17 research outputs found
Infinite horizon sparse optimal control
A class of infinite horizon optimal control problems involving -type
cost functionals with is discussed. The existence of optimal
controls is studied for both the convex case with and the nonconvex case
with , and the sparsity structure of the optimal controls promoted by
the -type penalties is analyzed. A dynamic programming approach is
proposed to numerically approximate the corresponding sparse optimal
controllers
Mean-Field Sparse Optimal Control
We introduce the rigorous limit process connecting finite dimensional sparse
optimal control problems with ODE constraints, modeling parsimonious
interventions on the dynamics of a moving population divided into leaders and
followers, to an infinite dimensional optimal control problem with a constraint
given by a system of ODE for the leaders coupled with a PDE of Vlasov-type,
governing the dynamics of the probability distribution of the followers. In the
classical mean-field theory one studies the behavior of a large number of small
individuals freely interacting with each other, by simplifying the effect of
all the other individuals on any given individual by a single averaged effect.
In this paper we address instead the situation where the leaders are actually
influenced also by an external policy maker, and we propagate its effect for
the number of followers going to infinity. The technical derivation of the
sparse mean-field optimal control is realized by the simultaneous development
of the mean-field limit of the equations governing the followers dynamics
together with the -limit of the finite dimensional sparse optimal
control problems.Comment: arXiv admin note: text overlap with arXiv:1306.591
Mean-Field Pontryagin Maximum Principle
International audienceWe derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov-type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables
Mean-Field Optimal Control
We introduce the concept of {\it mean-field optimal control} which is the
rigorous limit process connecting finite dimensional optimal control problems
with ODE constraints modeling multi-agent interactions to an infinite
dimensional optimal control problem with a constraint given by a PDE of
Vlasov-type, governing the dynamics of the probability distribution of
interacting agents. While in the classical mean-field theory one studies the
behavior of a large number of small individuals {\it freely interacting} with
each other, by simplifying the effect of all the other individuals on any given
individual by a single averaged effect, we address the situation where the
individuals are actually influenced also by an external {\it policy maker}, and
we propagate its effect for the number of individuals going to infinity. On
the one hand, from a modeling point of view, we take into account also that the
policy maker is constrained to act according to optimal strategies promoting
its most parsimonious interaction with the group of individuals. This will be
realized by considering cost functionals including -norm terms penalizing
a broadly distributed control of the group, while promoting its sparsity. On
the other hand, from the analysis point of view, and for the sake of
generality, we consider broader classes of convex control penalizations. In
order to develop this new concept of limit rigorously, we need to carefully
combine the classical concept of mean-field limit, connecting the finite
dimensional system of ODE describing the dynamics of each individual of the
group to the PDE describing the dynamics of the respective probability
distribution, with the well-known concept of -convergence to show that
optimal strategies for the finite dimensional problems converge to optimal
strategies of the infinite dimensional problem.Comment: 31 page