On the Robustness of the Integrable Trajectories of the Control Systems with Limited Control Resources

TL;DR

This paper investigates the robustness of trajectories in nonlinear control systems with limited control resources, demonstrating that trajectories remain stable and can be approximated even when control resources are partially exhausted.

Contribution

It establishes the robustness of system trajectories under resource constraints and shows they can be approximated by fully utilizing available control resources.

Findings

Trajectories are robust to remaining control resources.

Any trajectory can be approximated by using the full control resource.

The system's trajectories are stable under resource limitations.

Abstract

The control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed ball of the space $L_q\left(\Omega;\mathbb{R}^m\right),$ $q>1,$ with radius $r$ and centered at the origin. The trajectory of the system is defined as $p$-integrable multivariable function from the space $L_p\left(\Omega;\mathbb{R}^n\right),$ $\frac{1}{q}+\frac{1}{p}=1,$ satisfying the system's equation almost everywhere. It is shown that the system's trajectories are robust with respect to the remaining control resource. Applying this result it is proved that every trajectory can be approximated by the trajectory obtained by full consumption of the total control resource.