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Controllability and Stabilizability of the Linearized Compressible Navier-Stokes System with Maxwell's Law

Sakil Ahamed, Subrata Majumdar

Abstract

In this paper, we study the control properties of the linearized compressible Navier-Stokes system with Maxwell's law around a constant steady state $(ρ_s, u_s, 0), ρ_s>0, u_s>0$ in the interval $(0, 2π)$ with periodic boundary data. We explore the exact controllability of the coupled system by means of a localized interior control acting in any of the equations when time is large enough. We also study the boundary exact controllability of the linearized system using a single control force when the time is sufficiently large. In both cases, we prove the exact controllability of the system in the space $L^2(0,2π)\times L^2(0, 2π)\times L^2(0, 2π)$. We establish the exact controllability results by proving an observability inequality with the help of an Ingham-type inequality. Moreover, we prove that the system is exactly controllable at any time if the control acts everywhere in the domain in any of the equations. Next, we prove the small time lack of controllability of the concerned system. Further, using a Gramian-based approach demonstrated by Urquiza, we prove the exponential stabilizability of the corresponding closed-loop system with an arbitrary prescribed decay rate using boundary feedback control law.

Controllability and Stabilizability of the Linearized Compressible Navier-Stokes System with Maxwell's Law

Abstract

In this paper, we study the control properties of the linearized compressible Navier-Stokes system with Maxwell's law around a constant steady state in the interval with periodic boundary data. We explore the exact controllability of the coupled system by means of a localized interior control acting in any of the equations when time is large enough. We also study the boundary exact controllability of the linearized system using a single control force when the time is sufficiently large. In both cases, we prove the exact controllability of the system in the space . We establish the exact controllability results by proving an observability inequality with the help of an Ingham-type inequality. Moreover, we prove that the system is exactly controllable at any time if the control acts everywhere in the domain in any of the equations. Next, we prove the small time lack of controllability of the concerned system. Further, using a Gramian-based approach demonstrated by Urquiza, we prove the exponential stabilizability of the corresponding closed-loop system with an arbitrary prescribed decay rate using boundary feedback control law.
Paper Structure (31 sections, 37 theorems, 273 equations, 1 figure)

This paper contains 31 sections, 37 theorems, 273 equations, 1 figure.

Table of Contents

  1. Introduction and Main results
  2. Setting of the problem
  3. Interior controllability
  4. Boundary controllability
  5. Methodology of the proof of main results
  6. Stabilizability
  7. Organization of the paper
  8. Linearized operator
  9. Spectral analysis of the linearized operator
  10. Finite dimensional projected system
  11. Interior controllability
  12. Proof of \ref{['nmaxthm_pos']}
  13. The case of multiple eigenvalues
  14. Control acting everywhere in the domain
  15. Proof of \ref{['thm_pos']} :
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $f_2=0=f_3$ in nmaxeq3 and $\mathcal{O}_1\subseteq (0,2\pi)$. Then there exists a $T_0>0$ such that the system nmaxeq3 is exactly controllable in $L^{2}(0,2\pi) \times {\dot L^{2}(0,2\pi) \times \dot L^{2}(0,2\pi)}$ at time $T>T_0$, by an interior control $f_1\in L^2\left( 0,T;L^2(\mathcal{O}_1)

Figures (1)

  • Figure 1: Eigenvalues of $\mathcal{A}$ in the complex plane for $|n|$ varies from $1$ to $30$ when $\mu=\rho_s=u_s=b=1$ and k=1.

Theorems & Definitions (67)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 57 more