Floquet exponents and stabilizability in time-periodic parabolic systems
The paper deals with linear \(2\times 2\) parabolic systems of the form \[ U_ t(t)=A(a(t)U(t))+b(t)U(t), \] where A is some second order elliptic operator with suitable boundary conditions and a(\(\cdot)\), b(\(\cdot)\) are T-periodic matrix valued functions such that det(a(\(\cdot))\geq 0\), \(tr(a(\cdot))>0\). The authors give formulae for the accumulation points of the spectra of the evolution operators \(u(\cdot,s)\mapsto u(\cdot,s+T)\). In the case of invertible a(\(\cdot)\), this turns out to be only the origin, while for singular a(\(\cdot)\) there is an additional positive accumulation point. The results apply to questions of stabilizability of solutions of such problems and to the rates of stabilization. Examples include strongly damped wave equations, linear reaction-diffusion systems, and the linearized nerve axon equations.
scientific article; zbMATH DE number 4190266