55 research outputs found
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
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 . Consequently, the minimum time
function fails to be locally Lipschitz---never mind
semiconcave---near . Instead of such a regularity, we use an
exterior sphere condition for the hypograph of 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 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 HamiltonâJacobiâBellman 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 Ìn, 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 RankineâHugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows.âPlan Propio de InvestigaciĂłn, programa 9â (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 AndalucĂa and european FEDER funds)Project P18-RT-2422 (funded by Junta de AndalucĂa 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
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