Controllability of Semilinear Systems with Multiple Variable Delays in Control

Jerzy Klamka

doi:10.3390/math8111955

Abstract

In the paper semilinear, finite-dimensional, control systems with multiple time variable point delays in admissible controls are considered. Using Rothe’s fixed-point theorem, sufficient controllability conditions are formulated. The results of the paper are generalization to many time variable delays in control, of the results published recently.

Highlights

The last decades have seen a continually growing interest in controllability theory of dynamical control systems with different delays [1,2,3,4,5]
It should be stressed that using fixed-point techniques only sufficient, but not necessary controllability conditions may be formulated and proved
Let us recall that semilinear dynamical control systems contain linear and pure nonlinear parts in the differential state equations

Summary

Introduction

The last decades have seen a continually growing interest in controllability theory of dynamical control systems with different delays [1,2,3,4,5]. Up to the present time the problem of controllability for continuous-time and discrete-time linear dynamical systems has been extensively investigated in many papers and different controllability conditions are well known This is not true for the semilinear or generally for nonlinear dynamical control systems, especially with different types of delays in control and state variables. The main purpose of this paper is to study the relative controllability of semilinear control systems with multiple variable delays in admissible controls using Rothe’s fixed-point theorem. It should be pointed out, that the sufficient controllability conditions formulated and proved in the paper are a generalization to the many variable delays in the control case of the results recently published in [14]

System Description

Preliminaries

Controllability Conditions

Conclusions