Frequency domain conditions for the existence of Bohr almost periodic solutions in evolution equations
We consider a control problem for the heating process of an
elastic plate. The heat flux within the plate is modeled by the heat equation with nonlinear Neumann boundary conditions according to Newton's law. As input at a part of the boundary we take the nonlinearly transformed and modulated heat production of a separate heater which is given by a nonlinear Duffing-type ODE.
This ODE depends on measurements of the temperature within the plate and on Bohr resp. Stepanov almost periodic in time forcing terms. The physical problem is generalized to a bifurcation problem for non-autonomous evolution systems in rigged Hilbert spaces. Using Lyapunov functionals, invariant cones and monotonicity properties of the nonlinearities in certain Sobolev spaces, we derive frequency domain conditions for the existence
and uniqueness of an asymptotically stable and almost periodic in time temperature field.