State-Feedback Control of Logistic-Based Gene Regulatory Networks: Closed-Form Lyapunov Certificates, Monostabilization, and Delay-Uniform Stability

Abstract

Gene regulatory networks (GRNs) are high-value targets for therapeutic and synthetic-biology control. Classical Hill models carry a structural defect: the production term vanishes when the activator is absent, causing loss of controllability under multiplicative actuation and a collapse of network coupling under additive actuation -- precisely where biological operation is most common. Building on logistic functions as robust Hill alternatives, we develop an additive state-feedback framework for logistic-based GRNs, with two companion scalar results. A feedforward-plus-proportional law turns any positive setpoint into a closed-loop equilibrium, regardless of the uncontrolled dynamics. We prove local exponential stability under a Gershgorin gain bound and, via a common quadratic Lyapunov function built on the logistic sector bound, global exponential stability under the explicit condition $(\gamma_1+K_1)(\gamma_2+K_2)>\kappa_1\kappa_2\lambda^2/64$, with a closed-form rate. A diagonal Lyapunov certificate $P=\mathrm{diag}(B,A)$ yields an explicit settling-time bound (within $1\%$ of simulation) and an ISS ultimate-bound estimate. We further establish a parameter-uniform monostabilization budget $K^{*}=\kappa\lambda/4-\gamma$ for bistable self-activation switches, and a Halanay-type delay-uniform global exponential stability theorem under $\gamma+K>\kappa\lambda/4$, with two-sided closed-form rate bounds. Numerical comparison with the Hill counterpart confirms robust tracking in the nominal range while exposing the structural divergence near the boundary of the positive orthant: the Hill coupling $|[J_f]_{21}|=\Theta(x_{d,1}^{n-1})\to0$ as $x_{d,1}\to0$, whereas the logistic coupling stays strictly positive.