Global Stabilization of Uncertain Lotka–Volterra Systems via Positive Nonlinear State Feedback

This article deals with stabilization of Lotka–Volterra (LV) systems in the presence of interval uncertainty and a physical limitation on the control input, which restricts this input to be strictly positive. Considering the positiveness property of LV systems, a quasi-monomial structure for the state feedback based control input is proposed. Considering this structure, stability of the closed-loop system with no uncertainty is analyzed. This analysis leads to an algebraic inequality, whose satisfaction guarantees stability of the closed-loop system. To extend this result to uncertain LV systems with interval parameter uncertainty, a new approach, by which stability of the positive equilibrium point of the closed-loop uncertain LV systems can be checked in terms of feasibility of a component-wise linear matrix inequality (LMI), is introduced. To achieve a stabilizing controller, the unknown controller parameters are obtained by using the feasible solutions of the mentioned LMI. To overcome some restrictions on the proposed method, a new Lyapunov function is employed, which leads to a bilinear matrix inequalities (BMI) to ensure stabilization. This BMI approach reduces the conservativeness of the above-mentioned LMI-based approach. Also, it is shown that the BMI-based approach can be extended for global stabilization of time-varying LV systems. The efficiency of the proposed schemes is shown through some examples inspired from chemical/biological processes.

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