Point-Based Stabilization Control of Linear Parabolic Partial Differential Equation Systems on Space–Time Scales

The extensively studied partial differential equation (PDE) systems evolve on continuous time and continuous space, which restricts the investigation scope of PDE systems. A parabolic PDE system on space–time scales is constructed for the first time. This article focuses on global exponential stabilization (GES) of a linear parabolic PDE systems on space–time scales by using point-based collocated and noncollocated observations. First, we give the definitions of space–time scales, which contain four cases: continuous time and continuous space, continuous time and discrete space, discrete time and continuous space, and discrete time and discrete space. Wirtinger inequality on space scales is demonstrated, which provides mathematically analytical technique for PDE systems on space–time scales. Second, point-based collocated stabilization control strategy is developed for parabolic PDE systems on space–time scales. Third, considering the fact that sensors and actuators are always placed at different locations, a noncollocated point-based feedback controller is designed by using the observer estimation states. Different from the existing literature, the spatial domain does not need to be divided again according to the actuator and sensor locations. By using the Lyapunov function method, calculus on space–time scales, integration by parts, and Wirtinger inequality on space scales, sufficient criteria are developed such that the closed-loop systems can achieve GES. Finally, numerical simulation examples are presented to show the effectiveness of the developed point-based controllers.