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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Preliminaries

Abstract
For easy reference, we collect here some standard notation and results in functional analysis and partial differential equations that will be used throughout this work.
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### Chapter 2. Stabilization of Abstract Parabolic Equations

Abstract
In this chapter, we present a technique to design asymptotically exponentially stabilizing boundary proportional-type feedback controllers for nonlinear parabolic-like equations, namely equations for which their linear parts are generated by analytic $$C_0$$-semigroups. In what follows, we will simply refer to them as parabolic equations, in concordance with the title of this book. The feedback law’s main features are that it is expressed in an explicit simple form and has a finite-dimensional structure involving only the eigenfunctions of the linear operator obtained from the linearized equation. As we will see, these features will enable us to obtain the first results to appear in the literature regarding the stabilization of different equations, such as the stochastic heat equation, the Chan–Hilliard equations, and for boundary stabilization to nonsteady states for parabolic-type equations.
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### Chapter 3. Stabilization of Periodic Flows in a Channel

Abstract
Here we apply the control design algorithm from Chap. 2 to the Navier–Stokes equations, placed in a particular geometry, namely a semi-infinite channel. The high instability of the Navier–Stokes equations is well known as is the fact that the principal way to suppress the turbulence occurring in the dynamics of a fluid is to plug in a stabilizing feedback control. In addition, a Riccati-based robust controller is also constructed.
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### Chapter 4. Stabilization of the Magnetohydrodynamics Equations in a Channel

Abstract
Here we consider again a channel flow. But in addition to the assumptions of the previous chapter, we assume that the incompressible fluid is electrically conducting and affected by a constant transverse magnetic field. This kind of flow was first investigated both experimentally and theoretically by Hartmann [67]. The governing equations are the magnetohydrodynamics equations (MHD, for short), which are a coupling between the Navier–Stokes equations and the Maxwell equations.
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### Chapter 5. Stabilization of the Cahn–Hilliard System

Abstract
In this chapter, the Cahn–Hilliard system will be investigated. This system describes the process of phase separation, whereby the two components of a binary fluid spontaneously separate and form domains pure in each component. This phenomenon appears in many engineering and medical applications.
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### Chapter 6. Stabilization of Equations with Delays

Abstract
In this chapter, we consider equations with delays. Namely, the derivative of an unknown function at a certain time depends on the values of the function at previous times. More exactly, we consider in the model aftereffect phenomena by adding a memory term. Engineers conclude that actuators, sensors that are involved in feedback control, introduce, in addition, delays into the system. That is why from the control engineering point of view it is of great interest to consider control problems associated with equations with delays. Furthermore, special kinds of substances, such as viscoelastic fluids, may also impose such delays. We will prove here that the proportional feedback, designed in Chap. 2, still ensures stability for this kind of system.
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### Chapter 7. Stabilization of Stochastic Equations

Abstract
In this chapter, we consider stochastic PDEs. We address the boundary stabilization problem, which will be of two types: pathwise stabilization and stabilization in mean. Stochastic differential equations can be viewed as a generalization of dynamical systems theory to models with noise. This generalization arises naturally due to the fact that real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence. Clearly, noise perturbation complicates the problem considerably.
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### Chapter 8. Stabilization of Unsteady States

Abstract
In this chapter, we address the problem of stabilization of unsteady-state trajectories of time-dependent systems. In this case, the linear operator obtained from the linearization of the equation around the trajectory is time-dependent, so its spectrum is time-dependent as well. This means that the spectral method leaves out this case. We will follow the approach from Sect. 7.​2, Chap. 7. Namely, we will write the solution of the nonlinear equation in a mild formulation via a kernel and prove its stability.
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### Chapter 9. Internal Stabilization of Abstract Parabolic Systems

Abstract
In this chapter, we will reconsider the abstract parabolic equation framework from Chap. 2. This time, we will design an internal stabilizing proportional-type actuator. As in the boundary case, the feedback laws are of finite-dimensional nature, given in a simple form, and easy to manipulate from the computational point of view. And since we formulate the results in an abstract form, it is clear that for different types of precise models satisfying the imposed abstract hypotheses, these can be applied to the stabilization problem.
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### Backmatter

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