## Abstract

This paper employs a Takagi-Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial <formula><tex>$L^\infty$</tex></formula> norm <formula><tex>$\|\cdot\|_\infty$</tex></formula> for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe the complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov--Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincar\'{e} inequality in 1D spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of <formula><tex>$\|\cdot\|_\infty$</tex></formula>, if the sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance the 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.

Original language | English |
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Journal | IEEE Transactions on Fuzzy Systems |

DOIs | |

Publication status | Accepted/In press - 24 Feb 2018 |

## Keywords

- Agmon's inequality
- distributed parameter systems
- exponential stability
- Sampled-data control
- Takagi-Sugeno fuzzy partial differential equation model