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To stabilize PDE models, control laws require space-dependent functionalgains mapped by nonlinear operators from the PDE functional coefficients. Whena PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent,a gain-scheduling (GS) nonlinear design is the simplest approach to the designof nonlinear feedback. The GS version of PDE backstepping employs gainsobtained by solving a PDE at each value of the state. Performing such PDEcomputations in real time may be prohibitive. The recently introduced neuraloperators (NO) can be trained to produce the gain functions, rapidly in realtime, for each state value, without requiring a PDE solution. In this paper weintroduce NOs for GS-PDE backstepping. GS controllers act on the premise thatthe state change is slow and, as a result, guarantee only local stability, evenfor ODEs. We establish local stabilization of hyperbolic PDEs with nonlinearrecirculation using both a "full-kernel" approach and the "gain-only" approachto gain operator approximation. Numerical simulations illustrate stabilizationand demonstrate speedup by three orders of magnitude over traditional PDEgain-scheduling. Code (Github) for the numerical implementation is published toenable exploration.

Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation