Asymptotic behavior of ratio between the components of solutions of a linear system of ordinary differential equation and its applications to the determination of Lyapunov characteristic numbers

Summary In this paper we study the asymptotic directions of the solutions of a linear system of Ordinary Differential Equations. Theorems are given which generalize theorems of Smydtowna[4] and Massera. These theorems are used to generalize a theorem of Perron[3] which gives a formula for the determination of Liapunov's characteristic numbers in function of the coefficients of the equation. The results of this paper with less general hypotheses where presented as part of my doctoral dissertation by Escola de Engenharia de Sa? Carlos Universidade de S?o Paulo-S?o Paulo - Brasil This work was completed in the Center for Dynamical Systems, Brown University, while the author was on a fellowship from Conselho Nacional de Pesquisas Brazil. Entrata in Redazione il 13 maggio 1969.     

Necessary conditions for the optimality equation in average-reward Markov decision processes

An average-reward Markov decision process (MDP) with discretetime parameter, denumerable state space, and bounded reward function is considered. With such a model, we associate a family of MDPs. Then, we determinenecessary conditions for the existence of a bounded solution to the optimality equation for each one of the models in the family. Moreover,necessary andsufficient conditions are given so that the optimality equations have a bounded solution with an additional property.This research was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACYT) under Grant No. PCEXCNA-040640.Dedicated to Professor Eutimio Alberto Cuéllar-Goríbar on the occasion of his eightieth birthday     

Multicriteria dynamic programming with an application to the integer case

Fundamental dynamic programming recursive equations are extended to the multicriteria framework. In particular, a more detailed procedure for a general recursive solution scheme for the multicriteria discrete mathematical programming problem is developed. Definitions of lower and upper bounds are offered for the multicriteria case and are incorporated into the recursive equations to aid problem solution by eliminating inefficient subpolicies. Computational results are reported for a set of 0–1 integer linear programming problems.This research was supported in part by CONACYT (Consejo Nacional de Ciencia y Technologia), Mexico City, Mexico.     

Value iteration in average cost Markov control processes on Borel spaces

This paper deals with discrete-time Markov control processes withBorel state and control spaces, with possiblyunbounded costs andnoncompact control constraint sets, and the average cost criterion. Conditions are given for the convergence of the value iteration algorithm to the optimal average cost, and for a sequence of finite-horizon optimal policies to have an accumulation point which is average cost optimal.This research was partially supported by the Consejo Nacional de Ciencia y Tecnología (CONACyT) under grant 1332-E9206.     

Two Person Zero-Sum Semi-Markov Games with Unknown Holding Times Distribution on One Side: A Discounted Payoff Criterion

This paper deals with two person zero-sum semi-Markov games with a possibly unbounded payoff function, under a discounted payoff criterion. Assuming that the distribution of the holding times H is unknown for one of the players, we combine suitable methods of statistical estimation of H with control procedures to construct an asymptotically discount optimal pair of strategies. Work supported partially by Consejo Nacional de Ciencia y Tecnología (CONACyT) under Grant 46633-F.     

The lagrange approach to infinite linear programs

In this paper, we use the Lagrange multipliers approach to study a general infinite-dimensionalinequality-constrained linear program IP. The main problem we are concerned with is to show that thestrong duality condition for IP holds, so that IP and its dual IP* are both solvable and their optimal values coincide. To do this, we first express IP as a convex program with a Lagrangian function L, say. Then we show that the strong duality condition implies the existence of a saddle point for L, and that, under an additional, mild condition, theconverse is also true. Moreover, the saddle point gives optimal solutions for IP and IP*. Thus, our original problem is essentially reduced to prove the existence of a saddle point for L, which is shown to be the case under suitable assumptions. We use this fact to studyequality-constrained programs, and we illustrate our main results with applications to thegeneral capacity and themass transfer problems. This research was partially supported by the Consejo Nacional de Ciencia y Tecnología (CONACYT) grants 32299-E and 37355-E. It was also supported by CONACYT (for JRG and RRLM) and PROMEP (for JRG) scholarships.     

