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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: {zt = DΔz + b1(x)u1 + ⋯ + bm(x)um, t ≥ 0. z ∈ ℝn, {z = 0, on ∂Ω where Ω is a sufficiently smooth bounded domain in ℝN, bi ∈ L2 (Ω: ℝn), the control functions ui ∈ L2 (0, t1; 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 nγj × m matrices DPj and Pj B, i.e. Rank [Pj B: DPj B: D2 Pj B:... Dnγj-1 Pj B] ≠ nγj, where PjBu = Pjb1 u1 + ⋯ + Pj bm um. 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.