Contractor-Expander and Universal Inverse Optimal Positive Nonlinear Control

TL;DR

This paper develops new inverse optimal control frameworks for positive nonlinear systems using strict control Lyapunov functions, with applications demonstrated on biological predator-prey models and explicit stabilization formulas.

Contribution

It introduces two novel inverse optimal control frameworks tailored for positive systems with asymmetric costs, extending beyond traditional sign-unconstrained assumptions.

Findings

Frameworks produce inverse optimal stabilizers for positive orthants of arbitrary dimensions.

Explicit Sontag-like formulas are derived for positive system stabilization.

Biological models illustrate the practical relevance of the control constructions.

Abstract

For general control-affine nonlinear systems in the positive orthant, and with positive controls, we show how strict CLFs can be utilized for inverse optimal stabilization. Conventional ``LgV'' inverse optimal feedback laws, for systems with unconstrained states and controls, assume sign-unconstrained inputs and input penalties that are class-K in the input magnitude, hence symmetric about zero. Such techniques do not extend to positive-state-and-control systems. Major customizations are needed, and introduced in this paper, for positive systems where highly asymmetric (or unconventionally symmetric) costs not only on the state but also on control are necessary. With the predator-prey positive-state positive-input benchmark system as inspiration, using a strict CLF built in our previous paper, we prototype two general inverse optimal methodological frameworks that employ particular ``contractor and expander functions.'' One framework (A) employs a triple consisting of a CLF, a stabilizing feedback, and an expander, whereas the other framework (B) employs a pair of a CLF and a contractor function. Both frameworks yield inverse optimal stabilizer constructions, on positive orthants of arbitrary dimensions. A stronger construction results from a stronger CLF condition. Biological interpretation for the predator-prey model illuminates that such inverse optimal control constructions are bio-ecologically meaningful. In addition to general frameworks, we present two fully explicit designs: two Sontag-like universal formulae for stabilization of positive-orthant systems by positive feedback, one of them with inverse optimality.