On the Role of Convexity/Concavity in Vector Fields, Flows, and Stability/Stabilizability
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Description
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.
Notes
2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. M. Sassano and A. Astolfi, "On the Role of Convexity/Concavity in Vector Fields, Flows, and Stability/Stabilizability," IEEE Transactions on Automatic Control, vol. 69, no. 1, pp. 32-42, Jan. 2024, doi: 10.1109/TAC.2023.3277922.