Felix Klein Colloquium: Lyapunov's second method - news on a classic

In 1892, Lyapunov laid the foundations of modern stability theory of dynamical systems by first coining the concepts of stability (in Lyapunov's sense) and asymptotic stability. He then used generalized energy functions, now called Lyapunov functions, to provide sufficient conditions for asymptotic stability. A major advantage of his methodology is that no knowledge of the system trajectories is required, so that the method has found application for a large number of systems. In the western literature, his results were not recognized until late. Since the second half of the 20th century, however, applications, extensions and special cases of Lyapunov's results have been studied worldwide.

The methodology has been further developed for infinite-dimensional systems in particular. In 1970, Datko showed that exponential stability exists for strongly continuous semigroups on Hilbert spaces if a quadratic Lyapunov function exists. The crucial new infinite-dimensional aspect of this result is that in certain cases there can be no quadratic and simultaneously coercive Lyapunov function. A completely different approach was taken in the treatment of nonlinear partial differential equations, where in classical books direct Lyapunov theorems regularly assume coercivity. The corresponding proofs are then also simple generalizations of the finite-dimensional approach.

However, the question arises to what extent this assumption is justified in theory building and what significance the condition of coercivity really has. It turns out that a lot can be achieved without this condition.

The lecture will give an overview of the development of the theory of Lyapunov functions as well as some applications that go beyond stability issues. Finally, the importance of coercivity conditions in the finite and infinite dimensional case will be discussed.

Speaker: Prof. Dr. Fabian Wirth, University of Passau, Germany

(in cooperation with Andrii Mironchenko and Michael Schönlein)

Time: 17:15 - 18:30 o'clock

Place: Building 48, room 210

The lectures of the Felix Klein Colloquium will be held at 17:15 in room 210 of the Mathematics Building 48. Beforehand - from 16:45 - there will be an opportunity to meet the speaker at the colloquium tea in room 580.