Internal controllability of the Korteweg-de Vries equation on a bounded domain

This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval with a distributed control. When the control region is an arbitrary open subdomain, we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, the state function being controlled on the left part of the complement of the control region. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L2 space is also established

Internal control of the Schrödinger equation

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension $1$ using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the new difficulties when we consider the Nonlinear Schrödinger equation

Periodic switching strategies for an isoperimetric control problem with application to nonlinear chemical reactions

This paper deals with an isoperimetric optimal control problem for nonlinear control-affine systems with periodic boundary conditions. As it was shown previously, the candidates for optimal controls for this problem can be obtained within the class of bang-bang input functions. We consider a parametrization of these inputs in terms of switching times. The control-affine system under consideration is transformed into a driftless system by assuming that the controls possess properties of a partition of unity. Then the problem of constructing periodic trajectories is studied analytically by applying the Fliess series expansion over a small time horizon. We propose analytical results concerning the relation between the boundary conditions and switching parameters for an arbitrary number of switchings. These analytical results are applied to a mathematical model of non-isothermal chemical reactions. It is shown that the proposed control strategies can be exploited to improve the reaction performance in comparison to the steady-state operation mode.Comment: Submitted to "Applied Mathematical Modelling

Control in moving interfaces and deep learning

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de Lectura: 14-05-2021This thesis has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex

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