Boundary control and hidden trace regularity of a semigroup associated with a beam equation and non-dissipative boundary conditions

Irena Lasiecka, University of Virginia
Richard Marchand, United States Air Force Academy
T. I.M. Mcdevitt, Elizabethtown College


The aim of this work is to develop analyses suitable for studying beam and plate equations equipped with non-monotone feedback boundary conditions. While the analysis of monotone structures is well known by now and based on applications of a suitable version of monotone semigroup theory, in the non-monotone case detailed analysis (microlocal) on the boundary seems necessary. In fact, it is shown that boundary traces display a rather peculiar type of "hidden regularity" which is instrumental in showing that (i) the resulting semigroup is of Gevrey's class, and (ii) the associated control system is "well-posed" within a standard finite energy space and with controls that are not necessarily collocated. The result is valid for finite and infinite horizon control problems. This is the first control result of this type in hyperbolic-like dynamics and a non-collocated framework. The unexpected beneficial role of breaking monotonicity is proved to have critical influence on the well-posedness of the control system with control actuators placed at different boundary conditions than the damping. Numerical simulations reveal spectral properties of the operators complementing theoretical findings. ©Dynamic Publishers, Inc. Beam and plate equations, non-monotone feedback boundary conditions, hidden boundary regularity, Gevrey's class, spectral analysis, admissible control systems.