Boundary Controllability of Linearized Compressible Navier-Stokes System and Related Equations

Kumbhakar, Jiten (2024) Boundary Controllability of Linearized Compressible Navier-Stokes System and Related Equations. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Jiten Kumbhakar (17IP021)) 17IP021.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

In this thesis, we first study the controllability properties of the one dimensional linearized compressible Navier-Stokes equations for both barotropic and non-barotropic fluids using only one boundary control. In the barotropic case, the linearized system (around a positive constant steady states) consists of a transport equation (satisfied by the density of the fluid) coupled with a parabolic equation (satisfied by the velocity of the fluid) with first-order coupling. We consider three types of boundary conditions: (i) Periodic, where the control acts on the density (resp. velocity) component and is given by the difference of the values of the solution at both ends; (ii) Dirichlet, where the control acts on the density part through Dirichlet condition at the left end; (iii) A mixed-type, where we study two cases, one when the control acts on the density part through the difference of the solution at both ends with homogeneous Dirichlet conditions on the velocity, and the second when the control acts on the velocity part through Dirichlet condition at the left end with homogeneous Dirichlet condition on density. On the other hand, for non-barotropic fluids, the linearized system (around positive constant steady states) consists of a transport equation (satisfied by the density of the fluid) coupled with two parabolic equations (satisfied by the velocity and temperature) with the first-order couplings. Here, we consider only the periodic boundary conditions onto the system and study the null and approximate controllability properties using only one control acting either on density, velocity or temperature. In all of the above cases, we have proved optimal null controllability results for the linearized system with respect to the regularity of initial states for the velocity/ temperature case and with respect to time in the density case. Moreover, we obtain approximate controllability of the above systems at large time by using the null controllability and backward uniqueness property of the corresponding systems. These results are included in Chapter 3 and 4. Our proofs of null controllability results rely on the method of moments and an application of the Ingham-type inequalities. The spectral analysis of the associated adjoint operator plays a crucial role in this analysis and we will use this throughout the thesis. We also prove a new Ingham-type inequality in Chapter 4, which generalizes the earlier related results available in the literature. We prove all the controllability results presented in Chapter 3 using this newly obtained Ingham-type inequality, whereas, in Chapters 4, we use both the method of moments and the Ingham-type inequality. Furthermore, in Chapter 1, we give a brief overview of our main controllability results, and in Chapter 2, we present a detailed study of the basic results on controllability including the transport, heat and some nonlinear heat equations. Finally, in Chapter 5, we have considered a coupled system consisting of two nonlinear parabolic equations with square, product and non-local nonlinearities. In the system, a Neumann boundary control is applied to only one state while the other satisfies homogeneous Neumann boundary condition at the left end. On the other hand, at the right end of the interval, the states are coupled in terms of “equality condition of their normal derivatives” and a combined Robin-type condition. In this setup, we prove small-time local null controllability of the system by applying the so called “source term method”.

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