Mixed H ₂ ∕H _∞ sampled-data output feedback control design for a semi-linear parabolic PDE in the sense of spatial L ^∞ norm

This paper addresses distributed mixed H ₂ ∕H _∞ sampled-data output feedback control design for a semi-linear parabolic partial differential equation (PDE) with external disturbances in the sense of spatial L ^∞ norm. Under the assumption that a finite number of local piecewise measurements in space are available at sampling instants, a static sampled-data output feedback controller is suggested, where the sampling interval in time is bounded. The local piecewise measurements bring additional difficulty for the exponential stability and performance analysis since the existing Poincaré-Wirtinger inequality in 1D spatial domain is not applicable. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing Wirtinger's inequalities, a variant of Poincaré-Wirtinger inequality in 1D spatial domain, as well as Agmon's inequality, it is shown that the suggested static sampled-data output feedback controller not only guarantees the output exponential stability of the resulting closed-loop PDE in the spatial L ^∞ norm but also ensures the mixed H ₂ ∕H _∞ performance index defined in the spatial L ^∞ norm, if the sufficient conditions presented in terms of standard linear matrix inequalities (LMIs) are fulfilled. The satisfactory and better performance of the suggested sampled-data feedback controller is demonstrated by numerical simulation results of an illustrative example.