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On constructive extractability of measurable selectors of set-valued maps

Pavel Osinenko, Stefan Streif

TL;DR

This article analyzes under which particular conditions selectors are constructively extractable, and derives an algorithm derived from the selector theorems, and applied in a computational study with a three-wheel robot model.

Abstract

This paper investigates the possibility of constructive extraction of measurable selector from set-valued maps which may commonly arise in viability theory, optimal control, discontinuous systems etc. For instance, existence of solutions to certain differential inclusions, often requires iterative extraction of measurable selectors. Next, optimal controls are in general non-unique which naturally leads to an optimal set-valued function. Finally, a viable control law can be seen, in general, as a selector. It is known that selector theorems are non-constructive and so selectors cannot always be extracted. In this work, we analyze under which particular conditions selectors are constructively extractable. An algorithm is derived from the theorem and applied in a computational study with a three-wheel robot model.

On constructive extractability of measurable selectors of set-valued maps

TL;DR

This article analyzes under which particular conditions selectors are constructively extractable, and derives an algorithm derived from the selector theorems, and applied in a computational study with a three-wheel robot model.

Abstract

This paper investigates the possibility of constructive extraction of measurable selector from set-valued maps which may commonly arise in viability theory, optimal control, discontinuous systems etc. For instance, existence of solutions to certain differential inclusions, often requires iterative extraction of measurable selectors. Next, optimal controls are in general non-unique which naturally leads to an optimal set-valued function. Finally, a viable control law can be seen, in general, as a selector. It is known that selector theorems are non-constructive and so selectors cannot always be extracted. In this work, we analyze under which particular conditions selectors are constructively extractable. An algorithm is derived from the theorem and applied in a computational study with a three-wheel robot model.
Paper Structure (16 sections, 13 theorems, 31 equations, 2 figures, 1 algorithm)

This paper contains 16 sections, 13 theorems, 31 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof:
  4. Representable domains
  5. Proof:
  6. Proof:
  7. Proof:
  8. Selector theorem with constructive extractability
  9. Proof:
  10. Proof:
  11. Applications
  12. Computational study
  13. Conclusion
  14. Proof:
  15. Proof:
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathbb{X}$ be a measurable and $\mathbb{Y}$ a separable, completely metrizable topological space ($\mathbb{R}^n$ or any separable Banach space are examples), respectively. Let $F: \mathbb{X} \rightrightarrows \mathbb{Y}$ be a weakly measurable set-valued function with closed values. Then, $F$

Figures (2)

  • Figure 1: Exemplary section of the extracted selector, the value of the disassembled subgradient $\zeta_3$ at $x_3=-1$
  • Figure 2: Control and state of the three-wheel robot in a simulation with a practically stabilizing controller that uses: the analytic disassembled subgradients $\zeta_a$ of \ref{['eqn:NI-analyt-subgrad']} (left); selector-based disassembled subgradients $\zeta_s$ (right)

Theorems & Definitions (26)

  • Theorem 1: General measurable selector theorem
  • Example 1
  • Definition 1: Measurable function
  • Definition 2: Uniform convergence
  • Definition 3: Convergence almost uniformly
  • Definition 4: Convergence in measure
  • Definition 5: Cauchy in measure
  • Lemma 1: 6.36 in Ye2011-SF
  • Lemma 2: Countable reduction
  • Proposition 1
  • ...and 16 more