Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Approximate controllability of discrete semilinear systems and applications

Pages: 1803 - 1812, Volume 21, Issue 6, August 2016      doi:10.3934/dcdsb.2016023

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Hugo Leiva - Louisiana State University, Department of Mathematics, Baton Rouge, LA 70803, United States (email)
Jahnett Uzcategui - Universidad de Los Andes, Facualtad de Ciencias, Departamento de Matematica, Merida, 5101, Venezuela (email)

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.

Keywords:  Difference equations, approximate controllability, wave equation.
Mathematics Subject Classification:  Primary: 93B05; Secondary: 93C25.

Received: May 2015;      Revised: February 2016;      Available Online: June 2016.