method Stabilization Feedback inertial control

## Mathematics Subject Classifications (2010)

93C10 93B50 93D15 34D20 93D30 93B52

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