Mean-field optimal control and optimality conditions in the space of probability measures

Abstract

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 L 2 -calculus under additional regularity assumptions. We further justify the use of the L 2 -adjoint in numerical simulations by establishing a link between the adjoint in the space of probability measures and the adjoint corresponding to L 2 -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.

Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin Scheme

“…Moreover, it might be of interest to analyze how to handle for example uncertain initial states that are not differentiable with respect to the uncertainty. As soon as an adjoint framework for hyperbolic stochastic PDEs, i.e., in the spirit of [10], is established, we can also study the relation between the discontinuous stochastic Galerkin polynomial of the adjoint coefficients and a possible adjoint equation on PDE level.…”

Space mapping-based Receding Horizon Control for Stochastic Interacting Particle Systems: dogs herding sheep

“…Algorithm 2: Optimal Control Algorithm for the Coarse Problem Data: initial data for states and control, stopping tolerance opt , time steps K, desired destination Z * Result: optimal control u, optimal states y initialize; while u n+1 − u n L 2 > opt do solve deterministic state system (4); solve adjoint problem given in (6); compute gradient corresponding to (7); compute step size using the Armijo rule with projection; update controls by nonlinear conjugate gradient; end In our particular case, the projection P U has the explicit representation ( 8)…”

“…Based on this knowledge, the investigation of the interaction of crowds and external agents became of interest [2,21]. In particular, the idea of controlling crowds with the help of the external agents [8,7]. The corresponding optimal control problem (OCP) is then constrained by the dynamics of the respective ODE or SDE system.…”

Mean-field optimal control for biological pattern formation

“…In the past years, interacting particle or agent systems have been widely used to model collective behavior in biology, sociology and economics. Among the many examples of applications are biological phenomena such as animal herding or flocking [7,8,13], cell movement [18], as well as sociological and economical processes like opinion formation [15], pedestrian flow dynamics [6,27], price formation [4], robotics [26] and data science [22].…”

“…The control of the dynamics has recently become an active research area [1,9,12,17,24,25,28]. The impact of the control on pattern formation has been studied both on the level of agents as well as in the mean-field limit, and has been applied successfully to a wide range of applications including traffic flow [20] and herds of animals [7,8].…”