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### Measure-theoretic approach for minimum-time problems for multi-agent systems

##### Dr. Giulia Cavagnari, Rutgers University e Università degli studi di Pavia

Sala conferenze IMATI-CNR, Pavia - Tuesday, October 24, 2017 h.15:00

**Abstract. ** In this talk we deal with a systematization of the classical finite-dimensional time- optimal control theory in the case when the state of the system is known only up to some uncertainty or when it stands for the statistical distribution of particle masses. Both these situations are interesting for applications and motivate the study of the problem in the infinite-dimensional space of probability measures endowed with the Wasserstein metric. Here, tools from transport theory play a crucial role in the extension of classical results.

We consider a deterministic dynamics given by an (homogeneous) controlled continuity equation (PDE) ∂_{t}μ_{t} + div(v_{t}μ_{t}) = 0, where the vector field v_{t} has to be chosen among the L^{1}μ_{t}-selections of a given set-valued map F. This multifunction represents the admissible velocities of the single particles/agents and thus rules the underlying dynamics of characteristics in the form of a differential inclusion (ODE).

The main tool used is the Superposition Principle by Ambrosio-Gigli-Savaré, providing a connection between the (ODE) and the (PDE) even when the vector field is not regular. We prove also an extension of this result in the context of differential inclusions.

After providing a generalization of the minimum-time function, we present some results as counterparts to the classical ones: a Dynamic Programming Principle, existence of optimal trajectories, attainability and regularity results, and validity of a suitable Hamilton-Jacobi-Bellman equation.

To conclude we will discuss briefly an optimal-control problem where also some kind of interactions among the agents is considered, addressing in particular a control-sparsity problem.

These are joint works with Antonio Marigonda (University of Verona) and Benedetto Piccoli (Rutgers University - Camden, USA).

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