Approximate Controllability of a Semilinear Heat Equation

Abstract

We apply Rothe's type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation:

Controllability of the semilinear Benjamin-Bona-Mahony equation

“…In the particular case that a = 0 and b = 1 the operator H define by (4.4) is compact and Rang(H) is compact set (see [3]) , and as a consequence we obtain the interior approximate controllability of the semilinear heat equation (see [12]). …”

Approximate Controllability of Semilinear Impulsive Evolution Equations

“…The following lemma holds in general for a linear bounded operator : → between Hilbert spaces and (see [4,11,12]). …”

“…This paper has been motivated by the works done in Bashirov and Ghahramanlou [1], Bashirov and Jneid [2], and Bashirov et al [3], where a new technique to prove the controllability of evolution equations without impulses is used avoiding fixed point theorems, and the work done in [4].…”

“…To our knowledge, there are a few works on approximate controllability of impulsive semilinear evolution equations worth mentioning: Chen and Li [8] studied the approximate controllability of impulsive differential equations with nonlocal conditions, using measure of noncompactness and Monch fixed point theorem and assuming that the nonlinear term ( , ) does not depend on the control variable, and Leiva and Merentes in [4] studied the approximate controllability of the semilinear impulsive heat equation using the fact that the semigroup generated by Δ is compact.…”

“…Specifically, we can assume the following hypotheses: This case is similar to the semilinear impulsive heat equations studied in [4], where the authors use conditions (a) and (b), the compactness of the semigroup generated by the Laplacian operator Δ, and Rothe's fixed point theorem to prove the approximate controllability of the system on [0, ].…”