Optimal shape design of systems governed by variational inequalities

In this paper, we present some results concerning the optimal shape design problem governed by the variational inequalities of the fourth order. This problem can be considered as a model example for the design of the shape for elastic-plastic problem. The performance criterion is minimized by the material derivative method.     

Quadratic systems with a linear reflecting function

We construct a set of nonautonomous differential systems with quadratic right-hand side and with a linear reflecting function. We also construct a set of systems whose reflecting function coincides with that of a linear autonomous system.     

On quadratic differential systems with equal reflecting functions

For a two-dimensional quadratic system, we obtain necessary conditions for the existence of a triangular quadratic system with the same Mironenko reflecting function as the original system. We suggest an algorithm that permits establishing the coincidence of the reflecting functions of a quadratic nonstationary system and some stationary system.     

The differential systems with the same reflecting function

Using reflecting function, we give the conditions under which some complex differential systems are equivalent to the relatively simple differential system. The obtained results are applied to study the behavior of solutions of these complex differential systems.     

Optimal design of unreliable production–inventory systems with variable production rate

In this article, we study an economic manufacturing quantity (EMQ) problem for an unreliable production facility where the production rate is treated as a decision variable. As the stress condition of the machine changes with the production rate, the failure rate of the machine is assumed to be dependent on the production rate. The unit production cost is also taken as a function of the production rate, as the machine can be operated at different production rates resulting in different unit production costs. The basic EMQ model is formulated under general failure and general repair time distributions and the optimal production policy is derived for specific failure and repair time distributions viz., exponential failure and exponential repair time distributions. Considering randomness of the time to machine failure and corrective repair time, the model is extended to the case where certain safety stocks in inventory may be useful to improve service level to customers. Optimal production policies of the proposed models are derived numerically and the sensitivity of the optimal results with respect to those parameters which directly influence the machine failure and repair rates is also examined.     

On the existence of optimal control of temperature regimes

This paper considers the control problem with boundary condition for the one-dimensional parabolic equation on a bounded interval. For the quadratic cost functional, the author proves the existence of a minimizing function. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Optimal key systems

A key system is represented by an ordered rooted tree. We construct trees minimizing relevant cost functions. Our application concerns interactive voice response (IVR), or automated telephone operators, for navigation purposes.     

Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type “admissible strategy” against “admissible control”, and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman–Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadène, Hassani (Probab Theory Relat Fields 132:237–264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier. Partially supported by the NSF of P.R.China (No. 10701050; 10671112), Shandong Province (No. Q2007A04), and National Basic Research Program of China (973 Program) (No. 2007CB814904).     

Optimal design of multi-purpose structures

The paper presents the optimal (maximum transverse stiffness) design of an elastic, simply supported member of given volume that is to serve as a beam or as a column at different times during its design life. The optimal design can be interpreted in two ways. It is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection in beam action under a uniformly distributed load; it is also the design that has the least deflection at midspan under a uniformly distributed load, subject to a lower limit on its buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.The author is grateful to Professor W. Prager for suggesting several improvements to an earlier version of the paper.     

Optimal design of multi-purpose beam-columns

The paper deals with the optimal design of an elastic pinended member of given volume that is to serve as a beam for a part of its design life and as a column for the rest. The optimal design can be interpreted in two ways. Firstly, it is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection as a beam. Secondly, it is the design that has the least deflection as a beam under a midspan concentrated load, subject to a minimum permissible Euler buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.     

Optimal design of multi-purpose tie-beams

The paper presents a solution of the problem of minimizing the maximum deflection of a simply-supported beam under a transverse concentrated load. The volume (mass) of the beam is given, as is the maximum longitudinal elongation if the beam were to act as a tie. An optimal design for two requirements (beam and tie action) not only unifies the design procedure of mass-produced structural-mechanical elements, but also provides a practically acceptable design, in the sense that the resulting shape does not vanish at any point along its length, a drawback of many optimal designs for a single requirement. However, it is shown that, for cross sections of solid construction, as opposed to those of sandwich construction, the constraint on longitudinal elongation is weaker than on the (finite) maximum stress. The effectiveness of the optimal design is judged by comparing it with a prismatic member of the same volume.     

Optimal observation of multistage systems

The problem of optimal observation of a nonlinear step system is considered. The task is to obtain guaranteed estimates of the initial state from given incomplete and inaccurate measurements of the current states. The cases of fixed and unfixed transition times in a step system are examined. Algorithms for solving a posteriori and positional observation problems are proposed. The results are illustrated by examples.     

Optimal resource allocation in a two-sector economic model with singular regimes

The article considers the problem of resource allocation in a two-sector economic model with a nonlinear production function of a special type. The main mathematical apparatus is Pontryagin’s maximum principle, i.e., the theorem on necessary conditions of optimality. It is shown that in the given problem the maximum principle provides a necessary and sufficient condition of optimality. A possible singular solution of the problem is found. An extremum solution is constructed in explicit form under various assumptions about the initial values. A “sufficiently long” planning horizon is assumed. An alternative approach is described, which does not use the maximum principle and instead investigates the integral representation of the optimand functional. The detailed theoretical investigation of the problem is accompanied by numerous illustrations.     

Stationary and stationarizable regimes in normal stochastic differential systems

Narrow-sense stationary regimes are considered for multi-dimensional non-linear systems described by Ito stochastic differential equations with Wiener processes. The conditions for the existence of stationary and stationarizable one-dimensional distributions are derived. Exact expressions are obtained for stationary distributions in some mechanical systems.     

Optimal design for smoothing splines

In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.     

Optimal design and control of queues

We have divided this review into two parts. The first part is concerned with the optimal design of queueing systems and the second part deals with the optimal control of queueing systems. The second part, which has the lion’s share of the review since it has received the most attention, focuses mainly on the modelling aspects of the problem and describes the different kinds of threshold (control) policy models available in the literature. To limit the scope of this survey, we decided to limit ourselves to research on papers dealing with the three policies (N, T, and D), where a cost function is designed specifically and optimal thresholds that yield minimum cost are sought.