Approximate controllability of discrete semilinear systems and applications

Hugo  Leiva; Jahnett  Uzcategui

doi:10.3934/dcdsb.2016023

Article Contents

2016, Volume 21, Issue 6: 1803-1812. Doi: 10.3934/dcdsb.2016023

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Approximate controllability of discrete semilinear systems and applications

Hugo Leiva^1, and
Jahnett Uzcategui^2,

1.
Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803
2.
Universidad de Los Andes, Facualtad de Ciencias, Departamento de Matematica, Merida, 5101

Received: April 30, 2015

Revised: January 31, 2016

Published: July 2016

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Abstract

In this paper we study the approximate controllability of the following semilinear difference equation \[ z(n+1)=A(n)z(n)+B(n)u(n)+f(n,z(n),u(n)), \quad n\in \mathbb{N}^*, \] $z(n)\in Z$, $u(n)\in U$, where $Z$, $U$ are Hilbert spaces, $A\in l^{\infty}(\mathbb{N},L(Z))$, $B\in l^{\infty}(\mathbb{N},L(U,Z))$, $u\in l^2(\mathbb{N},U)$ and the nonlinear term $f:\mathbb{N} \times Z\times U\longrightarrow Z$ is a suitable function. We prove that, under some conditions on the nonlinear term $f$, the approximate controllability of the linear equation is preserved. Finally, we apply this result to a discrete version of the semilinear wave equation.

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Mathematics Subject Classification: Primary: 93B05; Secondary: 93C25.

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