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Iacopo P. Longo (Technical University Munich, Germany)

Strong and weak topologies for Carathéodory functions with applications in the study of non-autonomous ordinary differential equations

Carathéodory differential equations allow to treat phenomena which present discontinuities in time such as switching systems, control systems or time-dependent networks, among others. Nevertheless, due to the difficulty in constructing a continuous flow for such problems, a dynamical theory for non-linear Carathéodory equations remained unavailable for a long time.

We consider spaces of Carathéodory functions and give conditions guaranteeing the continuity of the skew product flow generated by Carathéodory differential equations (ODEs). In particular, we will show optimal theorems for both strong and weak topologies of $L^1_{loc}$-type.

As an application of such results, we show how suitable conditions on the solutions of an initial system $\dot x=f(t,x)$ allow to deduce the existence of a bounded pullback attractor for all the systems belonging to either the alpha limit set of $f$, the omega limit set of $f$, or the whole hull of $f$ and the implied consequences on the dynamics.

This is a joint work with Prof. Sylvia Novo and Prof. Rafael Obaya from the University of Valladolid.

date: 16 March 2021

time: 15:00

room: online presentation