Adaptive Neural Boundary Control Design for Nonlinear Flexible Distributed Parameter Systems

This paper proposes an adaptive neural boundary feedback control design for a nonlinear flexible distributed parameter system described by a nonuniform wave equation containing an unknown nonlinear function in the boundary conditions. In the proposed design method, a radial-basis-function neural network (NN) with an adaptive weight update law is used to approximate the unknown nonlinear term. By using this adaptive NN, an adaptive boundary feedback controller is constructed to ensure that the resulting closed-loop system is practically exponentially stable. The closed-loop stability of the system is analyzed by the Lyapunov direct method and technique of integration by parts, and the update law of an NN weight vector is determined simultaneously. The closed-loop well-posedness analysis is also provided by applying the C₀-semigroup approach. Moreover, the proposed design method is extended for a nonlinear flexible distributed parameter system modeled by a nonuniform Euler–Bernoulli beam equation containing an unknown nonlinear function in the boundary conditions. Finally, extensive numerical simulation results are given to demonstrate the performance of the proposed adaptive neural boundary control design method.