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Numerical Algebra, Control & Optimization

March 2017 , Volume 7 , Issue 1

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Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence
Valery Y. Glizer and Oleg Kelis
2017, 7(1): 1-20 doi: 10.3934/naco.2017001 +[Abstract](746) +[HTML](2) +[PDF](471.0KB)
Abstract:

We consider an infinite horizon zero-sum linear-quadratic differential game in the case where the cost functional does not contain a control cost of the minimizing player (the minimizer). This feature means that the game under consideration is singular. For this game, novel definitions of the saddle-point equilibrium and game value are proposed. To obtain these saddle-point equilibrium and game value, we associate the singular game with a new differential game for the same equation of dynamics. The cost functional in the new game is the sum of the original cost functional and an infinite horizon integral of the square of the minimizer's control with a small positive weight coefficient. This new game is regular, and it is a cheap control game. Using the solvability conditions, the solution of the cheap control game is reduced to solution of a Riccati matrix algebraic equation with an indefinite quadratic term. This equation is perturbed by a small parameter. Subject to a proper assumption, an asymptotic expansion of a stabilizing solution to this equation is constructed and justified. Using this asymptotic expansion, the existence of the saddle-point equilibrium and the value of the original game is established, and their expressions are derived. Illustrative example is presented.

A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items
Magfura Pervin, Sankar Kumar Roy and Gerhard Wilhelm Weber
2017, 7(1): 21-50 doi: 10.3934/naco.2017002 +[Abstract](997) +[HTML](4) +[PDF](716.2KB)
Abstract:

In this study, we develop an inventory model for deteriorating items with stock dependent demand rate. Shortages are allowed to this model and when stock on hand is zero, then the retailer offers a price discount to customers who are willing to back-order their demands. Here, the supplier as well as the retailer adopt the trade credit policy for their customers in order to promote the market competition. The retailer can earn revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. Besides this, we consider variable holding cost due to increase the stock of deteriorating items. Thereafter, we present an easy analytical closed-form solution to find the optimal order quantity so that the total cost per unit time is minimized. The results are discussed with the help of numerical examples to validate the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided in order to stabilize our model. The paper ends with a conclusion and an outlook to possible future studies.

Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence
Erik Kropat
2017, 7(1): 51-76 doi: 10.3934/naco.2017003 +[Abstract](447) +[HTML](3) +[PDF](612.3KB)
Abstract:

The homogenization of optimal control problems on periodic networks is considered. Traditional approaches for a homogenization of uncontrolled problems on graphs often rely on an artificial extension of branches. The main result shows that such an extension to thin domains is not required. A two-scale transform for network functions leads to a representation of the microscopic optimal control problem on the graph in terms of a two-scale transformed minimization problem that allows for a further homogenization. Here, the concept of $S$-homogenization is applied in order to prove the existence of an absolutely $S$-homogenized optimal control problem with respect to the superior domain and the microscopic scale encoded in the reference graph of the network. In addition, results on the $Γ$-convergence of optimal control problems on periodic networks are discussed.

Effective approximation method for solving linear Fredholm-Volterra integral equations
Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib and N. M. A. Nik Long
2017, 7(1): 77-88 doi: 10.3934/naco.2017004 +[Abstract](713) +[HTML](1) +[PDF](361.5KB)
Abstract:

An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given.

The soft landing problem for an infinite system of second order differential equations
Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias and Mai Zurwatul Ahlam Mohd Jaffar
2017, 7(1): 89-94 doi: 10.3934/naco.2017005 +[Abstract](648) +[HTML](2) +[PDF](285.4KB)
Abstract:

We study a soft landing differential game problem for an infinite system of second order differential equations. Control functions of pursuer and evader are subject to integral constraints. The pursuer tries to obtain equations $z(τ)=0$ and $\dot z(τ)=0$ at some time $τ > 0$ and the purpose of the evader is opposite. We obtain a condition under which soft landing problem is not solvable.

Adaptive order of block backward differentiation formulas for stiff ODEs
Z. B. Ibrahim, N. A. A. Mohd Nasir, K. I. Othman and N. Zainuddin
2017, 7(1): 95-106 doi: 10.3934/naco.2017006 +[Abstract](597) +[HTML](1) +[PDF](438.7KB)
Abstract:

In this paper, Adapative Order of Block Backward Differentiation Formulas (ABBDFs) are formulated using uniform step size for the numerical solution of stiff ordinary differential equations (ODEs). These ABBDF methods are of order four, five and six. The benefit of the ABBDF methods is the computation time in the computation of solutions. Numerical results are presented to demonstrate the advantage of implementing adaptive order selection in a single code.

Rank-based inference for the accelerated failure time model in the presence of interval censored data
Mostafa Karimi, Noor Akma Ibrahim, Mohd Rizam Abu Bakar and Jayanthi Arasan
2017, 7(1): 107-112 doi: 10.3934/naco.2017007 +[Abstract](484) +[HTML](1) +[PDF](277.3KB)
Abstract:

Semiparametric analysis and rank-based inference for the accelerated failure time model are complicated in the presence of interval censored data. The main difficulty with the existing rank-based methods is that they involve estimating functions with the possibility of multiple roots. In this paper a class of asymptotically normal rank estimators is developed which can be acquired via linear programming for estimating the parameters of the model, and a two-step iterative algorithm is introduced for solving the estimating equations. The proposed inference procedures are assessed through a real example. The results of applying the proposed methodology on the breast cancer data show that the algorithm converges after three iterations, and the estimations of model parameter based on Log-rank and Gehan weight functions are fairly close with small standard errors.

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