Attainability property for a probabilistic target in Wasserstein spaces

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called \emph{multiagent system} made of several possible interacting agents.Comment: Accepted for publication in DCDS-

Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems

In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a \emph{control sparsity} constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic Programming Principle, providing further applications in the Appendix.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Optimality conditions and regularity results for time optimal control problems with differential inclusions

We study the time optimal control problem with a general target $\mathcal S$ for a class of differential inclusions that satisfy mild smoothness and controllability assumptions. In particular, we do not require Petrov's condition at the boundary of $\mathcal S$. Consequently, the minimum time function $T(\cdot)$ fails to be locally Lipschitz---never mind semiconcave---near $\mathcal S$. Instead of such a regularity, we use an exterior sphere condition for the hypograph of $T(\cdot)$ to develop the analysis. In this way, we obtain dual arc inclusions which we apply to show the constancy of the Hamiltonian along optimal trajectories and other optimality conditions in Hamiltonian form. We also prove an upper bound for the Hausdorff measure of the set of all nonlipschitz points of $T(\cdot)$ which implies that the minimum time function is of special bounded variation.Comment: 23 pages, 1 figur

Random Lift of Set Valued Maps and Applications to Multiagent Dynamics

We introduce an abstract framework for the study of general mean field games and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set valued map expressing the admissible trajectories for the microscopical agents. The techniques used can be applied to consider a broad class of dependence between the trajectories of the single agent and the state of the system. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system

Dynamical Systems and Hamilton–Jacobi–Bellman Equations on the Wasserstein Space and their L2 Representations

Several optimal control problems in \BbbR d, like systems with uncertainty, control of flock dynamics, or control of multiagent systems, can be naturally formulated in the space of probability measures in \BbbR d . This leads to the study of dynamics and viscosity solutions to the Hamilton-- Jacobi--Bellman equation satisfied by the value functions of those control problems, both stated in the Wasserstein space of probability measures. Since this space can be also viewed as the set of the laws of random variables in a suitable L2 space, the main aim of the paper is to study such control systems in the Wasserstein space and to investigate the relations between dynamical systems in Wasserstein space and their representations by dynamical systems in L2, both from the points of view of trajectories and of (first-order) Hamilton--Jacobi--Bellman equations

Optimal control of multiagent systems in the Wasserstein space

Abstract This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in R^d in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on R^d equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem

Anisotropic tempered diffusion equations

We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by [F. Andreu, V. Caselles, J. M. Mazo ́n, Nonlinear Anal. 61 (2005), J. Eur. Math. Soc. 7 (2005)], therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine–Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.“Plan Propio de Investigación, programa 9” (funded by Universidad de Granada and european FEDER (ERDF) funds)Project RTI2018-098850-B-I00 (funded by MICINN and european FEDER funds)Project A-FQM-311-UGR18 (funded by Junta de Andalucía and european FEDER funds)Project P18-RT-2422 (funded by Junta de Andalucía and european FEDER funds

Anisotropic tempered diffusion equations

We introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by 4, 5, therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine-Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.Comment: 43 page

Controllability of Some Nonlinear Systems with Drift via Generalized Curvature Properties

We discuss the problem of local attainability for finite-dimensional nonlinear control systems with quite general assumption on the target set. Special emphasis is given to control-affine systems with a possibly nontrivial drift term. To this end, we provide some sufficient conditions ensuring local attainability, which involve geometric properties both of the target itself (such as a notion of generalized curvature), and of the Lie algebra associated to the control system. The main technique used is a convenient representation formula for the power expansion of the distance function along the trajectories, made at points sufficiently near to the target set