Dynamics and control for multi-agent networked systems: A finite-difference approach

Abstract

We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one.We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered.Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.2010 Mathematics Subject Classification. 34D20, 35B36, 65M06, 92D25, 93B05.

A Parabolic Approach to the Control of Opinion Spreading

“…• The result above holds in long time horizons of the order of N 2 T. This allows controlling the system with uniformly bounded controls. In case the control time horizon [0, T] were fixed, independent of T, as we shall see, the control would grow exponentially in N 2 , as it occurs in the linear setting (see [5]). • Similarly, the cost of control at time t = N 2 T for a nonlinearity G N (y) = (g N (y 1 )y 1 , · · · , g N (y N )y N ) T , independent of N, would also grow exponentially with N:…”

“…. Recently, in [5], these issues were addressed in the linear setting (G = 0) using spectral techniques [11], [10]. In particular, the following result was obtained:…”

“…Whether this reminder can be dropped to assure the exact reachability of consensus is an open problem. As pointed out in [5], this reminder can be avoided in the linear setting. • The result above holds when the control is active in both extremes.…”

“…Roughly speaking, the three systems presented above can be treated similarly and exhibit the same control properties. For the sake of clarity we present them in the context of the original control system (5).…”

“…• First of all, in Section 2, we present some well known results on the null controllability of parabolic equations, linear and semilinear, their finite-difference counterparts, and a brief summary of [5]. • In Section 3 we analyse the divergent behaviour of the control properties when the nonlinearity is scaled differently.…”