Hamilton-Jacobi equations for controlled gradient flows: existence and comparison principle for viscosity solutions

Seminario periodico del Dipartimento di Matematica
31 ottobre 2023
Orario di inizio 
PovoZero - Via Sommarive 14, Povo (Trento)
Aula Seminari "1"
Comunità universitaria
Comunità studentesca UniTrento
Ingresso libero
Andrea Pinamonti, Andrea Marchese, Giorgio Saracco, Gian Paolo Leonardi
Università degli Studi Trento 38123 Povo (TN) - Staff Dipartimento di Matematica
+39 04 61/281508-1625-1701-3786-1980
Luca Tamanini (Università Cattolica del Sacro Cuore)


In the pioneering works of Crandall and Lions the theory of viscosity solutions for Hamilton-Jacobi equations on infinitedimensional spaces was initiated in the setting of Hilbert spaces or Banach spaces having the Radon-Nikodym property.
But recent applications in large deviations, statistical mechanics and McKean-Vlasov control motivated the extension of the theory to metric spaces that need not be normed, most notably the Wasserstein space. In the first part, we will be interested in the Hamilton-Jacobi equation associated to a Mean Field control problem in which one linearly controls the gradient flow of an energy functional defined on an abstract metric space. Under fairly weak assumptions on the energy (in particular, no compactness on the sublevels is assumed, in contrast with the existing literature) we will show a comparison principle, by combining Ekeland's principle, the use of the Tataru distance and the regularizing effect of gradient flows in the EVI formulation. In the second part, we will look at a large class of energy functionals on the Wasserstein space (allowing for potential, internal and interaction energies), and leveraging again on the notion of EVI gradient flow we will discuss the problem of existence of viscosity solutions (based on a joint work with G. Conforti, R. Kraaij, and D. Tonon).

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