Lotka-Sharpe Neural Operators for Control of Population PDEs
This work provides a theoretical foundation and practical method for operator learning in control of age-structured population systems, addressing a key challenge in feedback design for ecological and biotechnological applications.
The authors prove Lipschitz continuity of the Lotka-Sharpe operator, enabling neural operator approximation for feedback control of age-structured predator-prey PDEs. They demonstrate that the approximate feedback law preserves stability and show online usage with estimated fertility/mortality functions.
Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar $ζ$, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to $ζ$. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.