On the Role of Convexity/Concavity in Vector Fields, Flows, and Stability/Stabilizability

Abstract

It is shown that strong convexity/concavity of a component of the vector field, as a function of the state variables, induces the same property on the corresponding component of the flow, as a function of the initial condition. Such an inherited property is then instrumental, for instance, for establishing several instability theorems, the proofs of which rely precisely on consequences of convexity/concavity of the flow with respect to the initial condition. Furthermore, the property of convexity/concavity permits the construction of a canonical Chetaev function to certify instability without explicitly resorting to the computation of the flow. Finally, necessary conditions for continuous stabilizability are derived, hence putting the properties of convexity/concavity of the vector field in relation to the well-known Brockett's theorem.