Lyapunov-based boundary control for 2 × 2 hyperbolic Lotka-Volterra systems

We consider a boundary feedback control problem for two coupled first-order hyperbolic partial differential equations with nonlinear coupling of Lotka-Volterra type. We use boundary control action on only one channel and design static and dynamic boundary controllers to drive the state at the end of the spatial domain to the desired constant reference values. The control design is based on a special Lyapunov functional related to an entropy function for Lotka-Volterra systems. The time derivative of the Lyapunov-based energy function can be made strictly negative by an appropriate choice of boundary conditions. We show global in time existence of the classical solutions and (asymptotic) exponential convergence of the state to the desired set point in the (C⁰) L²-norm. The design is illustrated with simulations.