About an Optimal Visiting Problem

Abstract

In this paper we are concerned with the optimal control problem consisting in minimizing the time for reaching (visiting) a xed number of target sets, in particular more than one target. Such a problem is of course reminiscent of the famous Traveling Salesman Problem and brings all its computational di culties. Our aim is to apply the dynamic programming technique in order to characterize the value function of the problem as the unique viscosity solution of a suitable Hamilton-Jacobi equation. We introduce some external variables, one per target, which keep in memory whether the corresponding target is already visited or not, and we transform the visiting problem in a suitable Mayer problem. This fact allows us to overcome the lacking of the Dynamic Programming Principle for the originary problem. The external variables evolve with a hysteresis law and the Hamilton-Jacobi equation turns out to be discontinuous.

A hybrid control framework for an optimal visiting problem

“…Furthermore, the dynamical nature of the problem poses some additional difficulties. It has already been observed in Bagagiolo and Benetton (2012) as the problem is not a mere sequence of minimal time problems. In particular, to recover the dynamic programming property, it requires a special framework able to include a "memory" of the targets already visited.…”

First-order mean-field games on networks and Wardrop equilibrium

“…However, prior methods for MFGs on networks are not valid for first-order MFGs, where a distinct set of phenomena occurs that includes the loss of smoothness for Hamilton-Jacobi (HJ) equations and lack of continuity for the value function at the vertices. First-order MFGs on networks were considered in [BM], [BFMP19], [BMP21] and [BB12], in particular in the context of optimal visiting problems where agents have multiple targets. The methods in those papers, because are applied to general time-dependent problems are quite different from ours, where we take full advantage of the stationary nature of the game.…”

Hybrid control for optimal visiting problems for a single player and for a crowd

“…If, for example, we consider the problem of visiting first target T 1 and then T 2 , we can easily observe that we would obtain a different problem just swapping the order of the visit. This is a consequence because no Dynamical Programming Principle would be available for the function v(x, t), since the only information brought by the state-position x does not give information about the already visited targets (see also [3]). Hence, at this level, we can not in general characterize the optimal visiting time as a solution of a Hamilton-Jacobi-Bellman equation.…”

“…In particular, the hybrid framework that we propose is related to hybrid control [15] and somehow to the mathematical switching hysteresis models [34]. The need of memory feature, associated with the optimal visiting and dynamic programming and Hamilton-Jacobi equations has been presented in [3], where a continuous hysteresis memory was introduced. The use of a switching/discontinuous/hybrid memory, as in the present paper, was instead used for a one-dimensional optimal visiting problem on a network in [5].…”