Vertex control of flows in networks

We study a transport equation in a network and control it in a single vertex. We describe all possible reachable states and prove a criterion of Kalman type for those vertices in which the problem is maximally controllable. The results are then applied to concrete networks to show the complexity of the problem

Bi-continuous semigroups for flows in infinite networks

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the $\mathrm{L}^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.Comment: 12 page

The vanishing viscosity limit for Hamilton-Jacobi equations on Networks

For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem

Transport of measures on networks

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

We consider an optimal control on networks in the spirit of the works of Achdou et al. (2013) and Imbert et al. (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. (2014) and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis (2016).Comment: ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, A Para\^itr

On the asymptotic behaviour of semigroups for flows in infinite networks

We study transport processes on infinite networks. The solution of these processes can be modeled by an operator semigroup on a suitable Banach space. Classically, such semigroups are strongly continuous and therefore their asymptotic behaviour is quite well understood. However, recently new examples of transport processes emerged where the corresponding semigroup is not strongly continuous. Due to this lack of strong continuity, there are currently only few results on the long-term behaviour of these semigroups. In this paper, we discuss the asymptotic behaviour for a certain class of these transport processes. In particular, it is proved that the solution semigroups behave asymptotically periodic with respect to the operator norm as a consequence of a more general result on the long-term behaviour by positive semigroups containing a multiplication operator. Furthermore, we revisit known results on the asymptotic behaviour of transport processes on infinite networks and prove the asymptotic periodicity of their extensions to the space of bounded measures.Comment: Correction of typos, rewritten introduction, some further simplifications of arguments, strengthened main result -- final versio

Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction

We consider a family of open star-shaped domains made of a finite number of non intersecting semi-infinite strips of small thickness and of a central region whose diameter is of the same order of thickness, that may be called the junction. When the thickness tends to 0, the domains tend to a union of half-lines sharing an endpoint. This set is termed "network". We study infinite horizon optimal control problems in which the state is constrained to remain in the star-shaped domains. In the above mentioned strips the running cost may have a fast variation w.r.t. the transverse coordinate. When the thickness tends to 0 we prove that the value function tends to the solution of a Hamilton-Jacobi equation on the network, which may also be related to an optimal control problem. One difficulty is to find the transmission condition at the junction node in the limit problem. For passing to the limit, we use the method of the perturbed test-functions of Evans, which requires constructing suitable correctors. This is another difficulty since the domain is unbounded

Hamilton–Jacobi equations for optimal control on junctions and networks

erratum de l'article dans ESAIM COCV, 2016, vol. 22 n° 2, pp. 539-542 ;doi : 10.1051/cocv/2016005We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. A notion of viscosity solution of Hamilton-Jacobi equations on the network has been proposed in earlier articles. Here, we propose a simple proof of a comparison principle based on arguments from the theory of optimal control. We also discuss stability of viscosity solutions. Résumé. On consid ere desprobì emes de contrôle optimal pour lesquels l'´ etat est contraint a rester sur un réseau. Une notion de solution de viscosité des equations de Hamilton-Jacobi associées a ´ eté proposée dans des travaux antérieurs. Ici, on propose une preuve simple d'un principe de comparaison. Cette preuve est basée sur des arguments de contrôle optimal. La stabilité des solutions de viscosité est aussí etudiée