Approximation of the Set of Integrable Trajectories of the Control Systems with Limited Control Resources

TL;DR

This paper develops a method to approximate the set of trajectories of affine control systems with limited control resources using finite piecewise-constant controls, ensuring accurate internal approximation under proper discretization.

Contribution

It introduces a step-by-step discretization approach to approximate the set of integrable trajectories for affine control systems with bounded controls.

Findings

Finite piecewise-constant controls approximate the trajectory set.

Proper discretization ensures internal approximation accuracy.

Method applicable to multivariable systems with Urysohn integral equations.

Abstract

In this paper an approximation of the set of multivariable and $L_2$ integrable trajectories of the control system described by Urysohn type integral equation is considered. It is assumed that the system is affine with respect to the control vector. The admissible control functions are chosen from the closed ball of the space $L_2$, centered at the origin with radius $\rho$. Step by step way, the set of admissible control functions is replaced by the set of controls, consisting of a finite number of piecewise-constant control functions. It is proved that under appropriate choosing of the discretization parameters, the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories.