2002/06/24 by Brian P. Ingalls, Brian Ingalls, Ingalls, Brian P. +5 · 5 citations
Computer Science · Engineering · Mathematics · #34A60 #Advanced Control Systems Optimization #Algorithm #Applied mathematics #Banach space #Computer science #Differential inclusion #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed-point theorem #Inclusion (mineral) #Initial value problem #Lipschitz continuity #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Physics #Picard–Lindelöf theorem #Pure mathematics #Relaxation (psychology) #Set (abstract data type) #Stability (learning theory) #Stability and Control of Uncertain Systems #State (computer science) #Thermodynamics #Value (mathematics) #Work (physics) #math.DS #math.OC #msc:34A60
paper · pdf · doi:10.48550/arxiv.math/0206251
published in arXiv (Cornell University) (Cornell University) · 11 pages
arxiv created 2002/06/24 · openalex publication_date 2002/06/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The fundamental Filippov-Wazwski Relaxation Theorem states that the solution set of an initial value problem for a locally Lipschitz inclusion is dense in the solution set of the same initial value problem for the corresponding relaxation inclusion on compact intervals. In our recent work, a complementary result was provided for inclusions with finite dimensional state spaces which says that the approximation can be carried out over non-compact or infinite intervals provided one does not insist on the same initial values. This note extends the infinite-time relaxation theorem to the inclusions whose state spaces are Banach spaces. To illustrate the motivations for studying such approximation results, we briefly discuss a quick application of the result to output stability and uniform output stability properties.