Adaptive and Neural Operator Control of Nonlinear Volterra Hyperbolic PDEs
Adaptive control learns the plant online; neural-operator control learns the control gains offline. We bring the two together for a class of nonlinear hyperbolic PDEs whose dynamics are governed by an unknown Volterra series of arbitrarily many kernels. An observer-based passive identifier learns a truncation of this series online. The infinite-dimensional map that synthesizes the backstepping kernels from the parameter estimates -- a cascade of PDEs on simplex domains of increasing dimension, prohibitive to solve in real time -- is approximated once, offline, by a neural operator. The closed loop then carries two learning processes in series: online learning of the plant feeds an offline-learned PDE solver, whose output is the online control gains. We prove closed-loop stability and asymptotic regulation of the plant state, observer state, and input, on a basin that recovers the exact-kernel basin as the neural-operator accuracy improves. With a single Lyapunov function we absorb at once the perturbations -- all vanishing -- of truncating an infinite Volterra series, of identifying the plant online, and of approximating the gains.