Global existence and boundedness for quasi‐variational systems

Giancarlo Cantarelli

doi:10.1155/s0161171299222818

Abstract

We consider quasi‐variational ordinary differential systems, which may be considered as the motion law for holonomic mechanical systems. Even when the potential energy of the system is not bounded from below, by constructing appropriate Liapunov functions and using the comparison method, we obtain sufficient conditions for global existence of solutions in the future and for their partial boundedness.

Highlights

Let G(u, p) and F (t, u) be scalar functions and Q(t, u, p) an Nvector (N ≥ 1)
We prove the theorem by the methods of [4, Thm. 1]
We show that Corollary 2.3 is a corollary of Theorem 2.2

Summary

Introduction

Let G(u, p) and F (t, u) be scalar functions and Q(t, u, p) an Nvector (N ≥ 1). We consider vector solutions u = (u1, . . . , uN ) of the quasi-variational ordinary differential system d dt ∇G(u, u)− ∇uG(u, u) + ∇u F (t, u) = Q(t, u, u), (1.1)where d/dt and “·” denote differentiation with respect to the independent variable t ∈ R+ = [0, ∞), and ∂ ∇= ∂p1 ∂pN ∇u = ∂u1. We obtain sufficient conditions for global existence in the future and for partial boundedness of the solutions of (1.1).

Results

Conclusion