Controllability of a system of parabolic equations with non-diagonal diffusion matrix

Abstract:

In this paper we give a necessary and sufficient algebraic condition for the approximate controllability of the following system of parabolic equations with Dirichlet boundary condition: {z<inf>t</inf> = DΔz + b<inf>1</inf>(x)u<inf>1</inf> + ⋯ + b<inf>m</inf>(x)u<inf>m</inf>, t ≥ 0. z ∈ ℝⁿ, {z = 0, on ∂Ω where Ω is a sufficiently smooth bounded domain in ℝ^N, b<inf>i</inf> ∈ L² (Ω: ℝ<inf>n</inf>), the control functions u<inf>i</inf> ∈ L² (0, t<inf>1</inf>; ℝ): i = 1, 2,..., m and D is an n × n non-diagonal matrix whose eigenvalues are semi-simple with positive real part. This algebraic condition is checkable since it is given in terms of the nγ<inf>j</inf> × m matrices DP<inf>j</inf> and P<inf>j</inf> B, i.e. Rank [P<inf>j</inf> B: DP<inf>j</inf> B: D² P<inf>j</inf> B:... D^nγj-1 P<inf>j</inf> B] ≠ nγ<inf>j</inf>, where P<inf>j</inf>Bu = P<inf>j</inf>b<inf>1</inf> u<inf>1</inf> + ⋯ + P<inf>j</inf> b<inf>m</inf> u<inf>m</inf>. Finally, this result can be applied to those systems of partial differential equations that can be rewritten as a diffusion system (see de Oliveira, 1998). © Institute of Mathematics and its Applications 2005: All rights reserved.

Año de publicación:

2005

Keywords:

• Parabolic equation
• Algebraic condition
• Approximate controllability

Fuente:

scopus

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