A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

doi:10.1093/imamci/20.4.393

IMA Journal of Mathematical Control and Information 2003 20(4):393-410; doi:10.1093/imamci/20.4.393
© 2003 by Institute of Mathematics and its Applications

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A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

Hugo Leiva¹

¹ School of Mathematics-CDSNS, Georgia Tech, Atlanta, GA 30332, USA

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following thermoelasticplate equation with Dirichlet boundary conditions

w_tt + ${Delta}$ ²w + ${alpha}$ ${Delta}$ ${theta}$ = a₁(x)u₁ + ... + a_m(x)u_m, t $≥$ 0, x ${Omega}$ ,

${theta}$ _t – ß ${Delta}$ ${theta}$ – ${alpha}$ ${Delta}$ w_t = d₁(x)u₁ + ... + d_m(x)u_m,t $≥$ 0, x ${Omega}$ ,

${theta}$ = w = ${Delta}$ w = 0, t $≥$ 0, x ${partial}$ ${Omega}$ ,

where ${alpha}$ $!=$ 0, ß > 0, ${Omega}$ is a sufficiently regular boundeddomain in R^N, a_i, d_i, L² ( ${Omega}$ ; R), the control functions u_i L²(0, t₁; R); i = 1, 2, ..., m. This condition is easy to checkand is given by

Rank [P_jB ${vdots}$ A_jP_jB ${vdots}$ A²_jP_jB ${vdots}$ ... A^3_j–1_jP_jB] = 3_j,BU=b₁U₁+...+b_mU_m,b_i=[0, a_i, d_i], A_j=[0, – ${lambda}$ ²_j, 0, 1, 0, – ${alpha}$ ${lambda}$ _j, 0, ${alpha}$ ${lambda}$ _j, –ß ${lambda}$ _j]P_j, j $≥$ 1,

where ${lambda}$ _j, S are the eigenvalues of – ${Delta}$ with Dirichlet boundarycondition and P_j, S are the projections on the correspondingeigenspace.

Keywords: thermoelastic plate equation; algebraic condition; approximate controllability.

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