On group strategy-proof mechanisms for a many-to-one matching model

For the many-to-one matching model in which firms have substitutable and quota q–separable preferences over subsets of workers we show that the workers-optimal stable mechanism is group strategy-proof for the workers. Therefore, in centralized markets like entry-level professional labor markets if the proposed matching is the workers-optimal stable matching then, no group of workers can never benefit by reporting untruthfully their preference relations. We exhibit an example showing that this property fails if the preferences of firms are substitutable but not quota q–separable.Revised August 2004We are grateful to Flip Klijn, Howard Petith, William Thomson, a referee and an associate editor of this journal for helpful comments. The work of R. Martínez, A. Neme, and J. Oviedo is partially supported by the Universidad Nacional de San Luis through Grant 319502, by the Consejo Nacional de Investigaciones Científicas y Técnicas CONICET, through Grant PICT-02114, and by the Agencia Nacional de Promoción Científica y Técnica, through Grant 03-10814. The work of J. Massó is partially supported by the Spanish Ministry of Science and Technology, through Grant BEC2002–2130, and by the Generalitat de Catalunya, through Grant 2001SGR-00162 and the Barcelona Economics Program (CREA). All authors acknowledge financial support from the Grant PCI España-Iberoamérica 2003 (Programa de Cooperación Interuniversitaria de la Agencia Española de Cooperación Internacional-AECI).     

A new technique for generating quadratic programming test problems

This paper describes a new technique for generating convex, strictly concave and indefinite (bilinear or not) quadratic programming problems. These problems have a number of properties that make them useful for test purposes. For example, strictly concave quadratic problems with their global maximum in the interior of the feasible domain and with an exponential number of local minima with distinct function values and indefinite and jointly constrained bilinear problems with nonextreme global minima, can be generated.Unlike most existing methods our construction technique does not require the solution of any subproblems or systems of equations. In addition, the authors know of no other technique for generating jointly constrained bilinear programming problems.Support of this work has been provided by the Instituto Nacional de Investigação Científica de Portugal (INIC) under contract 89/EXA/5 and by the Natural Sciences and Engineering Research Council of Canada operating grant 5671.Much of this paper was completed while this author was on a research sabbatical at the Universidade de Coimbra, Portugal.     

Application of average dynamic programming to inventory systems

We show the existence ofaverage cost (AC-) optimal policy for an inventory system withuncountable state space; in fact, the AC-optimal cost and an AC-optimal stationary policy areexplicitly computed. In order to do this, we use a variant of thevanishing discount factor approach, which have been intensively studied in recent years but the available results not cover the inventory problem we are interested in.The work of the first author (OVA) was partially supported by Fondo del Sistema de Investigación del Mar de Cortéz under grant SIMAC/94/CT-005. The work of the second author (RMdO) was partially supported by Consejo Nacional de Ciencia y Tecnologia (CONACyT) under grant 0635P-E9506.     

Recursive adaptive control of Markov decision processes with the average reward criterion

We are concerned with Markov decision processes with Borel state and action spaces; the transition law and the reward function depend on anunknown parameter. In this framework, we study therecursive adaptive nonstationary value iteration policy, which is proved to be optimal under thesame conditions usually imposed to obtain the optimality of other well-knownnonrecursive adaptive policies. The results are illustrated by showing the existence of optimal adaptive policies for a class of additive-noise systems with unknown noise distribution.This research was supported in part by the Consejo Nacional de Ciencia y Tecnología under Grants PCEXCNA-050156 and A128CCOEO550, and in part by the Third World Academy of Sciences under Grant TWAS RG MP 898-152.     

Generalized diagonal dominance in connection with the accelerated overrelaxation (AOR) method

With the definition of generalized diagonal dominant matrices we improve the known results about the intervals of convergence of the (AOR) method for linear systems. We consider this problem for different kinds of matrices and we get some important results forH-matrices.Supported by Instituto Nacional de Investigação Cientifica.     

Existence of optimal stationary policies in average reward Markov decision processes with a recurrent state

We consider discrete-timeaverage reward Markov decision processes with denumerable state space andbounded reward function. Under structural restrictions on the model the existence of an optimal stationary policy is proved; both the lim inf and lim sup average criteria are considered. In contrast to the usual approach our results donot rely on the average regard optimality equation. Rather, the arguments are based on well-known facts fromRenewal Theory.This research was supported in part by the Consejo Nacional de Ciencia y Tecnologia (CONACYT) under Grants PCEXCNA 040640 and 050156, and by SEMAC under Grant 89-1/00ifn$.     

