Mauro Garavello

We consider a min-max problem for strictly concave conservation laws on a 1-1 network, with inflow controls acting at the junction. We investigate the minimization problem for a functional measuring the total variation of the flow of the solutions at the node, among those solutions that maximize the time integral of the flux. To formulate this problem we establish a regularity result showing that the total variation of the boundary-flux of the solution of an initial-boundary value problem is controlled by the total variation of the initial datum and of the flux of the boundary datum. In the case the initial datum is monotone, we show that the flux of the entropy weak solution at the node provides an optimal inflow control for this min-max problem. We also exhibit two prototype examples showing that, in the case where the initial datum is not monotone, the flux of the entropy weak solution is no more optimal.

This paper deals with the concepts of measure controls and of measure vector fields, within the mathematical framework of measure differential equations (MDEs), recently proposed in~\cite{piccoli_measure_2019}. Measure controls can be seen as a generalization of relaxed control. Moreover, they are particularly suitable for studying dynamics with uncertainty. The main results of this paper include establishing the existence and well-posedness of control systems with measure controls and proving the equivalence between measure controls and measure vector fields. The stability and closure properties of the trajectory set are also studied.

We present a macroscopic traffic flow model where standard vehicles coexist with vehicles informed on the traffic distribution. The resulting mixed nonlocal-local integro-differential PDEs is proved to generate a locally Lipschitz continuous semigroup whose orbits are uniquely characterized as solutions to the system, according to a natural definition of solution. The norms and function spaces adopted are intrinsic to the different nature of the equations.

Motivated by a vaccination coverage problem, we consider here a zero-sum differential game governed by a differential system consisting of a hyperbolic partial differential equation (PDE) and an ordinary differential equation (ODE). Two players act through their respective controls to influence the evolution of the system with the aim of minimizing their objective functionals [Formula: see text] and [Formula: see text], under the assumption that [Formula: see text]. First we prove a well-posedness and a stability result for the differential system, once the control functions are fixed. Then we introduce the concept of nonanticipating strategies for both players, and we consider the associated value functions, which solve two infinite-dimensional Hamilton–Jacobi–Isaacs equations in the viscosity sense.

We consider a min-max problem for strictly concave conservation laws on a 1-1 network, with inflow controls acting at the junction. We investigate the minimization problem for a functional measuring the total variation of the flow of the solutions at the node, among those solutions that maximize the time integral of the flux. To formulate this problem we establish a regularity result showing that the total variation of the boundary-flux of the solution of an initial-boundary value problem is controlled by the total variation of the initial datum and of the flux of the boundary datum. In the case the initial datum is monotone, we show that the flux of the entropy weak solution at the node provides an optimal inflow control for this min-max problem. We also exhibit two prototype examples showing that, in the case where the initial datum is not monotone, the flux of the entropy weak solution is no more optimal.

The formation, movement and gluing of clusters can be described through a system of nonlocal conservation laws. Here, the well-posedness of this system is obtained, as well as various stability estimates. Remarkably, qualitative properties of the solutions are proved, providing information on stationary solutions and on the propagation speed. In some cases, fragmentation leads to clusters developing independently. Moreover, these equations may serve as an encryption/decryption tool. This poses new analytical problems and asks for improved numerical methods.