Constructive Extractability of Measurable Selectors in Set-Valued Maps

Core Concepts

The author explores the constructive extractability of measurable selectors from set-valued maps, providing a new algorithmic approach to this problem.

Abstract

This paper delves into the possibility of constructively extracting measurable selectors from set-valued maps, crucial in viability theory and optimal control. The study introduces a new algorithm based on a theorem for constructing measurable selectors. Applications in dynamical systems and practical stabilization are discussed, showcasing the significance of this work in control theory.

Stats

For instance, existence of solutions to certain differential inclusions often requires iterative extraction of measurable selectors.
Theorem 1 states that a weakly measurable set-valued function with closed values admits a measurable selector.
The study presents Algorithm 1 for extracting measurable selectors up to any approximation error.
The computational study demonstrates the application of Algorithm 1 in practically stabilizing a three-wheel robot model.

Quotes

"The current work is somewhat motivated by these concerns and suggests to investigate a possible algorithmic content of a measurable selector theorem." - Tsiotras and Mesbahi [20]
"An important property of representable domains consisting of mutually disjoint sets is that piece-wise constant maps on them are measurable." - Fact 1 [29]
"In general, it is important, however, to consider relaxed, i.e., measurable controls." - [37]

Key Insights Distilled From

On constructive extractability of measurable selectors of set-valued maps

by Pavel Osinen... at arxiv.org 03-12-2024

https://arxiv.org/pdf/2403.05858.pdf

Deeper Inquiries

What implications does the constructive extractability of measurable selectors have on real-world control applications

The constructive extractability of measurable selectors has significant implications for real-world control applications. By providing a method to constructively extract measurable selectors from set-valued maps, this work enables the practical implementation of control strategies that rely on selector extraction. In systems where the existence of solutions or optimal controls depends on the availability of measurable selectors, having a constructive approach ensures that these solutions can be effectively computed and implemented in real-time scenarios.
For instance, in sliding mode control systems, where chattering-free stabilization is crucial, the ability to constructively extract measurable selectors allows for more efficient and reliable control strategies. Similarly, in viability theory and differential inclusions, which often require iterative extraction of selectors for finding feasible solutions or trajectories, having a constructive method enhances the feasibility and effectiveness of these approaches in real-world applications.
Overall, the constructive extractability of measurable selectors enhances the robustness and reliability of control systems by enabling verifiable computation methods for selector extraction.

How can non-constructive selector theorems impact the efficiency and reliability of control systems

Non-constructive selector theorems can have significant impacts on the efficiency and reliability of control systems. These non-constructive results imply that while theoretical existence proofs may establish the presence of certain properties like measurable selectors within set-valued maps, they do not provide practical algorithms or methods to actually compute these selectors effectively.
In real-world applications where precise computation and implementation are essential for system performance and stability, relying solely on non-constructive selector theorems can lead to challenges. The inability to efficiently extract measurable selectors can result in computational inefficiencies, increased complexity in system design and tuning processes, as well as potential limitations in achieving desired control objectives.
Furthermore, non-constructive selector theorems may introduce uncertainties into control systems due to their reliance on theoretical proofs without direct applicability to practical implementations. This lack of constructibility hinders optimization efforts and limits adaptability to dynamic environments where quick decision-making based on computed measures is necessary.

How can the concept of representable domains be extended to other areas beyond control theory

The concept of representable domains extends beyond its application in control theory to various other areas across different disciplines. In mathematics and optimization problems involving complex functions or mappings with specific properties such as semi-concavity or convexity constraints,
representable domains offer a structured framework for defining regions over which certain operations hold true consistently.
In machine learning models,
representable domains could be used
to define input spaces with specific characteristics that facilitate effective training
and generalization capabilities.
In economics,
the notion
of representable domains could aid
in modeling decision-making processes under uncertainty by delineating feasible solution spaces based
on given constraints.
Moreover,
in physics simulations,
representable domains might play a role
in defining permissible states or configurations within physical systems subject
to particular laws or principles.
Overall,
the extension
of representable domains outside traditional control theory opens up new avenues for applying this concept across diverse fields where structured domain definitions are essential
for problem-solving
and analysis purposes