Naïla Hayek

Infinite-horizon optimal control in the discrete-time framework

1. Presentation of the problems and tools of the finite horizon.- 2. Infinite horizon theorems.- 3. The special case of the bounded processes.- Related topics. Appendix A : Sequences.- Appendix B: Static optimization.- References.

Infinite horizon multiobjective optimal control problems in the discrete time case

This article studies multiobjective optimal control problems in the discrete time framework and in the infinite horizon case. The functions appearing in the problems satisfy smoothness conditions. This article generalizes to the multiobjective case results obtained for single-objective optimal control problems in that framework. The dynamics are governed by difference equations or difference inequations. Necessary conditions of Pareto optimality are presented, namely Pontryagin maximum principles in the weak form and in the strong form. Sufficient conditions are also provided. Other notions of Pareto optimality are defined when the infinite series do not necessarily converge.

Second-Order Necessary Conditions for the Infinite-Horizon Variational Problems

We give new proofs of first-order and second-order necessary conditions for the infinite-horizon variational problems in the framework of the continuous time. We study the notion of conjugate points and we build new second-order necessary conditions. We also translate all of our results into the Hamiltonian formalism. Moreover, we apply these general results to the special Lagrangians in the form

Infinite Horizon Discrete Time Control Problems for Bounded Processes

e^{-\delta i} lx, \dot x

Terrorists’ Eradication Versus Perpetual Terror War

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Pontryagin principles for bounded discrete-time processes

We establish Pontryagin Maximum Principles in the strong form for infinite horizon optimal control problems for bounded processes, for systems governed by difference equations. Results due to Ioffe and Tihomirov are among the tools used to prove our theorems. We write necessary conditions with weakened hypotheses of concavity and without invertibility, and we provide new results on the adjoint variable. We show links between bounded problems and nonbounded ones. We also give sufficient conditions of optimality.

A Generalization of Mixed Problems with an Application to Multiobjective Optimal Control

We study an infinite-horizon sequential dynamic game where the players are a government and an international terrorist organization. We provide conditions for the existence of equilibria in which the terrorists’ resources are totally destroyed by a government’s strike. Specifically, we study strong eradication equilibria in which the government’s strike annihilates the terrorists’ resources, preventing the terrorists from acting. We also pay attention to weak eradication equilibria in which the terrorists’ resources are destroyed but in which the initial value of the terrorists’ strike is nevertheless positive. We also show the existence of an equilibrium in which war is perpetual between the government and the terrorists. Perpetual war can only coexist with weak eradication equilibria. For these cases, we provide conditions under which the government would be better off in a weak eradication equilibrium.

An existence theorem for the Scheinkman-Weiss economic model

Abstract We establish necessary conditions and sufficient conditions of optimality in the form of Pontryagin principles for infinite-horizon discrete-time optimal control problems governed by a difference inequation.

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper generalizes to multiobjective optimization the notion of mixed problems as Philippe Michel calls it for single-objective optimization. This notion is then applied to a multiobjective control problem under constraints in the discrete time framework to obtain strong Pontryagin maximum principles in the finite-horizon case. The infinite-horizon case is also treated with conditions ensuring that the multipliers associated to the objective functions are not all zero.

A NOTE ON LUENBERGER'S ZERO-MAXIMUM PRINCIPLE FOR CORE ALLOCATIONS

The Scheinkman-Weiss model and later works by Conze-Lasry-Scheinkman provide insights on cycles and correlations of economies with incomplete markets, namely with borrowing constraints. This work gives a mathematical solution in the case of high relative risk aversion, which has not yet been solved. The existence of an equilibrium results from the solution of a system of nonlinear functional differential equations. High risk aversion leads to new mathematical difficulties, not present in previous papers on this subject.