Feedback stabilization and controllability of some linear and nonlinear coupled PDEs

Majumdar, Subrata (2022) Feedback stabilization and controllability of some linear and nonlinear coupled PDEs. PhD thesis, Indian Institute of Science Education and Research, Kolkata.

Text (PhD thesis of Subrata Majumdar (18RS016)) 18RS016.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

This thesis studies the boundary controllability and stabilizability issues of some coupled systems related to the linearized compressible Navier-Stokes system as well as some related nonlinear systems. At first, in Chapter 1, we give an overall introductory study of the problems considered in this thesis. Next, we discuss a brief preliminaries on the theory of control of partial differential equations (PDEs) in Chapter 2. We divide our work into two parts. First part is devoted to the exponential stabilization of some linear and non linear coupled ODE-parabolic systems. And in the second part, we have studied the boundary exact controllability and feedback stabilization of a mixed hyperbolic-elliptic coupled partial differential equation. In Chapter 3, we consider the linearized compressible Navier-Stokes equations for viscous barotropic fluid. Exponential stabilization of this system has been studied for different possible boundary conditions. We employ the well-known backstepping method to establish the feedback stabilization of the concerned system by means of Dirichlet or Neumann boundary control acting only at the right end point of the domain. It has been proved that the closed loop system is stabilizable with exponential decay e⁻ω⁰t, where ω⁰ is the accumulation point of one branch of the spectrum of the underlying spatial operator. One of the novelties of this work is that the critical decay rate ω⁰ has been achieved by the feedback law without using the spectrum of the corresponding spatial operator. Chapter 4 is devoted to the study of local exponential stabilization of two similar kind of nonlinear models coming from biologocial phenomenon, namely Rogers-McCulloch (RM) and FitzHugh-Nagumo (FHN) systems. Note that, the intrinsic properties of the above systems (when we restrict our study for linear case) are same. We explore the method of backstepping directly to these nonlinear models and we conclude the exponential stabilization results provided the initial data are small enough. For the FHN model, the feedback control law achieve the critical decay rate e⁻δt, where δ is the accumulation point of one branch of the spectrum of the corresponding linearized operator. But for the RM equation, we can reach the decay up to e⁻ωt, ω < δ. we can reach the decay up to e−ωt, ω < δ. Next, in Chapter 5, we study the control properties of a first-order hyperbolic-elliptic mixed-class coupled system in the interval (0, 1). More explicitly, we consider the linearized compressible Navier-Stokes system in the case of creeping flow with the Dirichlet boundary conditions. Exact controllability of this system has been studied at any time T ≥ 1. Duality between controllability of main system and observability of the adjoint has been employed to prove the exact controllability of the system. At first, we achieve an auxiliary observability inequality for the adjoint system using multiplier approach. Compactness-uniqueness method is useful to absorb a compact error term from the auxiliary inequality and provide the required observability inequality. For the exact controllability of the system at the optimal control time T = 1, we implement the method of moments. We have also shown the system is not null controllable at any time T < 1. Finally, using Riccati-based feedback law, rapid exponential stabilization has been studied for this system. Finally in Chapter 6, we conclude the thesis by mentioning some research problems which can be taken into account in future research works. To sum up, in this thesis, we essentially focus on the three problems listed below: Stabilizability of ODE-parabolic model • Boundary feedback stabilizability of the linearized compressible Navier-Stokes equations by backstepping approach. • Local exponential stabilization of the Rogers-McCulloch equation and FitzHugh-Nagumo equations by the method of backstepping. Control and stabilization of hyperbolic-elliptic model • Boundary exact controllability and stabilizability of a first-order hyperbolic-elliptic equations.

Actions (login required)

View Item