Premium
Controllability of Semilinear Systems of Parabolic Equations with Delay on the State
Author(s) -
Carrasco A.,
Leiva Hugo,
Merentes N.,
Sanchez J. L.
Publication year - 2015
Publication title -
asian journal of control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.769
H-Index - 53
eISSN - 1934-6093
pISSN - 1561-8625
DOI - 10.1002/asjc.1162
Subject(s) - bounded function , controllability , mathematics , eigenvalues and eigenvectors , diagonal matrix , domain (mathematical analysis) , diagonal , matrix (chemical analysis) , operator (biology) , state (computer science) , mathematical analysis , nonlinear system , function (biology) , pure mathematics , physics , geometry , quantum mechanics , biochemistry , materials science , chemistry , repressor , algorithm , evolutionary biology , biology , transcription factor , composite material , gene
In this paper we prove the approximate controllability of the following semilinear system parabolic equations with delay on the state variable∂z ( t , x ) ∂t= D Δz + L z t + Bu ( t , x ) + f ( t , z ( t , x ) , u ( t , x ) ) ,t ∈0 , r,∂z ∂η= 0 ,t ∈0 , r,x ∈ ∂ Ω ,z ( 0 , x ) = φ 0 ( x ) ,x ∈ Ω ,z ( s , x ) = ψ 0 ( s , x ) ,s ∈− τ , 0,x ∈ Ω ,where Ω is a bounded domain inR N , D is a n × n non diagonal matrix whose eigenvalues are semi‐simple with non negative real part, the control u belongs toL 2 ( [ 0 , r ] ; U )( U = L 2 ( Ω , R m ) ) and B is a n × m matrix. Here τ ≥0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L 2 ([− τ ,0]; Z )→ Z is linear and bounded with Z = L 2 ( Ω ; R n ) and the nonlinear function f :[0, r ] × I R n × I R m → I R n is smooth and bounded.