DE Seminar: Madison Christ (UMBC)

Title: Stability and Periodicity in Coupled Hyperbolic Systems

Abstract: We are interested in the emergence and control of resonance in coupled wave-like systems. In uncoupled wave problems, we broadly understand which situations give rise to the (non)existence of resonant solutions. When dealing with a coupled problem, avoiding resonance is less straightforward. We also know that in the context of PDE control, the geometry of wave domains is central issue for PDE analysis. Here we look at systems of adjacent wave chambers with a lower dimensional common interface. In examining the stability properties of the coupled homogeneous Cauchy problem, we can gain insight into the periodic well-posedness (PWP) of the forced system; that is: determining sufficient conditions preventing resonance. By utilizing classical results such as the Lumer-Phillips, Lax-Milgram, and Holmgren’s Unique Continuation theorems, as well as spectral analysis, we will show well-posedness of the Cauchy system and strong stability of the induced semigroup. We then apply a seminal result of Galdi et al. to infer a type of periodic well-posedness. Subsequently, we discuss the challenges of using spectral analysis and/or control estimates to improve our result through the explicit characterization of the class of forcing functions for which resonance will not occur. 

The DE Seminar will take place during the 11am hour, with public questions concluding at 12pm. The committee will examine and confer during the 12pm hour. 

Committee:

Arum Lee 

Matthew Kvalheim 

Andrei Draganescu 

Animikh Biswas 

Justin Webster (committee chair and advisor)