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The maximum principle for discrete-time control systems and applications to dynamic games

Alberto Domínguez Corella, Onésimo Hernández-Lerma

Abstract

We study deterministic nonstationary discrete-time optimal control problems in both finite and infinite horizon. With the aid of Gateaux differentials, we prove a discrete-time maximum principle in analogy with the well-known continuous-time maximum principle. We show that this maximum principle, together with a transversality condition, is a necessary condition for optimality; we also show that it is sufficient under additional hypotheses. We use Gateaux differentials as a natural setting to derive first-order conditions. Additionally, we use the discrete-time maximum principle to derive the discrete-time Euler equation and to characterize Nash equilibria for discrete-time dynamic games.

The maximum principle for discrete-time control systems and applications to dynamic games

Abstract

We study deterministic nonstationary discrete-time optimal control problems in both finite and infinite horizon. With the aid of Gateaux differentials, we prove a discrete-time maximum principle in analogy with the well-known continuous-time maximum principle. We show that this maximum principle, together with a transversality condition, is a necessary condition for optimality; we also show that it is sufficient under additional hypotheses. We use Gateaux differentials as a natural setting to derive first-order conditions. Additionally, we use the discrete-time maximum principle to derive the discrete-time Euler equation and to characterize Nash equilibria for discrete-time dynamic games.
Paper Structure (12 sections, 18 theorems, 96 equations)

This paper contains 12 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Maximum Principle
  3. The Optimal Control Model
  4. Necessary Conditions
  5. Sufficient Conditions
  6. Examples
  7. Proof of Lemma \ref{['DV']}
  8. Variants of The Maximum Principle
  9. Finite Horizon Optimal Control Problems
  10. Markov Strategies
  11. The Euler Equation
  12. Dynamic Games

Key Result

Proposition 3

Let $\mathcal{X}$ be a linear space and $\mathcal{V}$ a subset of $\mathcal{X}$. Let $p\in\mathcal{V}$ be an internal point in the direction $q\in\mathcal{X}$ and $h:\mathcal{V}\to\mathbb R$ a given function. If the Gateaux differential of $h$ at $p$ in the direction $q$ exists and $h$ has a maximum

Theorems & Definitions (37)

  • Definition 2
  • Proposition 3
  • proof
  • Lemma 5
  • Theorem 6
  • proof
  • Remark 7
  • Proposition 8
  • proof
  • Corollary 9
  • ...and 27 more