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Energy decay and a transmission problem in electromagneto-elasticity. (English) Zbl 1052.93025

The authors study a problem which is of great importance in future designs of structures made of “smart materials”, specifically of the controllability of multilayered isotropic materials having piecewise constant coefficients in a bounded domain. The dynamic system considered here obeys the equations: \[ \rho^k u^k_{tt}= -\sum_{ij} \partial/\partial x_i(A^k_{ij} \partial u^k/\partial x_j)+ \alpha(x)\,\text{curl}(a^k(x) E^k)= 0,\;x\in \Omega,\;t\in [0,\infty). \] Here \(\alpha\) is the linking coefficient, \(a\) is the electric permeability, and \(\rho\) is the density. Superscript \(k\) denotes the \(k\)th layer of the material.
The remaining equations are the classical Maxwell equations in a medium with variable (distributed) properties: \[ E_t- \text{curl}(b(x)H)- \text{curl}(a(x)u_t)= 0,\;H_t- \text{curl}(a(x)E)= 0,\;\text{div}(E)= \text{div}(H)= 0. \] The elastic operator \(L= \sum_{ij} \partial/\partial x_i(A^k_{ij}\partial u^k/\partial x_j)\), can also be written in terms of the Lamé coefficients \(\lambda\) and \(\mu: L= \mu\Delta+ (\lambda+ \mu)\nabla\text{\,div}\), somewhat simplifying the arguments. All coefficients are assumed to be piecewise constant positive functions.
Using D. L. Russel’s ideas of proving controllability by linking it to stability, the authors prove exact boundary controllability for this system.

MSC:

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
74F15 Electromagnetic effects in solid mechanics
78A25 Electromagnetic theory (general)
93D20 Asymptotic stability in control theory
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