We consider a class of optimal control problems for measure-valued nonlinear transport equations describing traffic flow problems on networks. The objective is to minimise/maximise macroscopic quantities, such as traffic volume or average speed, controlling few agents, e.g. smart traffic lights and automated cars. The measure theoretic approach allows to study in a same setting local and non-local drivers interactions and to consider the control variables as additional measures interacting with the drivers distribution. We also propose a gradient descent adjoint-based optimisation method, obtained by deriving first-order optimality conditions for the control problem, and we provide some numerical experiments in the case of smart traffic lights for a 2–1 junction.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

By modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the ‘unmodified’ theory.

We consider a discontinuous Sturm–Liouville equation together with two supplementary transmission conditions at the point of discontinuity. We suggest our own approach for finding asymptotic approximation formulas for the eigenvalues of such discontinuous problems.

We investigate time harmonic Maxwell equations in heterogeneous media, where thepermeability μ and the permittivity ε are piecewise constant. Theassociated boundary value problem can be interpreted as a transmission problem. Ina very natural way the interfaces can have edges and corners. We give a detaileddescription of the edge and corner singularities of the electromagnetic fields.