Minimal Time of Null Controllability for the 1D Heat Equation by a Strategic Zone Profile

Cheikh Seck,Mame Libasse Laye Ane

doi:10.4236/jamp.2021.97113

Abstract

The aim of this work is to improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators. In fact, motivated by the work of Khodja on the null controllability of the heat equation and of El Jai on the controllability by the use of strategic zone actuators, we managed, in this work, to improve the minimal time of null controllability to the 1D heat equation. However, the restrictions and difficulties to establish the inequality of coercivity of the parabolic operator, require to seek other methods of internal control. Thus in this paper, a mixed method combining the method of moments and the notion of strategic profile was used to find a better minimal time of null controllability of the 1D heat equation.

Highlights

One of the objectives of the control theory of partial differential equations of evolution is to be interested in how to act on such dynamic systems
The aim of this work is to improve the minimum time of null controllability of the 1D heat equation by using the notion of strategic zone actuators
A mixed method combining the method of moments and the notion of strategic profile was used to find a better minimal time of null controllability of the 1D heat equation

Summary

Introduce the Problem

One of the objectives of the control theory of partial differential equations of evolution is to be interested in how to act on such dynamic systems. The exact controllability of distributed systems has attracted a lot of interest in recent years. This thanks to one of the pioneers J.L. Lions [1] [2] who developed the HUM method (Hilbert Uniqueness Methods). For hyperbolic problems, this method has given important results (Lions [2], Niane [3] [4], Seck et al [5] [6]). By Khodja [11], Tucsnak [12] and Avdonin [13], a means of calculating a minimum cost associated with this minimum time of null controllability has been established

Problem Statement

F T containing

Notations and Definition

Main Theorem

Controllability on the FT Space

Conclusions and Perspectives