Carathéodory's existence theorem

Carathéodory's existence theorem guarantees a local absolutely continuous solution to an ordinary differential equation when its right-hand side is measurable in time, continuous in the state, and locally bounded by an integrable function.

Carathéodory function: A function measurable in one variable and continuous in another, with suitable local integrable bounds. The theorem applies this mixed regularity to the differential equation's right-hand side.
Lebesgue measurability: A property that lets sets and functions be integrated within Lebesgue measure theory. Measurability in time is one of the theorem's central assumptions.
Differential inclusion: A differential relation in which the derivative belongs to a set of permitted values rather than equaling one prescribed value. Existence methods for differential inclusions often extend Carathéodory's measurable-time framework.
Constantin Carathéodory: A Greek mathematician whose work included calculus of variations, thermodynamics, and foundational analysis. The theorem bears his name and reflects his contributions to analysis.
Absolutely continuous function: A function whose changes over disjoint intervals have arbitrarily small total size when their combined length is sufficiently small. The theorem guarantees solutions in this regularity class, allowing derivatives to exist almost everywhere.
Lebesgue integrability: A condition ensuring a function has a finite Lebesgue integral over the domain in question. The local bound on the differential equation's right-hand side must be integrable in time.
Control theory: The study of systems shaped toward desired behavior through inputs or feedback. Controlled systems often produce time-dependent right-hand sides with limited regularity.
Giuseppe Peano: An Italian mathematician known for axiomatic work and an existence theorem for continuous differential equations. Peano's theorem provides an earlier existence result under a different regularity assumption.
Peano existence theorem: An existence theorem for ordinary differential equations with a continuous right-hand side. It offers a related existence result under continuity rather than measurable-time dependence.
Almost-everywhere convergence: Convergence that holds except on a set of measure zero. The solution's differential equation is required to hold almost everywhere, not at every time.

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