On the Global Controllability of Certain Nonlinear Partial Di!erential Equations

Mondal, Debanjit (2026) On the Global Controllability of Certain Nonlinear Partial Di!erential Equations. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Debanjit Mondal (20IP005)) 20IP005.pdf - Submitted Version Restricted to Repository staff only Download (6MB)

Abstract

This thesis is devoted to the study of global controllability for several nonlinear partial differential equations (PDEs). Chapter 1 provides an overview of the problems addressed in this work, highlighting a key strategy based on the use of large controls over small time intervals. In Chapter 2, we present the necessary preliminaries on the controllability theory of nonlinear ordinary and partial differential equations. The main results are organized into two parts. The first part concerns the global controllability of single nonlinear PDEs driven by finite-dimensional external forces. The analysis relies on the Agrachev–Sarychev approach in geometric control theory, and, in certain cases, is combined with the return method. The second part is devoted to a coupled nonlinear system, where global controllability is achieved using a single finite-dimensional control, again based on the Agrachev–Sarychev framework. Chapter 3 is divided into two parts. First, we study the Camassa–Holm equation on the periodic domain T and prove that it is globally approximately controllable in Hs (T) for s> 3/2 in any positive time by means of a three-dimensional control. Moreover, this control prevents finite-time blow-up of solutions. In the second part, we establish global approximate controllability of the Kawahara equation on T in H₀s (T) for s≥ 0, using a two-dimensional control. In Chapter 4, we investigate third- and fifth-order nonlinear dispersive equations on T with additive finite-dimensional controls. Approximate controllability is obtained by exploiting the controllability of the inviscid Burgers equation, linearized around a suitably constructed return trajectory. In addition, the resulting controls depend continuously on the initial and target states. Chapter 5 addresses a Boussinesq system of BBM–BBM type on the one-dimensional torus. The system is driven by a single five-dimensional control acting on one component. We prove that both components can be simultaneously steered arbitrarily close to any desired target state in any given time, and that the control prevents finite-time blow-up. Finally, Chapter 6 summarizes the results and outlines several open problems and directions for future research.

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