Null Controllability of Certain Continuous and Discrete PDE Control Systems

Kumar, Manish (2025) Null Controllability of Certain Continuous and Discrete PDE Control Systems. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Manish Kumar (19IP001)) 19IP001.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

This thesis work is based on the study of two types of controllability problems, as listed below. T-1 Studying null controllability for some control systems described by coupled partial differential equations (PDEs), using the method of moments. T-2 Studying a suitable analogous notion of null controllability for some discrete control systems that approximate some evolutionary PDE control system, by deriving suitable discrete Carleman estimate. For the first type of problem, we have considered two different coupled PDE control systems; one coupling the Kuramoto-Sivashinsky-Korteweg-de Vries equation (fourth-order parabolic eq.) with heat equation (second order parabolic eq.), while the other one involves coupling between the linear Kuramoto-Sivashinsky-Korteweg-de Vries equation and the transport equation. The study of null controllability of these systems has been done under the action of one control, which acts either as a localized interior control or as a boundary control. In the second category of problems, we have studied the notion of ɸ-null controllability for a time-discrete parabolic system and a fully discrete linear KdV equation using boundary controls. To begin with, we start this thesis by introducing the basics of control theory in an abstract setup for evolutionary PDE control systems and describing the two methods that will be used in this thesis to study the above-mentioned problems, along with some illustrations in Chapter 1. In Chapter 2, we give a general overview of all the control problems considered in the thesis with all the associated results. The main content of the thesis starts with Chapter 3, where we establish the null controllability of the stabilized Kuramoto-Sivashinsky equation (KS-KdV equation coupled with heat equation via first order derivatives), posed on a bounded domain with periodic boundary conditions. In this study, we have analyzed four different problems as mentioned below: • localized interior control acting via KS-KdV component • boundary control acting via zeroth order derivative of KS-KdV component • localized interior control acting via heat component • boundary control acting via zeroth order derivative of heat component To study these control problems, we first establish global null controllability results for the associated linear control systems, obtained by linearizing them around the origin. Using these global controllability results for the linearized system and source term method along with Banach fixed point theorem, we finally conclude local null controllability results for the main nonlinear stabilized KS equation. In order to obtain the controllability results for the linear system, we first deduce the equivalent moment problem associated with each of the four control problems in consideration, using the eigenfunctions and eigenvalues of the adjoint of the underlying differential operator. And then to solve these four moment problems, we construct a suitable biorthogonal family and conclude the desired controllability result directly. Thus, the main task in this study is to obtain a suitable biorthogonal family. Concerning Chapter 4, we study similar four control problems, but now for a linear parabolichyperbolic coupled system, coupling the linear KS-KdV equation with the transport equation via first order derivatives. The methodology we employ in this study is the same as the one used in the last chapter. More precisely, we derive four moment problems associated with each of the underlying control problems and then construct a suitable family of biorthogonal using the same idea as before, to solve the reduced moment problems. Further, we study the notion of ɸ-null controllability (an analogous notion of null controllability in a discrete setting) for a time-discretized parabolic system posed on a bounded domain, coupling two heat equations via Kirchhoff-type boundary condition in Chapter 5. In this case, the control acts on the system through the boundary, and its study is done via a duality approach, where we first derive some relaxed observability estimate for the time-discrete adjoint system, associated with the considered discrete control system, and then conclude about ɸ-null controllability result using the standard variational approach. In order to obtain the relaxed observability estimate, we derive a suitable discrete Carleman estimate for the time-discretized adjoint differential operator. Moreover, using the uniformly bounded discrete control function obtained in the study of ɸ-null controllability of the discrete control system and the associated discrete solution, we obtain an approximation to the control-solution pair of the main parabolic control system. In Chapter 6, we study the notion of ɸ-null controllability for a fully-discrete approximation of linear KdV boundary control system, obtained by using the finite difference method in both variables. Like the previous chapter, we follow the duality approach where we first establish a suitable discrete Carleman estimate for a fully-discrete differential operator and then use it along with some properties of the adjoint system to deduce the desired relaxed observability estimate. Lastly, we use the standard variational argument to conclude the desired ɸ-null controllability result. In the concluding chapter, Chapter 7, we wrap up this thesis by summarizing all the research works and mentioning the related unsolved open problems that demand further exploration.

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