In the study of control and stabilization of dynamic elastic systems, a significant challenge is the ability to rigorously address whether linked dynamic structures can be controlled using boundary feedback alone. When a structure is composed of a number of interconnected elastic elements or is modelled by a system of coupled partial differential equations, the behavior becomes much harder to both predict and to control. Structures composed of multiple layers or components of different dimensions pose serious challenges because the energy transferred through the interfaces between components can lead to uncontrollable behavior. This talk focuses on issues that arise when attempting to understand the control and stability of such complex systems.

In the talk, we will consider a multi-layered structure-fluid interaction, in which the structure and fluid PDE components each evolve on distinct domains. Here, the structure is composed of a thick and thin layer: the thick layer is described by a three dimensional wave equation; the thin layer is described by a two dimensional wave equation and constitutes the boundary interface between the "thick" wave equation and three dimensional heat or fluid component equation. With respect to this model, preliminary results of wellposedness and asymptotic stability will be presented.

This talk is concerned with the existence of periodic solutions of a viscous Burgers’ equation when a forced oscillation is prescribed. We establish the existence theory by contraction mapping in L-2 space on [0,1] with smooth property. Asymptotical periodicity is obtained as well, and the periodic solution is achieved by passing time limit to infinity.

This talk focuses on a structural acoustic interaction system consisting of a semilinear wave equation defined on a smooth bounded domain \( \Omega\subset {\mathbb R}^3 \) which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of \( \Omega \). In particular, the source terms acting on the wave and plate equations are allowed to have arbitrary growth order. We employ a standard Galerkin approximation scheme to establish a rigorous proof of the existence of local weak solutions. In addition, under some conditions on the parameters in the system, we prove such solutions exist globally in time and depend continuously on the initial data.

The MGT equation is a model describing acoustic wave propagation and arises as a model of high-frequency ultrasound (HFU) waves. The dynamic response of MGT depends on the physical parameters, in particular, the relaxation parameter \(\tau\), which accounts for the finite speed of propagation. Since τ is relatively small, it is important to trace the dynamics with vanishing parameter \(\tau \rightarrow 0\). It is shown that the decay rates for the finite energy are preserved uniformly. The corresponding result provides not only a robust stabilizing mechanism for HFU waves but also leads to a new ”higher energy” stability estimates.

I will discuss a joint work with M. Stanislavova where we use semigroup methods to establish polynomial in time (and depending on the spectrum of the generator, possibly uniform) stability in the \(L^2\) operator norm for semigroups generated by certain Hamiltonian linearizations.

The nonlinear Schrodinger equation (NLS) and the KdV equation are two dispersive PDE that admit a nonlinear smoothing effect, which can be used to obtain polynomial-in-time bounds on the solution to the equation, as proven by Erdogan and Tzirakis. In this talk, I will discuss a nonlinear smoothing effect for the Benjamin-Ono equation, an analogue of the KdV with lower dispersion. This smoothing effect also leads to polynomial-in-time bounds on the solution to the Benjamin-Ono equation. This is a joint work with Dionyssios Mantzavinos, Seungly Oh, and Atanas Stefanov.