Invisible Control of Self-Organizing Agents Leaving Unknown Environments

Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures Invisible Control of Self-Organizing Agents Leaving Unknown Environments Mattia Bongini (TUM), Massimo Fornasier (TUM) Problem and Goals • We are concerned with the multiscale modeling, control and simulation of self-organizing agents leaving an unknown area. • In [1] we explore the possibility of sparsely controlling these systems with a bottom-up approach, where control on the crowd (followers) is obtained by means of very few aware agents (leaders), hidden in the crowd and not recognized by followers. • In [3] we compute optimality conditions for optimal controls when the interaction among agents is of mean-field type. The microscopic model We propose a microscopic model for a human crowd leaving an unknown environment under limited visibility. Due to their lack of information about the positions of exits, agents need to explore the environment first. Let Ω denote the area to be evacuated, x be the exit and Σ be its visibility zone (i.e., followers can see the exit inside Σ). The microscopic dynamics described by NF followers and NL leaders is: for i = 1, . . . ,NF and k = 1, . . . ,NL,  ẋi = vi, v̇i = A(xi, vi) + ∑NF j=1H(xi, vi, xj , vj ; x, y) + ∑NL `=1H(xi, vi, y`,w`; x, y), ẏk = wk = ∑NF j=1K (yk, xj) + ∑NL `=1K (yk, y`) + uk. (1) • A is a self-propulsion term of the form A(x , v ) := (1− χΣ(x))Cz(z − v ) + χΣ(x)Cτ ( x − x |xτ − x | − v ) + Cs(s − |v |)v } {{ } :=S(x ,v ) , where s ≥ 0 is a given characteristic cruise speed,z is a 2-dimensional random vector with normal distribution N (0, σ2), and Cz , Cτ , Cs > 0. • The interactions follower-follower and follower-leader are given by H(x , v , y ,w ; x, y) := −C F r Rγ,r(x , y ) + (1− χΣ(x))χBN (x ;x,y)(y ) Ca N ∗ (w − v ) , for given positive constants C F r ,Ca, r , and γ, and where Rγ,r(x , y ) = { e−|y−x | γ y−x |y−x | if y ∈ Br(x)\{x}, 0 otherwise. The function Rγ,r models a metrical repulsive force, while the second term accounts for the topological alignment force, which vanishes inside Σ. Followers do not distinguish between other followers and leaders! • BN (x ; x, y) is the minimal ball centered at x encompassing at least N agents, and N ∗ is the actual number of agents in BN (x ; x, y). Computing BN (x ; x, y) requires the knowledge of the positions of all agents, given by x and y. • The interactions leader-follower and leader-leader reduce to a mere (metrical) repulsion, i.e., K (x , y ) = C L r Rζ,r(x , y ), where C L r , ζ > 0 are in general different from C F r and γ. Here the repulsion force is a velocity field, while for followers it is an acceleration. • uk is the control chosen in two ways: as the unit vector pointing towards the exit (fixed strategy), or as a solution in the set of admissible controls Uadm of min u(·)∈Uadm {t > 0 | xi(t) / ∈ Ω, ∀i = 1, . . . ,N}, subject to (1). (2) The mesoscopic model • Our interest in (1) lies in the case NL NF, that is the population of followers exceeds by far the one of leaders. • When NF is so large, a microscopic description of both populations is no more a viable option, thus we consider the evolution of the distribution of followers, denoted by f (x , v ), together with the microscopic equations for the leaders (whose number is still small), with distribution g(x , v ). • Their evolution is described by the following system (studied in [3]) { ∂ ∂tf + v · ∇xf = −∇v · (G [f , g] f ) + 1 2σ 2C 2 z (1− χΣ)∆v f , ẏk = wk = ∫ R2d K (yk, x)f (x , v )dxdv + ∑NL `=1K (yk, y`) + uk, (3) where G [f , g] (x , v ) = S(x , v ) + ∫ R2d H(x , v , x̂ , v̂ ;π1f , π1g) (f (x̂ , v̂ ) + g(x̂ , v̂ )) d x̂d v̂ . • To simulate the above coupled ODE-PDE system we first derive a Boltzmanntype dynamics obtained by the binary interactions follower-follower and followerleader, then we recover the Fokker-Planck operator in (3) by means of a grazing interaction limit, as in [8], and we use a Monte Carlo-based method, see [7]. Simulations of the microscopic model