Monotonicity results for non-autonomous dynamical systems: The case of a general convex cone

Abstract

Let K K be a closed convex cone in the state space R n \mathbb {R}^n . This note characterizes the K K -monotonicity of a non-autonomous dynamical system x ˙ ( t ) = f ( t , x ( t ) ) \dot x (t) = f(t,x(t)) governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field f f is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone K K is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity.