Optimization Processes of Tangible and Intangible Networks through the Laplace Problems for Regular Lattices with Multiple Obstacles along the Way

A systematic approach is proposed to the theme of safety,reliability and global quality of complex networks(material and immaterial)by means of special mathematical tools that allow an adequate geometric characterization and study of the operation,even in the presence of multiple obstacles along the path.To that end,applying the theory of graphs to the problem under study and using a special mathematical model based on stochastic geometry,in this article we consider some regular lattices in which it is possible to schematize the elements of the network,with the fundamental cell with six,eight or 2(n+2)obstacles,calculating the probability of Laplace.In this way it is possible to measure the“degree of impedance”exerted by the anomalies along the network by the obstacles examined.The method can be extended to other regular and/or irregular geometric figures,whose union together constitutes the examined network,allowing to optimize the functioning of the complex system considered.

Ehrhart Polynomials of 3-Dimensional Simple Integral Convex Polytopes

The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry.

General Convex Integral Control

In this paper, a fire-new general integral control, named general convex integral control, is proposed. It is derived by defining a nonlinear function set to form the integral control action and educe a new convex function gain integrator, introducing the partial derivative of Lyapunov function into the integrator and resorting to a general strategy to transform ordinary control into general integral control. By using Lyapunov method along with the LaSalle s invariance principle, the theorem to ensure regionally as well as semi-globally asymptotic stability is established only by some bounded information. Moreover, the lemma to ensure the integrator output to be bounded in the time domain is proposed. The highlight point of this integral control strategy is that the integral control action seems to be infinity, but it factually is finite in the time domain. Therefore, a simple and ingenious method to design the general integral control is founded. Simulation results showed that under the normal and perturbed cases, the optimum response in the whole control domain of interest can all be achieved by a set of control gains, even under the case that the payload is changed abruptly.