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Optimal control problems with free right end point

Nico Tauchnitz

TL;DR

This work develops an elementary, constructive proof of Pontryagin's maximum principle for optimal control problems with free terminal time, using simple needle variations and extending the standard PMP to a suite of advanced problem classes: optimal multiprocesses, infinite horizon, time delays, and Volterra integral dynamics. It unifies the treatment by representing switching as a control variable via interval decompositions and derives adjoint equations, transversality conditions, and Hamiltonian maximization in each setting. The paper further provides Arrow-type sufficiency conditions across these contexts and illustrates the theory with economic and strategic examples, including maintenance, advertising, and Nash/Differential games. Collectively, the results broaden the applicability of PMP to systems with memory, delays, and discrete switching, while offering economic interpretations through shadow prices and opportunity costs. The methodologies have potential impact on long-horizon planning, resource management, and dynamic strategic optimization under complex temporal structures.

Abstract

This paper is dedicated to the elementary proof of Pontryagin's maximum principle for problems with free right end point. The proof for the standard problem is taken from the monography of Ioffe and Tichomirov. We assume piecewise continuous controls and the proof turns out to be very simple. We generalize the concept to the problem of optimal multiprocesses, to control problems with delays and to the control of Volterra integral equations. Furthermore, we discuss the problem on infinite horizon. Moreover, we state Arrow type sufficiency conditions. The optimality conditions are demonstrated on illustrative examples.

Optimal control problems with free right end point

TL;DR

This work develops an elementary, constructive proof of Pontryagin's maximum principle for optimal control problems with free terminal time, using simple needle variations and extending the standard PMP to a suite of advanced problem classes: optimal multiprocesses, infinite horizon, time delays, and Volterra integral dynamics. It unifies the treatment by representing switching as a control variable via interval decompositions and derives adjoint equations, transversality conditions, and Hamiltonian maximization in each setting. The paper further provides Arrow-type sufficiency conditions across these contexts and illustrates the theory with economic and strategic examples, including maintenance, advertising, and Nash/Differential games. Collectively, the results broaden the applicability of PMP to systems with memory, delays, and discrete switching, while offering economic interpretations through shadow prices and opportunity costs. The methodologies have potential impact on long-horizon planning, resource management, and dynamic strategic optimization under complex temporal structures.

Abstract

This paper is dedicated to the elementary proof of Pontryagin's maximum principle for problems with free right end point. The proof for the standard problem is taken from the monography of Ioffe and Tichomirov. We assume piecewise continuous controls and the proof turns out to be very simple. We generalize the concept to the problem of optimal multiprocesses, to control problems with delays and to the control of Volterra integral equations. Furthermore, we discuss the problem on infinite horizon. Moreover, we state Arrow type sufficiency conditions. The optimality conditions are demonstrated on illustrative examples.

Paper Structure

This paper contains 38 sections, 23 theorems, 439 equations, 4 figures.

Table of Contents

  1. Eine Standardaufgabe mit Wiedergewinnungswert
  2. Die Aufgabenstellung und das Pontrjaginsche Maximumprinzip
  3. Der Beweis des Maximumprinzips
  4. Ökonomische Deutung des Maximumprinzips
  5. Hinreichende Bedingungen nach Arrow
  6. Instandhaltungsmanagement
  7. Optimale Werbestrategien
  8. Kapitalismusspiel
  9. Optimale Multiprozesse
  10. Vorbereitende Betrachtungen
  11. Die Aufgabenstellung und das Pontrjaginsche Maximumprinzip
  12. Der Beweis des Maximumprinzips
  13. Hinreichende Bedingungen nach Arrow
  14. Aufgaben mit unendlichem Zeithorizont
  15. Pathologien und Variierbarkeit
  16. ...and 23 more sections

Key Result

Theorem 1.2

Sei $(x_*(\cdot),u_*(\cdot)) \in \mathscr{A}^{\mathcal{S}}_{\rm adm} \cap \mathscr{A}^{\mathcal{S}}_{\rm Lip}$. Ist $(x_*(\cdot),u_*(\cdot))$ ein starkes lokales Minimum der Aufgabe (PMPeinfach1)--(PMPeinfach3), dann existiert eine Vektorfunktion $p(\cdot) \in PC_1([t_0,t_1],{\mathbb R}^n)$ derart, erfüllt sind.

Figures (4)

  • Figure 1: Variation $u(t;v,\tau,\lambda)$.
  • Figure 2: Zeitbereich $\Theta$ und die Trapezaufteilung.
  • Figure 3: Für jedes $z \in U_\delta(x_0)$ gehört $x_z(\overline{\tau})$ zu $U_\varrho(x(\overline{\tau}))$.
  • Figure 4: Die Fortsetzungen $y_z(\cdot)$ von $x_z(\cdot)$.

Theorems & Definitions (42)

  • Definition 1: Stückweise Stetigkeit
  • Definition 2: Stückweise stetige Differenzierbarkeit
  • Definition 3: Stückweise Stetigkeit
  • Definition 4: Stückweise stetige Differenzierbarkeit
  • Theorem 1.2: Pontrjaginsches Maximumprinzip
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2: $k$-fache Zerlegung
  • Theorem 2.3: Pontrjaginsches Maximumprinzip
  • Theorem 2.5
  • ...and 32 more