On the existence of an optimal solution of the epsilon variational problem

It has been shown that known sufficient conditions for existence of optimal control imply existence of an optimal solution of the nondynamic epsilon variational problem as formulated by Balakrishnan.This work forms a part of the doctoral dissertation submitted by the author in early 1969 at UCLA and was supported in part by the Air Force Office of Scientific Research under Grant No. 69–1646. The author expresses his gratitude to Professor A. V. Balakrishnan for many useful discussions in the preparation of this work.     

Application of the epsilon technique to the identification of distributed-parameter systems

The aim of this paper is to present and discuss the application of the epsilon technique to the identification of distributed-parameter systems. After the problem formulation, attention is focused on computational aspects. The numerical results obtained for two typical systems are presented.This work was supported by ENEL. The authors are indebted to Professor A. Ruberti for having suggested the topic and for useful advice.     

On a pursuit-evasion problem under a linear change of the pursuer resource

We study a pursuit-evasion problem in the case when an integral constraint is imposed on the pursuer control class which is a generalization of integral as well as geometric constraints and only a geometric constraint is imposed on the evader control class. We prove the theorem of alternative. The optimal pursuit problem is solved by a generalized parallel pursuit strategy, and lower bounds for the distance between the pursuer and the evader are established in the pursuit problem.     

A simple pursuit-evasion problem on a ball of a Riemannian manifold

Mathematical Notes - Apursuit-evasion differential game with simple motions, in which points (players) move on a ball of a Riemannian manifold, is studied. It is assumed that all players have the...     

An optimal solution technique to the single-vendor multi-buyer integrated inventory supply chain by incorporating some realistic factors

This paper develops two generalized integrated inventory models to deliver a single product from a vendor to multiple buyers. To minimize the total cost of set up, ordering, inventory holding and transportation, the production flow is synchronized by transferring the lot with equal and/or unequal (either all are equal or all are unequal or a combination of equal and unequal) sized batches (sub-lots), each of which incurs a transportation cost. For easy implementation of the models, we relax some unrealistic assumptions in the existing models such as unlimited capacities of the transport equipment and buyers’ storage, insignificant set up and transportation times, unlimited lead time and batch sizes. A common optimal solution technique to the models is derived and their performances are analyzed. Potential significances of the solution method are highlighted with solutions of some numerical problems. The importance of the relaxed factors and limitation of the models are discussed.     

A variational technique for optimal boundary control in a hyperbolic problem

We investigate the problem of controlling the boundary functions in a one dimensional hyperbolic problem by minimizing the functional including the final state. After proving the existence and uniqueness of the solution to the given optimal control problem, we get the Frechet differential of the functional and give the necessary condition to the optimal solution in the form of the variational inequality via the solution of the adjoint problem. We constitute a minimizing sequence by the method of projection of the gradient and prove its convergence to the optimal solution.     

A multimethod technique for solving optimal control problem

A multimethod algorithm for solving optimal control problems is implemented in the form of parallel iteration processes of optimization with the choice of the best approximation. The multimethod algorithm is based on a sequence of different methods to provide fast convergence to an optimal solution. Such a technology allows one to take into account some particularities of the problem at all stages of its solution process and improve the efficiency of optimal control search.     

Application of optimal control to the cardiac defibrillation problem using a physiological model of cellular dynamics

Optimal control techniques are investigated with the goal of terminating reentry waves in cardiac tissue models. In this computational study the Luo–Rudy phase-I ventricular action potential model is adopted which accounts for more biophysical details of cellular dynamics as compared to previously used phenomenological models. The parabolic and ordinary differential equations are solved as a coupled system and an AMG preconditioner is used to solve the discretized elliptic equation. The numerical results demonstrate that defibrillation is possible by delivering a single strong shock. The optimal control approach also leads to successful defibrillation and demands less total current. The present study motivates us to further investigate optimal control techniques on realistic geometries by incorporating the structural heterogeneity in the cardiac tissue.     

