Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems

It is shown that, if a parametrized fämily of extremals F can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity.     

Maximin share and minimax envy in fair-division problems

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.     

Upper and lower bounds for the energy of bipartite graphs

Using Lagrange's multiplier rule, we find upper and lower bounds of the energy of a bipartite graph G, in terms of the number of vertices, edges and the spectral moment of fourth order. Moreover, the upper bound is attained in a graph G if and only if G is the graph of a symmetric balanced incomplete block design (BIBD). Also, we determine the graphs for which the lower bound is sharp.     

Optimal Control of Magnetohydrodynamic Equations with State Constraint

This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of viscous incompressible fluid flow and the Maxwell equation of electromagnetic field. An integral type state constraint is considered.     

Sequent calculi for propositional star-free likelihood logic

We consider classical, multisuccedent intuitionistic, and intuitionistic sequent calculi for propositional likelihood logic. We prove the admissibility of structural rules and cut rule, invertibility of rules, correctness of the calculi, and completeness of the classical calculus with respect to given semantics.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 3–21, January–March, 2005.     

非线性不适定问题一种双循环的牛顿型迭代格式

本文研究非线性算子方程F(x)=y的解,结合最速下降法,Newton-Landweber迭代格式及正则化思想,在F满足适当的条件下,构造出新的双循环迭代格式。本文对格式的收敛性进行了严格论证,并估计出迭代格式的收敛精度。     

Optimal Path Planning Based on Visibility

Various problems associated with optimal path planning for mobile observers such as mobile robots equipped with cameras to obtain maximum visual coverage of a surface in the three-dimensional Euclidean space are considered. The existence of solutions to these problems is discussed first. Then, optimality conditions are derived by considering local path perturbations. Numerical algorithms for solving the corresponding approximate problems are proposed. Detailed solutions to the optimal path planning problems for a few examples are given.     

Load-Carrying Capacity of Circular Plates Made of a Fiber-Reinforced Composite

The load-carrying capacity of circular plates made of a fiber-reinforced composite and loaded with axisymmetric transverse forces under axisymmetric boundary conditions is investigated. The plates can be freely supported or clamped along their contour. To the known yield condition used, in the plane of main bending moments, there correspond irregular hexagons, which depend on the mechanical properties of composite constituents and some characteristic geometrical parameters. It is shown that, depending on the ratio of ultimate radial and circumferential bending moments, in the plastic state, the circular plates (the central part of the clamped plates) take the form of a cone or a frustum of cone. For each of the cases, the limit load is found, which is a function of the above-mentioned mechanical parameters. The statically allowable fields of bending moments and the corresponding allowable fields of deflection rates are determined. The solutions found are exact. Special cases for the freely supported plates are analyzed, and the results obtained are presented in the form of graphs.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 177–192, March–April, 2005.     

An alternative approach for non-linear optimal control problems based on the method of moments

We propose an alternative method for computing effectively the solution of non-linear, fixed-terminal-time, optimal control problems when they are given in Lagrange, Bolza or Mayer forms. This method works well when the nonlinearities in the control variable can be expressed as polynomials. The essential of this proposal is the transformation of a non-linear, non-convex optimal control problem into an equivalent optimal control problem with linear and convex structure. The method is based on global optimization of polynomials by the method of moments. With this method we can determine either the existence or lacking of minimizers. In addition, we can calculate generalized solutions when the original problem lacks of minimizers. We also present the numerical schemes to solve several examples arising in science and technology.     

Optimal stopping problems by two or more decision makers: a survey

A review of the optimal stopping problem with more than a single decision maker (DM) is presented in this paper. We classify the existing literature according to the arrival of the offers, the utility of the DMs, the length of the sequence of offers, the nature of the game and the number of offers to be selected. We enumerate various definitions for this problem and describe some dynamic approaches. Fouad Ben Abdelaziz is on leave from the Institut Superieur de Gestion, University of Tunis, Tunisia e-mail: foued.benabdelaz@isg.run.tn.