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Lecture "Viability Properties of Mean Field Type Control Systems"

In January 22 Yurii Averboukh delivered a lecture «Viability Properties of Mean Field Type Control Systems»

Abstract: The control theory examines dynamical systems affected by some decision maker who can choose control parameters. It has been studied since 1950s. Among various concepts studied within the control theory, the viability property plays an important role. Viability of the set means the capability of the system to remain in the given set. This concept provides the bridge between the dynamical properties of the controlled system and the differential structure of sets in the phase space. The famous dynamic programming principle in the nonsmooth form can be deduced as a corollary of the viability theorem. I will give a very short introduction to the finite dimensional control theory and the viability theory in this case. The main part of my talk will be concerned with the mean field type control theory. It studies the systems consisting of many identical small agents governed by one decision maker in the limit case when the number of agents tends to infinity. The additional assumption of the mean field interaction between agents reduces such systems to the control systems in the space of probability measures. I will discuss the approaches to definition of strategies. The main result is the viability theorems those provide the equivalent form of the viability property in the terms of tangent and normal cones. To this end we develop non-smooth analysis in the space of probability measures.
Yurii Averboukh.pdf