From PDEs with Boundary Control to the Abstract State Equation with an Unbounded Input Operator: A Tutorial

Abstract

This tutorial paper explains in detail how mathematical models of.dynmical systems described by linear partial difflrential equations (PDEs) with controls in the boundary conditions can be generalized as abstract boundary control systems and then how the latter lead to abstract linear state equations with an unbounded input operator. We start with two simple examples: the one-dimensional heat equation with control in the Neumann boundary condition and the one-dimensional wave equation with control in the Dirichlet-Neumann boundary condition. These two examples will serve as a motivation for a more general setup for a boundary control system from which we will eventually derive the abstract state equation with an unbounded input operator. We intend to give a comprehensive introduction for control theorists who are not experts in infinite-dimensional systems theory but who want to read and understand papers in this field.