A nagumo type result for linear dynamical systems

Abstract

Nagumo's condition, also known as the sub-tangency condition, characterizes the set invariance with respect to the trajectories of dynamical systems. The general formulation of Nagumo's condition considers a comprehensive framework, referring to time-variant nonlinear systems and generic sets with arbitrary shapes. For time-variant linear dynamical systems and sets defined by arbitrary norms, we prove that Nagumo's condition has an equivalent form expressing the nonpositiveness of the system-matrix measure. This equivalent form is more attractive for application than the general formulation, since it points out the role of the time-dependent model coefficients. We also show that, for time-invariant linear systems and some particular cases of sets defined by weighted p-norms, p ∈ {1, 2, ∞} , Nagumo's condition is equivalent with well-known algebraic properties (Lyapunov inequality, location of Gershgorin's disks) satisfied by the constant system-matrix. Finally, we evaluate the key ideas of the paper as a potential resource for developing educational materials dedicated to advanced studies in Control Engineering.