Eszter Sikolya

Variational and Semigroup Methods for Waves and Diffusion in Networks

We study diffusion and wave equations in networks. Combining semigroup and variational methods we obtain well-posedness and many nice properties of the solutions in general Lp-context. Following earlier articles of other authors, we discuss how the spectrum of the generator can be connected to the structure of the network. We conclude by describing asymptotic behavior of solutions to the diffusion problem.

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.

Asymptotic behavior of flows in networks

Abstract Using functional analytical and graph theoretical methods, we extend the results of [Kramar M. und Sikolya E.: Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249 (2005), 139–162] to more general transport processes in networks allowing space dependent velocities and absorption. We characterize asymptotic periodicity and convergence to an equilibrium by conditions on the underlying directed graph and the (average) velocities.

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size (

Semigroups of max-plus linear operators

N

The norm convergence of a Magnus expansion method

) and can be written as

Spectral properties and asymptotic periodicity of flows in networks

\dot u_N=A_N u_N

Flows in networks with dynamic ramification nodes

. Our results rely on the convergence of the transition matrices

Maximal Controllability for Boundary Control Problems

A_N

Asymptotic periodicity of recurrent flows in infinite networks

to an operator