Title: Global Classical Solvability of Initial-Boundary Problems for Hyperbolic Lotka-Volterra Systems and Lyapunov-Based Boundary Control (1 hour)
Abstract: We consider a mixed initial-boundary value problem for semilinear hyperbolic systems with nonlinear reaction of Lotka-Volterra type. Such nonlinearity is often used to model biological systems and networks, predator-prey systems, coupled wave equations for Raman scattering, or evolutionary dynamics of species. The reaction nonlinearity is not globally Lipschitz in L2, but has Lipschitz properties depending on an L∞-norm bound. We reformulate the problem in an abstract setting as a modified Cauchy problem with homogeneous boundary conditions and apply Banach contraction mapping theorem. We show that global existence of classical solutions in L2 holds if a uniform a-priori bound on the L∞-norm of the solution and boundary term exists. As an application we consider a boundary feedback control problem for two coupled first-order hyperbolic partial differential equations with nonlinear coupling of Lotka-Volterra type, arising in Raman optical amplifiers. We use boundary control action on 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 classical solutions and (asymptotic) exponential convergence in the (C0) L2-norm.