The paper studies initial value problems (IVPs) with dynamics ruled by discontinuous ordinary differential equations (ODEs), focusing on the class of IVPs that possess a unique solution.
The authors identify a precise class of such systems, called "solvable initial value problems", and prove that for this class, the unique solution can always be obtained analytically via transfinite recursion.
The key idea is to relax the continuity requirement on the right-hand term of the ODE, while still ensuring the existence of a unique solution. The authors introduce the concept of "solvable functions" - functions of class Baire one where the set of discontinuity points forms a closed set for any closed subset of the domain.
The authors present a "search method" called the (α)Monkeys approach, which generalizes the "Ten Thousand Monkeys" algorithm for continuous ODEs. This method allows constructing a sequence of continuous functions that converge to the unique solution in the limit, even for discontinuous right-hand terms.
The main result is a transfinite recursion procedure that is guaranteed to obtain the unique solution of the IVP in at most a countable number of steps, as long as the right-hand term is a solvable function. The authors also provide a nontrivial example where the solution at an integer time encodes the halting set of Turing machines, showcasing the connection between solvable systems and ordinal Turing computations.
To Another Language
from source content
arxiv.org
Key Insights Distilled From
by Olivier Bour... at arxiv.org 05-02-2024
https://arxiv.org/pdf/2405.00165.pdfDeeper Inquiries