About the controlability of some equations in Cardiology, Biology, Fluid Mechanics, and Viscoelasticity

In this thesis we analyze the properties of controllability and observability for selected partial differential equations which model various phenomena in cardiology, biology, fluid mechanics and viscoelasticity. We begin, in chapter 2, with the analysis of the uniform controllability of families of linear coupled parabolic systems approximating parabolic-elliptic systems. We prove, under appropriate assumptions on the coupling terms, the uniform, with respect to the degenerating parameter, null controllability of the family when only one control is acting on the system. In chapter 3, we analyze the uniform null controllability of a family of nonlinear reaction-diffusion systems approximating a nonlinear parabolic-elliptic system model- ing electrical activity in the cardiac tissue. Combining Carleman estimates and energy inequalities, we prove the uniform null controllability of the family by means of a single control. Chapter 4 studies the controllability of the parabolic Keller-Segel system of chemo- taxis which converges to its parabolic-elliptic version. We show that this nonlinear coupled parabolic system is locally uniformly controllable around a solution of the parabolic-elliptic system when the control is acting on the chemical component. In chapter 5, we consider the wave equation with both a viscous Kelvin-Voigt and a frictional damping as a model of viscoelasticity. Decomposing the system in its parabolic and hyperbolic parts, we prove the null controllability of the system when the control region, driven by the flow of an ODE, covers the whole domain. Finally, in chapter 6, we study the cost of controlling the Stokes system to zero. Using a new controllability result for a hyperbolic system with a pressure term and the control transmutation method, we show that the cost of driving the Stokes system to rest at a time $T >0$ is of order $e^{C/T}$ when $T \to 0^+$, as in the case of the heat equation