6 research outputs found
Optimal strategies for driving a mobile agent in a guidance by repulsion model
We present a guidance by repulsion model based on a driver-evader interaction
where the driver, assumed to be faster than the evader, follows the evader but
cannot be arbitrarily close to it, and the evader tries to move away from the
driver beyond a short distance. The key ingredient allowing the driver to guide
the evader is that the driver is able to display a circumvention maneuver
around the evader, in such a way that the trajectory of the evader is modified
in the direction of the repulsion that the driver exerts on the evader. The
evader can thus be driven towards any given target or along a sufficiently
smooth path by controlling a single discrete parameter acting on driver's
behavior. The control parameter serves both to activate/deactivate the
circumvention mode and to select the clockwise/counterclockwise direction of
the circumvention maneuver. Assuming that the circumvention mode is more
expensive than the pursuit mode, and that the activation of the circumvention
mode has a high cost, we formulate an optimal control problem for the optimal
strategy to drive the evader to a given target. By means of numerical shooting
methods, we find the optimal open-loop control which reduces the number of
activations of the circumvention mode to one and which minimizes the time spent
in the active~mode. Our numerical simulations show that the system is highly
sensitive to small variations of the control function, and that the cost
function has a nonlinear regime which contributes to the complexity of the
behavior of the system, so that a general open-loop control would not be of
practical interest. We then propose a feedback control law that corrects from
deviations while preventing from an excesive use of the circumvention mode,
finding numerically that the feedback law significantly reduces the cost
obtained with the open-loop control
Instantaneous control of interacting particle systems in the mean-field limit
Controlling large particle systems in collective dynamics by a few agents is
a subject of high practical importance, e.g., in evacuation dynamics. In this
paper we study an instantaneous control approach to steer an interacting
particle system into a certain spatial region by repulsive forces from a few
external agents, which might be interpreted as shepherd dogs leading sheep to
their home. We introduce an appropriate mathematical model and the
corresponding optimization problem. In particular, we are interested in the
interaction of numerous particles, which can be approximated by a mean-field
equation. Due to the high-dimensional phase space this will require a tailored
optimization strategy. The arising control problems are solved using adjoint
information to compute the descent directions. Numerical results on the
microscopic and the macroscopic level indicate the convergence of optimal
controls and optimal states in the mean-field limit,i.e., for an increasing
number of particles.Comment: arXiv admin note: substantial text overlap with arXiv:1610.0132
Asymptotic behavior and control of a "guidance by repulsion" model
Electronic version of an article published as Mathematical Models and Methods in Applied Sciences 30.04 (2020): 765-804, https://doi.org/10.1142/S0218202520400047 © 2020 World Scientific Publishing CompanyWe model and analyze a guiding problem, where the drivers try to steer the evaders' positions toward a target region while the evaders always try to escape from drivers. This problem is motivated by the guidance-by-repulsion model [R. Escobedo, A. Ibañez and E. Zuazua, Optimal strategies for driving a mobile agent in a guidance by repulsion model, Commun. Nonlinear Sci. Numer. Simul. 39 (2016) 58-72] where the authors answer how to control the evader's position and what is the optimal maneuver of the driver. First, we analyze well posedness and behavior of the one-driver and one-evader model, assuming of the same friction coefficients. From the long-time behavior, the exact controllability is proved in a long enough time horizon. Then, we extend the model to the multi-driver and multi-evader case. We assumed three interaction rules in the context of collective behavior models: flocking between evaders, collision avoidance between drivers and repulsive forces between drivers and evaders. These interactions depend on the relative distances, and each agent is assumed to be undistinguishable and obtained an averaged effect from the other individuals. In this model, we develop numerical simulations to systematically explore the nature of controlled dynamics in various scenarios. The optimal strategies turn out to share a common pattern to the one-driver and one-evader case: the drivers rapidly occupy the position behind the target, and want to pursuit evaders in a straight line for most of the time. Inspired by this, we build a feedback strategy which stabilizes the direction of evadersThis project has received funding from the European Research Council (ERC) under the Euro-pean Union’s Horizon 2020 research and innovation programme (grant agreement No. 694126-DyCon). The work of the first author has been supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE-2016-0035. The work of the second author has been funded by the Alexander von Humboldt-Professorship program, the European Union’s Horizon 2020 research andinnovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex, grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, ICON of the French ANR and Nonlocal PDEs: Analysis, Control and Beyond, AFOSR Grant FA9550-18-1-024
Mean-field optimal control and optimality conditions in the space of probability measures
We derive a framework to compute optimal controls for problems with states in
the space of probability measures. Since many optimal control problems
constrained by a system of ordinary differential equations (ODE) modelling
interacting particles converge to optimal control problems constrained by a
partial differential equation (PDE) in the mean-field limit, it is interesting
to have a calculus directly on the mesoscopic level of probability measures
which allows us to derive the corresponding first-order optimality system. In
addition to this new calculus, we provide relations for the resulting system to
the first-order optimality system derived on the particle level, and the
first-order optimality system based on -calculus under additional
regularity assumptions. We further justify the use of the -adjoint in
numerical simulations by establishing a link between the adjoint in the space
of probability measures and the adjoint corresponding to -calculus.
Moreover, we prove a convergence rate for the convergence of the optimal
controls corresponding to the particle formulation to the optimal controls of
the mean-field problem as the number of particles tends to infinity