A generalized initial value problem for ODE's

TL;DR

This paper introduces the generalized initial value problem for ODEs, where initial conditions are mapped to subsets of the domain, enabling systems to alter their properties and potentially improve stability.

Contribution

It presents the first formulation of the generalized initial value problem for ODEs, expanding the scope of initial conditions and system analysis.

Findings

Generalized initial value problem broadens ODE analysis.

Mapping initial conditions can remove obstructions to stability.

Potential for new system behaviors and stability properties.

Abstract

This paper introduces, up to the author's knowledge, for the first time the generalized initial value problem. In this problem, given an ordinary differential equation defined in some set, the initial conditions are mapped to a subset of that domain. This generalization allows some systems to radically change their properties, in fact some obstructions to asymptotic stability are get rid of by mapping the initial conditions set using modularity.