TY - JOUR
A2 - Grinfeld, Michael
AU - Leiva, Hugo
AU - Merentes, N.
AU - Sanchez, J.
PY - 2013
DA - 2013/11/03
TI - Approximate Controllability of a Semilinear Heat Equation
SP - 424309
VL - 2013
AB - We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: zt(t,x)=Δz(t,x)+1ωu(t,x)+f(t,z(t,x),u(t,x)) in (0,τ]×Ω,z=0, on (0,τ)×∂Ω,z(0,x)=z0(x), x∈Ω, where Ω is a bounded domain in ℝN (N≥1), z0∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and the nonlinear function f:[0,τ]×ℝ×ℝ→ℝ is smooth enough, and there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that |f(t,z,u)-az|≤c|u|β+b, for all u,z∈ ℝ,|u|,|z|≥R. Under this condition, we prove the following statement: for all open nonempty subset ω of Ω, the system is approximately controllable on [0,τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the final state z1 at time τ>0.
SN - 2356-7082
UR - https://doi.org/10.1155/2013/424309
DO - 10.1155/2013/424309
JF - International Journal of Partial Differential Equations
PB - Hindawi Publishing Corporation
KW -
ER -