Numerical solution of a relaxed control problem by the epsilon method

A relaxed control problem is solved using the epsilon method of Balakrishnan. The singularity of the problem with respect to one of the controls presents no difficulty. However, solutions are sensitive to the initial guess of the states and controls. For a system with true dynamics, terminal conditions may generally be approximated, but cannot be satisfied exactly.This paper is based on the MS Thesis of K. Shmueli at the Department of Aeronautical Engineering, Technion-Israel Institute of Technology, Haifa, Israel.     

Application of the hypodifferential descent method to the problem of constructing an optimal control

In this paper the problem of optimal control of a nonlinear ODE system with given boundary conditions and the integral restriction on control is considered. With the help of the theory of exact penalty functions the original problem is reduced to the problem of unconstrained minimization of a nonsmooth functional. The necessary minimum conditions in terms of hypodifferentials are found. A class of problems for which these conditions are also sufficient is distinguished. On the basis of these conditions the hypodifferential descent method is applied to the considered problem. Under some additional assumptions the hypodifferential descent method converges in a certain sense.     

Application of a conformal-mapping technique to a boundary-value problem of current distribution in plain conductors

Using the conformal-mapping technique, we have found analytical solutions of the current-distribution problem for different configurations of plain conductors.     

Computing a global optimal solution to a design centering problem

In this paper we present a method for solving a special three-dimensional design centering problem arising in diamond manufacturing: Find inside a given (not necessarily convex) polyhedral rough stone the largest diamond of prescribed shape and orientation. This problem can be formulated as the one of finding a global maximum of a difference of two convex functions over ³ and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated. Here we prove that with the information available on the rough stone and on the reference diamond, evaluating the objective function at a pointx amounts to computing the distance, with respect to a Minkowski gauge, fromx to a finite number of planes. We propose a method for finding these planes and we report some numerical results.     

On a problem of optimal search

Summary A class of one-dimensional search problems is considered. The formulation results in a functional-minimization equation of the dynamic programming type. In the case of a uniform a priori distribution for the location of the hidden object an optimality criterion is established and the optimal search procedure is found.

---

Zusammenfassung Nach allgemeinen Betrachtungen über das Spektrum von Suchproblemen wird eine Klasse eindimensionaler Suchprobleme betrachtet. Die Formulierung führt zu einer Funktionalgleichung vom Typ der dynamischen Optimierung. In dem Fall einer uniformen a priori Verteilung für das Auffinden eines versteckten Objekts wird ein Optimalitätskriterium begründet und das optimale Suchverfahren angewandt.

     

Analytically measurable selection of epsilon optimal transition kernals

The problem of existence of ah ε-optimal transition kernel for a canonical continuous time stochastic process with a general cost variable is considered. An analytically measurable, ε-optimal kernel exists if the state space is a compact Banach space, the cost variable is lower-semi-analytic, and the graph of the admissibility function is an analytic set. The result is applied to a problem in which the controller is to optimally select transition probabilities for a non-Markovian step process based on statistical estimates of holding time distributions.     

Asymptotics of a solution to an optimal boundary control problem

A problem of optimal boundary control of solutions to an elliptic-type equation with a small coefficient at the highest derivative and integral constraints on the control is considered. Asymptotic estimates for solutions to a problem that approximates the original problem are obtained.     

On a simple optimal stopping problem

L.A. Shepp has posed and analyzed the problem of optimal random drawing without replacement from an urn containing predetermined numbers of plus and minus balls. Here Shepp's results are extended by improving the bounds on values of perturbed urns, deriving an exact algorithm for the urn values and computing the stopping boundary for urns of up to 200 balls.     

The optimal solution to the problem of complex extrapolation of a first-order scheme

Given the linear stationary first-order iterative scheme $\text{x(}{}^{\mathrm{m+1}}\text{= Tx(}{}^{\mathrm{m}}\text{+ c}$ for the solution of the linear complex system (I ? T)x = c, its extrapolated complex scheme $\text{x(}{}^{\mathrm{m+1}}\text{) = T}{}_{\mathrm{\omega }}\text{x(}{}^{\mathrm{m}}\text{) + ωc [T}{}_{\mathrm{\omega }}\text{≡ (1 ? ω)I + ωT]}$ is considered. The problem which is studied and solved is that of determining an optimum value for ω, over the set of complex numbers, such that the extrapolated scheme considered converges asymptotically as fast as possible.