Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semi-linear Parabolic PDE Systems

This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in thesense of spatial L^∞ norm ∥⋅∥_∞ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensorsare discretely distributed in space. Initially, a T-S fuzzy PDE model isassumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a staticsampled-data fuzzy local state feedback controller is constructed based on theT-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger’s inequalities, avariation of vector-valued Poincaré–Wirtinger inequality in 1D spatial domain, as well as avector valued Agmon’s inequality, it is shown that the suggested sampled-datafuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of ∥⋅∥_∞, if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) arefulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally,the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.