Abstract-In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

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Backstepping, on the other hand, requires little background beyond calculus for users to understand the design and the stability analysis. This book is designed to be used in a one semester course on backstepping techniques for boundary control of PDEs. In Fall we offered such a course at the University of California, San Diego. The course attracted a large group of postgraduate, graduate, and advanced undergraduate students. Due to the diversity of backgrounds of the students in the class, we developed the course in a broad way so that students could exercise their intuition no matter their backgrounds, whether in fluids, flexible structures, heat transfer, or control engineering. The course was a success and at the end of the quarter two of the students, Matthew Graham and Charles Kinney, surprised us with a gift of a typed version of the notes that they took in the course. We decided to turn these notes into a textbook, with Matt's and Charles' notes as a starting point, and with the homework sets used during the course as a basis for the exercise sections in the textbook. We have kept the book short and as close as possible to the original course so that the material can be covered in one semester or quarter. Although short, the book covers a very broad set of topics, including most major classes of PDEs. We present the development of backstepping controllers for parabolic PDEs, hyperbolic PDEs, beam models, transport equations, systems with actuator delay, Kuramoto-Sivashinsky-like and Korteweg-de Vries-like linear PDEs, and Navier-Stokes equations. We also cover the basics of motion planning and parameter-adaptive control for PDEs, as well as observer design with boundary sensing. Short versions of a course on boundary control, based on preliminary versions of the book, were taught at the

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