Boundary Stabilization of Parabolic Equations

1. Front Matter

   Pages i-xii

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2. Preliminaries

   • Ionuţ Munteanu

   Pages 1-17

3. Stabilization of Abstract Parabolic Equations

   • Ionuţ Munteanu

   Pages 19-48

4. Stabilization of Periodic Flows in a Channel

   • Ionuţ Munteanu

   Pages 49-76

5. Stabilization of the Magnetohydrodynamics Equations in a Channel

   • Ionuţ Munteanu

   Pages 77-92

6. Stabilization of the Cahn–Hilliard System

   • Ionuţ Munteanu

   Pages 93-107

7. Stabilization of Equations with Delays

   • Ionuţ Munteanu

   Pages 109-126

8. Stabilization of Stochastic Equations

   • Ionuţ Munteanu

   Pages 127-169

9. Stabilization of Unsteady States

   • Ionuţ Munteanu

   Pages 171-185

10. Internal Stabilization of Abstract Parabolic Systems

    • Ionuţ Munteanu

    Pages 187-206

11. Back Matter

    Pages 207-214

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This monograph presents a technique, developed by the author, to design asymptotically exponentially stabilizing finite-dimensional boundary proportional-type feedback controllers for nonlinear parabolic-type equations. The potential control applications of this technique are wide ranging in many research areas, such as Newtonian fluid flows modeled by the Navier-Stokes equations; electrically conducted fluid flows; phase separation modeled by the Cahn-Hilliard equations; and deterministic or stochastic semi-linear heat equations arising in biology, chemistry, and population dynamics modeling.


The text provides answers to the following problems, which are of great practical importance:

• Designing the feedback law using a minimal set of eigenfunctions of the linear operator obtained from the linearized equation around the target state
  
• Designing observers for the considered control systems
  
• Constructing time-discrete controllers requiring only partial knowledge of the state
  

After reviewing standard notations and results in functional analysis, linear algebra, probability theory and PDEs, the author describes his novel stabilization algorithm. He then demonstrates how this abstract model can be applied to stabilization problems involving magnetohydrodynamic equations, stochastic PDEs, nonsteady-states, and more.

 
Boundary Stabilization of Parabolic Equations will be of particular interest to researchers in control theory and engineers whose work involves systems control. Familiarity with linear algebra, operator theory, functional analysis, partial differential equations, and stochastic partial differential equations is required.

• Parabolic Partial Differential Equations
• Boundary Stabilization
• Magnetohydrodynamics equations
• Cahn-Hilliard System
• Stochastic Partial Differential Equations
• Feedback control
• Control theory
• partial differential equations

“This book will be of particular interest to researchers in control theory and engineers whose work involves systems control. It also provides an extensive bibliography to guide those who wish to delve further into these matters.” (Larbi Berrahmoune, Mathematical Reviews, March 2, 2020)

“This book is well written and clear, it’s a nice reference for the boundary stabilization of parabolic equations.” (Kaïs Ammari, zbMATH 1421.93066, 2019)

• Faculty of Mathematics, Alexandru Ioan Cuza University, Iaşi, Romania

  Ionuţ Munteanu

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