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:where Ω is a bounded domain in R N , D is an n × n nondiagonal matrix whose eigenvalues are semi-simple with nonnegative real part, the controlHere 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, and φ 0 ∈ Z , φ ∈ L 2 ([−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 = − ∂ ∂ 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.

Controllability of Semilinear Systems of Parabolic Equations with Delay on the State

“…For the proof of the following results presented in the equality plays an important role. Theorem The system is approximately controllable on [0, δ ] if, and only if, each of the following systems is approximately controllable on [0, δ ] …”

“…In this section we choose a Hilbert space where system can be written as an abstract functional differential equation; to this end, we consider the following additional hypothesis : H3.The matrix D is semi simple (block diagonal) and the eigenvalues d i ∈ C of D satisfy Re( d i )≥0.…”

“…Now, we shall write the system as an ordinary differential equations in the space (See ): If we define the following operators by where the equation can be written as an ordinary differential equation in the Hilbert space as follows: and the corresponding linear equation as follows …”

“…In this section we choose a Hilbert space where system (I.1) can be written as an abstract functional differential equation; to this end, we consider the following additional hypothesis [3,7,9,13]:…”

Approximate controllability and regularity for nonlinear differential equations

“…The controllability results for linear control systems have been proved by many authors, and several authors have extended these concepts to infinite dimensional semilinear system (see [6][7][8]). In recent years, as for the controllability for semilinear differential equations, Carrasco and Lebia [9] discussed sufficient conditions for approximate controllability of parabolic equations with delay, and Naito [10] and other authors [6][7][8]11] proved the approximate controllability under the range conditions of the controller B.…”