Design of optimization-based reliable boundary control for mode-dependent parabolic PDE systems

The objective of this study is to address the finite-time stabilization problem for mode-dependent partial differential equation systems governed by the parabolic model with parameter uncertainties and external disturbances under a reliable boundary control protocol. Specifically, the approach of finite-time boundedness and input-output finite-time stability are deployed in tandem to limit both state and output, respectively, during certain transients. Moreover, to enhance the system’s operational effectiveness and reduce the cost function, a particle swarm optimization algorithm is employed. Furthermore, a mode-dependent proportional-integral observer is put forth to estimate the states of a mode-dependent parabolic partial differential equation systems. Thereafter, reliable boundary control is developed to attain the intended outcomes. Specifically in the design of control, the faulty actuator model which includes both linear and nonlinear aspects, are considered, to enhance the robustness of the controller. Subsequently, through the implementation of the mode-dependent Lyapunov functions and the integral-based Wirtinger’s inequality, the requisite criteria are established within the framework of linear matrix inequalities to ascertain both input-output finite-time stabilization and finite-time boundedness of the assessed system. Simultaneously, the established criteria facilitate the determination of both the developed controller and the proportional-integral observer gain matrices. Eventually, two numerical examples are provided to validate the effectiveness and applicability of the devised control protocol.