Regresar

Approximate controllability of a system of parabolic equations with delay

Abstract:

In this paper we give necessary and sufficient conditions for the approximate controllability of the following system of parabolic equations with delay:{(frac(∂ z (t, x), ∂ t) = D Δ z + L z<inf>t</inf> + B u (t, x), t ∈ (0, r],; frac(∂ z, ∂ η) = 0, x ∈ ∂ Ω, t ∈ (0, r],; z (0, x) = φ{symbol}<inf>0</inf> (x), x ∈ Ω,; z (s, x) = φ{symbol} (s, x), s ∈ [- τ, 0), x ∈ Ω,) where Ω is a bounded domain in RN, D is an n × n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the control u ∈ L2 ([0, r] ; U) = L2 ([0, r] ; L2 (Ω, Rm)) and B ∈ L (U, Z) with U = L2 (Ω, Rm), Z = L2 (Ω ; Rn). The standard notation z<inf>t</inf> (x) defines a function from [- τ, 0] to Rn (with x fixed) by z<inf>t</inf> (x) (s) = z (t + s, x), - τ ≤ s ≤ 0. Here τ ≥ 0 is the maximum delay, which is supposed to be finite. We assume that the operator L : L2 ([- τ, 0] ; Z) → Z is linear and bounded, and φ{symbol}<inf>0</inf> ∈ Z, φ{symbol} ∈ L2 ([- τ, 0] ; Z). To this end: First, we reformulate this system into a standard first-order delay equation. Secondly, the semigroup associated with the first-order delay equation on an appropriate product space is expressed as a series of strongly continuous semigroups and orthogonal projections related with the eigenvalues of the Laplacian operator (A = - frac(∂, ∂2)); this representation allows us to reduce the controllability of this partial differential equation with delay to a family of ordinary delay equations. Finally, we use the well-known result on the rank condition for the approximate controllability of delay system to derive our main result. © 2008 Elsevier Inc. All rights reserved.