On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay

An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued $C^{0}$- and BV-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using the Lyapunov's indirect method. The singular limit $\tau \to 0$ is further studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.Comment: 34 pages, 4 figures, 1 set of Matlab code

Control Strategies for the Fokker-Planck Equation

Using a projection-based decoupling of the Fokker-Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup

On the Stabilization of Infinite Dimensional Semilinear Systems

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given