Distributed Parameter Systems: Recent Episodes
University of Twente
In this series of movies we give an overview of the ideas and concepts introduced in the text book ``An Introduction to Infinite-Dimensional Linear Systems Theory'', by R.F. Curtain and H.J. Zwart, Springer Verlag, 1995. Roughly the first 5 chapter of this book are treated. In the last movie we make a link to port-Hamiltonian systems theory.
In this lecture we illustrate the theory of the previous movies on the class of port-Hamiltonian systems. We begin by showing that a large class of pde's can be written as a port-Hamiltonian systems. Next we give a simple characterization for the existence of a contraction semigroup. There are also simple sufficient conditions for the exponential stability of the semigroup.
In this lecture we define the concept of stability. We link the concept of exponential stability to the existence of a positive solution to the Lyapunov equation. Furthermore, we show that the location of spectrum of the infinitesimal generator does not imply the stability of the associated semigroup. If the solution of the homogeneous abstract differential equation converses to zero, then the system is said to be strongly stable. The system (with an input) is stabilizable if there exists a feed
In this lecture we introduce the concept of a transfer function for an abstract linear systems. We show that the transfer function of a pde can be easily obtained. Its inverse Laplace transform defines the impulse response. For scalar transfer functions the classical design rules of Nyquist and Bode still apply.
In this lecture we introduce input and output equations to the abstract differential equation of the previous lecture. If we choose the input as a function of the state, we obtain a feedback. Inputs applied to a pde on the boundary of its spatial domain, lead to boundary control systems.
In this lecture we introduce the class of distributed parameter systems by means of some real life examples. In the second part we introduce the concept of semigroups and infinitesimal generators. A special class of strongly continuous semigroups is formed by the class of contraction semigroups. There are simple conditions for an operator to be the infinitesimal generator of such a semigroup. Finally, we how how an abstract differential equation and its solutions can related to a pde.