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Super-duality and necessary optimality conditions of order "infinity" in optimal control theory

Nikolay Pogodaev, Maxim Staritsyn

Abstract

We systematically introduce an approach to the analysis and (numerical) solution of a broad class of nonlinear unconstrained optimal control problems, involving ordinary and distributed systems. Our approach relies on exact representations of the increments of the objective functional, drawing inspiration from the classical Weierstrass formula in Calculus of Variations. While such representations are straightforward to devise for state-linear problems (in vector spaces), they can also be extended to nonlinear models (in metric spaces) by immersing them into suitable linear "super-structures". We demonstrate that these increment formulas lead to necessary optimality conditions of an arbitrary order. Moreover, they enable to formulate optimality conditions of "infinite order", incorporating a kind of feedback mechanism. As a central result, we rigorously apply this general technique to the optimal control of nonlocal continuity equations in the space of probability measures.

Super-duality and necessary optimality conditions of order "infinity" in optimal control theory

Abstract

We systematically introduce an approach to the analysis and (numerical) solution of a broad class of nonlinear unconstrained optimal control problems, involving ordinary and distributed systems. Our approach relies on exact representations of the increments of the objective functional, drawing inspiration from the classical Weierstrass formula in Calculus of Variations. While such representations are straightforward to devise for state-linear problems (in vector spaces), they can also be extended to nonlinear models (in metric spaces) by immersing them into suitable linear "super-structures". We demonstrate that these increment formulas lead to necessary optimality conditions of an arbitrary order. Moreover, they enable to formulate optimality conditions of "infinite order", incorporating a kind of feedback mechanism. As a central result, we rigorously apply this general technique to the optimal control of nonlocal continuity equations in the space of probability measures.

Paper Structure

This paper contains 49 sections, 39 theorems, 339 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Circle of ideas
  3. Organization of the paper. Contribution and novelty
  4. Notations and conventions
  5. (Almost) classical setup
  6. Generalized controls
  7. Immersion
  8. Exact increment formula
  9. 1-variation and PMP
  10. 2-variation and second-order optimality conditions
  11. $\infty$-variation and feedback optimality conditions
  12. Mean-field control setup
  13. Differentiability in $\mathcal{P}_2$
  14. Test functions
  15. Metric properties of flows
  16. ...and 34 more sections

Key Result

Proposition 2.9

Let $u \in \mathcal{U}$ be fixed, $(x, p)$ be the corresponding state and adjoint trajectories, and $\bm p$ be defined by LCadj. Then, the equality $\nabla \bm{p}_t(x(t)) = p(t)$ holds for all $t \in I$.

Figures (3)

  • Figure 1: Pushforward of vectors and pullback of covectors.
  • Figure 2: The curve $\gamma_s$ on the reachable set $\mathcal{R}_T(\vartheta)$ obtained by switching from $u$ to $\bar{u}$ at the time moment $s$.
  • Figure 3: Relation between $\mu[t_0,\vartheta_0]$ and its lift $\gamma[t_0,\vartheta]$.

Theorems & Definitions (81)

  • Remark 2.1
  • Remark 2.2: Operator notation
  • Remark 2.3: Random initial state. Double relaxation
  • Remark 2.4: Alternative domain choices
  • Remark 2.5: ${L}^2_{\mu}$-regularity of $\nabla \bar{\bm p}$
  • Remark 2.6: Connection to the Weierstrass formula
  • Remark 2.7
  • Remark 2.8: Geometric language
  • Proposition 2.9
  • Lemma 2.10: ABressan_BPiccoli_2007a
  • ...and 71 more