Discrete Jacobi sub‐equation method for nonlinear differential–difference equations

We will propose a unified algebraic method to construct Jacobi elliptic function solutions to differential–difference equations (DDEs). The solutions to DDEs in terms of Jacobi elliptic functions sn, cn and dn have a unified form and can be presented through solving the associated algebraic equations. To illustrate the effectiveness of this method, we apply the algorithm to some physically significant DDEs, including the discrete hybrid equation, semi‐discrete coupled modified Korteweg–de Vries and the discrete Klein–Gordon equation, thereby generating some new exact travelling periodic solutions to the discrete Klein–Gordon equation. A procedure is also given to determine the polynomial expansion order of Jacobi elliptic function solutions to DDEs. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Dynamic Programming Approach for the 3<Emphasis Type="Italic">D</Emphasis> Navier–Stokes Equations

The dynamic programming approach for the control of a 3D flow governed by the stochastic Navier–Stokes equations for incompressible fluid in a bounded domain is studied. By a compactness argument, existence of solutions for the associated Hamilton–Jacobi–Bellman equation is proved. Finally, existence of an optimal control through the feedback formula and of an optimal state is discussed. This paper has been written at Scuola Normale Superiore di Pisa and at école Normale Supérieure de Cachan, Antenne de Bretagne.     

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces

In this article, a recent nonlinear alternative for contraction maps in Fréchet spaces due to Frigon and Granas [1998, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.     

Deterministic impulse control in native forest ecosystems management

This paper illustrates an application of impulse control systems analysis to the management of renewable resources. We obtain an explicit definition for the impulse optimal strategy in the case of a nonlinear natural system which models certain types of forests. The result is in agreement with an econo-ecological conservative action on the forest. The examples developed in this paper associate new theories on impulse or jump control with natural forest management. In addition, we compare our results with the classical ones. We also show that a nonoptimal managing action on the forest is frequently the cause for significant medium-term losses.This work was supported by CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, and by the Universidad Nacional del Litoral, Argentina.     

Lubrication Approximation for Thin Viscous Films: Asymptotic Behavior of Nonnegative Solutions

We use standard regularized equations and adapted entropy functionals to prove exponential asymptotic decay in the H ¹ norm for nonnegative weak solutions of fourth-order nonlinear degenerate parabolic equations of lubrication approximation for thin viscous film type. The weak solutions considered arise as limits of solutions for the regularized problems. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means. The problems investigated here are one-dimensional in space, with power-law nonlinearities. Our approach is direct and natural, as it is adapted to deal with the more complex nonlinear terms occurring in the regularized, approximating problems.     

An envelope theorem and some applications to discounted Markov decision processes

In this paper, an Envelope Theorem (ET) will be established for optimization problems on Euclidean spaces. In general, the Envelope Theorems permit analyzing an optimization problem and giving the solution by means of differentiability techniques. The ET will be presented in two versions. One of them uses concavity assumptions, whereas the other one does not require such kind of assumptions. Thereafter, the ET established will be applied to the Markov Decision Processes (MDPs) on Euclidean spaces, discounted and with infinite horizon. As the first application, several examples (including some economic models) of discounted MDPs for which the et allows to determine the value iteration functions will be presented. This will permit to obtain the corresponding optimal value functions and the optimal policies. As the second application of the ET, it will be proved that under differentiability conditions in the transition law, in the reward function, and the noise of the system, the value function and the optimal policy of the problem are differentiable with respect to the state of the system. Besides, various examples to illustrate these differentiability conditions will be provided. This work was partially supported by Benemérita Universidad Aut ónoma de Puebla (BUAP) under grant VIEP-BUAP 38/EXC/06-G, by Consejo Nacional de Ciencia y Tecnología (CONACYT), and by Evaluation-orientation de la COopération Scientifique (ECOS) under grant CONACyT-ECOS M06-M01.     

Approximation of average cost optimal policies for general Markov decision processes with unbounded costs

The aim of the paper is to show that Lyapunov-like ergodicity conditions on Markov decision processes with Borel state space and possibly unbounded cost provide the approximation of an average cost optimal policy by solvingn-stage optimization problems (n = 1, 2, ...). The used approach ensures the exponential rate of convergence. The approximation of this type would be useful to find adaptive procedures of control and to estimate stability of an optimal control under disturbances of the transition probability.Research supported in part by Consejo Nacional de Ciencia y Tecnologia (CONACYT) under grant 0635P-E9506.Research supported by Fondo del Sistema de Investigatión del Mar de Cortés under Grant SIMAC/94/CT-005.     

Wave solutions for variants of the KdV–Burger and the K(n,n)–Burger equations by the generalized G′/G‐expansion method

An application of the ‐expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the Korteweg–de Vries–Burger and the K(n,n)–Burger equations. The generalized ‐expansion method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. It is shown that the generalized ‐expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems. Copyright © 2017 John Wiley & Sons, Ltd.