Finite dimensional observers and compensators for thermoelastic systems with boundary controls and point observations

A thermoelastic control system defined on a bounded domain Ω ⊂ Rⁿ is considered. The free dynamics is generally unstable. The control actions are defined via bending moments applied to the edge of the plate and the observations taken are pointwise measured in the interior of Ω. Both control operator and observation operator are fully unbounded, with a combined degree of unboundedness greater than one, the super-critical case. We construct an approximation theory, based on FEM, which leads to a finite-dimensional dynamic compensator, possessing all the expected desirable stability properties, once it is inserted into the original system. In particular, it preserves the same margin of stability of the continuous problem, except, perhaps, for an arbitrary ε>0