Exact interior controllability of magnetoelastic plates by means of purely magnetic actuation

Abstract

We establish exact interior controllability for a two-dimensional magnetoelastic plate system with control acting solely in the magnetic field equation. The main result shows that exact controllability of the coupled system is achievable in arbitrarily small time $T > 0$, despite the control acting only the magnetic dynamics. This extends the principle of indirect control - previously demonstrated for thermoelastic systems [Avalos-2000] - to the magnetoelastic regime, revealing that steering the mechanical plate displacement through magnetic actuation alone is possible. The analysis employs the operator-theoretic multiplier method adapted to handle vectorial fields with divergence-free constraints and non-self-adjoint coupling. The approach requires several technical components: norm equivalences for divergence-free vector fields, analysis of non-self-adjoint coupling operators, integration by parts identities for $\rot\rot$ systems, and trace regularity results. The proof follows three steps: establishing a trace regularity result for the adjoint system, deriving an energy estimate via the multiplier method, and using a compactness-uniqueness argument to eliminate lower-order terms. This work provides the first controllability result for magnetoelastic systems and extends the indirect control framework from thermoelasticity to this setting. The techniques developed here are applicable to the control-theoretic investigation of magnetically-coupled elastic systems, with potential applications in smart materials, damping devices, and electromagnetic actuators.