Global existence and boundedness for quasi-variational systems (Q1301920)

The author studies the quasi-variational ordinary differential system \[ \frac{d}{dt}(\nabla_p G(u,\dot{u}))-\nabla_u G(u,\dot{u}) +\nabla_u F(t,u)=Q(t,u,\dot{u}), \] where \(G(u,p)\) is a function of class \(C^1\), strictly convex in the variable \(p \in \mathbb{R}^N\) for every \(u \in {\mathbb R}^N\), with \(G(u,0)=0\) and \(\nabla G(u,0)=0\) on \(\mathbb{R}^N.\) This system may be considered as the motion law for holonomic mechanical systems. Sufficient conditions for the global existence of solutions to system (1) in the future and for their partial boundedness are obtained by constructing appropriate Lyapunov functions and using the comparison method.