Stability Analysis Of Networks Of Wave Equations With Boundary Switch Feedback Controls

As a kind of control thought, switch has already been applied in control field, among them, the most typical representation is the Bang-Bang control, whose control variable is set from the constraint boundary values. Because the switch systems can greatly reduce the time which the controllers take and shorten the output of the power, it achieves the effect of energy saving. In addition, the wave network has the rich application background whether in the real life or the production, such as, elastic network in the field of engineering,microelectronics technology and circuit design, neurobiology and many other fields. So, the study of the wave network system with switch feedback controls has been a popular topic.In this paper, we consider the system with two different kinds of switch controls.The system is described by the wave equations with the structure of binary tree, of which one end is fixed and the other two ends are damped by the velocities feedback controls?(t)u2t(x, t), ?(t)u3t(x, t) respectively, where the ?(t), ?(t) are both Heaviside-type functions of periodic 2-T. According to two different cases, we analyze the stability of the closed-loop system. The first one is that the two controls are on-off simulatively, the other case is that there is only one controller working at anytime. Using the D'Alembert formula, we get the explicit expression of the solution for the closed loop system. And then, we analyze the spectrum of the iterate matrices, from which we obtained the properties of the solution.Finally, we get the exponential stability and the decay rate.