Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume

An equivalent condition of continuous metric selection

AbstractAn intrinsic characterization is given of those finite-dimensional subspaces whose metric projections admit continuous selections

Best approximation properties in spaces of measurable functions

We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fr\'echet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces $d(w,1)$.Comment: 26 page

Dissipative Lipschitz dynamics

In this dissertation we study two related important issues in control theory: invariance of dynamical systems and Hamilton-Jacobi theory associated with optimal control theory. Given a control system modelled as a differential inclusion, we provide necessary and sufficient conditions for the strong invariance property of the system when the dynamic satisfies a dissipative Lipschitz condition. We show that when the dynamic is almost upper semicontinuous and satisfies the dissipative Lipschitz property, these conditions can be expressed in terms of approximate Hamilton-Jacobi inequalities, which subsumes the classic infinitesimal characterization of strongly invariant systems given under the Lipschitz assumtion. In the important case when the dynamic of the system is the sum of a maximal dissipative and a Lipschitz multifunction, the approximate inequalities turn into an exact mixed type inequality that involves the lower and upper Hamiltonian of the dissipative and the Lipschitz piece respectively. We then extend this Hamiltonian characterization to nonautonomous systems by assuming a potentially discontinuous differential inclusion whose right-hand side is the sum of an almost upper semicontinuous dissipative and an almost lower semicontinuous dissipative Lipschitz multifunction. Finally, a Hamilton-Jacobi theory is developed for the minimal time problem of a system with possibly discontinuous monotone Lipschitz dynamic. This is achieved by showing the minimal time function associated to an upper semicontinuous and a monotone Lipschitz data is characterized as the unique proximal semi-solution to an approximate Hamilton-Jacobi equation satisfying an analytical boundary condition

Existence of nash equilibrium in atomless games

Master'sMASTER OF SCIENC

Differential, energetic, and metric formulations for rate-independent processes

We consider different solution concepts for rate-independent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banach-space setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rate-independent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces

Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page