Approximate Controllability of the Burgers Equation With Impulses and Delay

Abstract

The semilinear heat equation with non instantaneous impulses (NII), memory and delay is considered and its approximate controllability is obtained. This is done by employing a technique that avoids fixed point theorems and pulls back the control solution to a fixed curve in a short time interval. We demonstrate, once again, that the controllability of a system is robust under the influence of non instantaneous impulses, memory and delays. In support, a numerical example with simulation for a linear heat equation is given to validate the obtained controllability result for the linear part. Finally we present some open problems and a possible general framework to study the controllability of non instantaneous impulsive (NII) semilinear equations. Contents 11 6.3. Open Problem 12 References 12

Controllability of the Burgers Equation Under the Influence of Impulses, Delay and Nonlocal Conditions

“…In this section, we shall prove the main result of this paper, the interior approximate controllability of the Burguers equation with impulses, delay and nonlocal conditions given by (1), which is equivalent to prove the approximate controllability of the system (8). In this regard, according to [1,12,15,16,17], for all φ ∈ C and u ∈ L 2 (0, τ ; U ) the nonlocal Cauchy problem (8) admits only one mild solution z ∈ P C t 1 ..t P ([−r, τ ]; Z) given by…”

“…are smooth enough such that the system (1) admits a unique mild solutions for each control u according to [15,16,17], and satisfy the following estimates for z, w, u, v ∈ R:…”

Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

“…[36,44], etc). In the past, several authors investigated the existence results of the impulsive and functional (delay) differential equations, see for example, [3,4,24,25,33], etc and the references therein.…”

“…Therefore, it is important to study the problem of approximate controllability for the infinite dimensional control systems. In the past several years, there have been many developments on the approximate controllability of the deterministic and stochastic impulsive systems with delay, see for instance, [3,22,25,31,33,43,47,48,55], etc. In these works, the set of sufficient conditions of the approximate controllability of semilinear systems were established by invoking the so-called resolvent operator condition introduced in [8], whenever the corresponding linear system is approximately controllable.…”

“…with the control u λ defined in (33). It is clear from the definition of F λ that for λ > 0, the fixed point of F λ is a mild solution of the system (1).…”