Boundary Control Of The Wave Equation With Spatially Varying Propagation Speed And The KS-like Equation

In this paper, boundary control design for the Wave Equation with Spatially Varying Propagation Speed and the Kuramoto-Sivashinsky-like (KS-like) Equation in the distributed parameter systems is considered.For the control of the Wave Equation with Spatially Varying Propagation Speed , by one linear invertible transformation, we first convert the system into a new sys-tem that the propagation speed is constant. Then based on the design of the new system, the observer and output feedback controllers of the original system are de-signed to achieve exponential stabilization of the closed-loop system. In the design process, an Volterra invertible transformation is introduced to establish the relation between the new system and the target system, then state feedback controller is designed through the backstepping method to make the system achieve exponential stability. In the process of apply transformation, one kernel function is generated. The kernel function satisfy a Klein-Gordon-type hyperbolic partial differential equa-tion (PDE), then the hyperbolic PDE is converted into equivalent integral equation, By the method of successive approximations, the well posedness of the kernel func-tion is proved. Observer and output-feedback controller are designed according to the characteristics of the system when only boundary measurements are available. the exponential stabilization of the closed-loop system is proved.KS-like equation is high order PDE that resembles KS equation, it behaves almost the same as the linearized KS equation when some conditions are satisfied, so it is worth to be studied. The control of the KS-like equation need to design two controllers. In order to design controller, we utilize variable transformation to reduce order. To put the reduced order system into a strict-feedback form,the first controller is designed. Then an Volterra invertible transformation is introduced to establish the relation between the new system that is reduced order and the target system. First, the second state feedback controller is designed through the back-stepping method to make the system achieve exponential stability. Then according to the characteristics of the system, an observer and output-feedback controllers are designed when only boundary measurements are available. The well posedness of the kernel function that from the process of transformation and the exponential stabilization of the closed-loop system consists of the control plant, the observer, and the controllers are proved.