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Chapter 1. Examples of Control Systems

Feedback control is a process of regulating a physical system to a desired output based on the feedback of actual outputs of the system. Feedback control systems are ubiquitous around us, including trajectory planning of a robot manipulator, guidance of a tactical missile toward a moving target, regulation of room temperature, and control of string vibrations. These control systems can be abstracted by Figure 1.1. Here a plant subject to a disturbance d is given, and a controller is to be designed so that the output y tracks the reference r. The plant can be modeled by either a system of ordinary differential equations (finite dimensional systems) or partial differential equations (infinite dimensional systems).
Weijiu Liu

Chapter 2. Elementary Functional Analysis

Elementary functional analysis, such as Hilbert spaces, Sobolev spaces, and linear operators, is collected in this chapter for the reader’s convenience. Since elementary stabilization results are presented without using functional analysis, readers may skip this chapter and come back when needed. In this book, ℕ denotes the set of all nonnegative natural numbers, ℂ denotes the set of complex numbers, ℝ n denotes the n-dimensional Euclidean space, and ℝ = ℝ1 denotes the real line. Points in ℝ n will be denoted by x = (x 1, …, x n ), and its norm is defined by
$$\|\mathbf{x}\| = \left(\sum_{i = 1}^{n}x_{i}^{2}\right)^{\frac{1}{2}}.$$
Weijiu Liu

Chapter 3. Finite Dimensional Systems

Because many control concepts and theories in finite dimensional systems have been transplanted to partial differential equations, we present a brief introduction to feedback control of finite dimensional systems. We start with the control systems without disturbances and address the control systems subject to a disturbance in Section 3.10. Linear finite dimensional control systems have the following general form
$$\dot{\mathbf{x}} = \mathbf{Ax} + \mathbf{Bu},$$
$$\mathbf{y} = \mathbf{Cx} + \mathbf{Du},$$
$$\mathbf{x}(0) = \mathbf{x}_{0},$$
where x =(x 1, x 2, …, x n ) (hereafter ∗ denotes the transpose of a vector or matrix) is a state vector, x 0 is an initial state caused by external disturbances, y = (y 1, …, y l)∗ is an output vector, u = (u 1, …, u m )∗ is a control vector, and A, B, C, D are n × n, n × m, l × n, l × m constant matrices, respectively. The equation (3.1) is called a state equation. A simple example of such control systems is the mass-spring system (1.2), which is repeated as follows
$$\left[\begin{array}{r}\dot{x}_{1}\\ \dot{x}_2\end{array}\right] = \left[\begin{array}{rl}0 & 1\\ -\frac{k}{m} & 0\end{array}\right] \left[\begin{array}{r}x_{1}\\ x_2\end{array}\right] + \left[\begin{array}{r}0\\ \frac{1}{m}\end{array}\right] u,$$
$$y = \left[1\quad0\right] \left[\begin{array}{r}x_{1}\\ x_2\end{array}\right].$$
Weijiu Liu

Chapter 4. Linear Reaction-Convection-Diffusion Equation

In this chapter, we discuss the control problem of the linear reaction-convectiondiffusion equation
$$\frac{\partial u}{\partial t} = \mu \nabla^{2} u + \nabla \cdot (u\mathbf{v}) + au.$$
Depending on a particular real problem, u can represent a temperature or the concentration of a chemical species. The constant μ > 0 is the diffusivity of the temperature or the species, the vector v(x) = (v 1(x), …, vn(x)) is the velocity field of a fluid flow, and a(x) is a reaction rate. ∇2 is defined by
$$\nabla^{2} u = \frac{\partial^{2}u}{\partial x_{1}^{2}} + \ldots + \frac{\partial^{2} u}{\partial x_{n}^{2}},$$
$$\nabla u = \left(\frac{\partial u}{\partial x_{1}}, \ldots, \frac{\partial u}{\partial x_{n}}\right).$$
\(\nabla \cdot (u\mathbf{v}) = {\rm div} (u\mathbf{v}) = \sum\nolimits_{i = 1}^{n} \frac{\partial (uv_{i})}{\partial x_{i}}\) denotes the divergence of the vector u v.
Weijiu Liu

Chapter 5. One-dimensional Wave Equation

In this chapter, we study the control problem of the one-dimensional wave equation
$$\frac{\partial^2 u}{\partial t^{2}} = c^2 \frac{\partial^2 u}{\partial x^2}.$$
A typical physical problem modeled by the wave equation is the vibration of a string. In this problem, u = u(x, t) represents the vertical displacement of the string from its equilibrium and the positive constant c (m/s) is a wave speed. In what follows, for convenience, we will use the subscripts u t , u x or \(\frac{\partial u}{\partial t}, \frac{\partial u}{\partial x}\) interchangeably to denote the derivatives of u with respect to t,x, respectively.
Weijiu Liu

Chapter 6. Higher-dimensional Wave Equation

In this chapter, we study the control problem of the linear wave equation
$$\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u.$$
This equation can serve as a mathematical model for many physical problems, such as the vibration of a membrane. In the membrane problem, u = u(x, y, t) represents the vertical displacement of the membrane from its equilibrium and the positive constant c (m/s) is a wave speed. In this chapter, we use the following notation. Let Ω be a bounded open set in ℝ n and x 0 ∈ ℝ n . Set (see Figure 6.1)
$$\Gamma = \partial\Omega,$$
$$\mathbf{m}(\mathbf{x}) = \mathbf{x} - \mathbf{x}^0 = (x_1 - x_{1}^{0}, \ldots, x_{n} - x_{n}^{0}),$$
$$\Gamma (\mathbf{x}^{0}) = \{\mathbf{x} \in \Gamma : \mathbf{m}(\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) > 0\},$$
$$\Gamma_{\ast}(\mathbf{x}^0) = \Gamma - \Gamma (\mathbf{x}^0) = \{\mathbf{x} \in \Gamma : \mathbf{m}(\mathbf{x}) \cdot \mathbf{n}(\mathbf{x}) \leq 0\},$$
where n denotes the unit normal pointing towards the outside of Ω.
Weijiu Liu


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