Mean-field optimal control (Q2926570)

In this very important paper the authors study the discrete (finite-dimensional-continuum) infinite dimensional limit for \(N \rightarrow \infty\) of ODE constrained control problems of the type:NEWLINENEWLINE(1) \(\dot{x}_i = v_i, \dot{v}_i = (H * \mu_N)(x_i v_i)+ f(t, x_i, v_i), i=1,\dots,N, t \in [0,T]\),NEWLINENEWLINEwhere \(\mu_N=\frac{1}{N} \sum^N_{j=1} S_{(x_i v_i)}\), is the empirical atomic measure supported on the agents states \((x_i, v_i)\in \mathbb R^{2d}\), controlled by the minimizer of the cost functionalNEWLINENEWLINE(2) \(\varepsilon^N_v (f): = \int ^T_0 \int_{\mathbb R^{2d}} (L(x,v,\mu_N (t)) + \psi (f(t,x,v)))d \mu_N (t) (x,v)dt\).NEWLINENEWLINEThe authors prove the existence of controls for (1) and (2) based on compactness arguments for the considered class of feedback control functions. The novelty with respect to the usual closed loop control problems stems precisely from the feedback from the control in terms of a locally Lipschitz continuous function \(f(t,.,.)\) of the state variables \((x_j, v_j)\) for \(j=1,2,\dots,N\) in order to grasp, although only intuitively, this fundamental difference.NEWLINENEWLINEThe main result of the proposed work is to clarify in which sense the finite dimensional solutions of (1) and (2) converge for \(N \rightarrow \infty\) to a solution of the PDE constrained problemNEWLINENEWLINE(3) \((\partial \mu / \partial t) + v \cdot \nabla_x \mu = \nabla_v \cdot [H * \mu + f) \mu]\),NEWLINENEWLINEcontrolled by the minimizer \(f\) of the cost functionalNEWLINENEWLINE(4) \(\varepsilon_\psi (f): = \int ^T_0 \int_{\mathbb R^{2d}} (L(x,v,\mu (t)) + \psi (f(t,x,v)))d \mu (t) (x,v)dt\).NEWLINENEWLINEThe authors' arguments are based on the combination of the concept of mean-field, using techniques of optimal transport in order to connect (1) to (3) and the concept of \(\Gamma\)-limit in order to connect the minimizations of (2) and (4) (mean field optimal control).