一类半线性抛物型偏微分方程描述的分布参数系统的边界控制

Boundary control of the distributed parameter systems described by a class of semi-linear parabolic partial differential equations

针对一类由半线性抛物型偏微分方程描述的分布参数系统,考虑系统边界为Robin或混合边界条件的情况,提出基于边界控制的控制策略,并研究其镇定问题.首先,根据无穷维抽象发展方程理论和Lumer-Phillips理论证明闭环系统的适定性;其次,通过传感器对系统状态进行测量,并将数据传递给控制器,根据测量方式的不同,分为平均域测量和边界值测量;再次,基于Lyapunov稳定性理论,采用Lyapunov直接法,并借助于线性矩阵不等式(LMI)方法,设计满足系统稳定条件的有效控制器;然后通过在系统边界处分别施加基于平均域测量和边界值测量的输出反馈控制作用,使原本不稳定的开环分布参数系统状态在较短的时间内到达稳定状态;最后,通过仿真实验验证了所设计控制器的有效性.

The boundary control strategy of the distributed parameter systems described by a class of semi-linear parabolic partial differential equations with Robin or mixed boundary conditions is proposed, and the corresponding stabilization problem is also studied. According to the theory of the infinite dimensional abstract development equation and the theory of Lumer-Phillips, the well-posedness of closed loop system can be proved. By means of different measurement methods,the states of the system can be obtained by domain-averaged measurement and boundary-valued measurement respectively and then transmitted to the controllers. Employing the Lyapunov’s direct method and the linear matrix inequality(LMI)technique, the effective controllers satisfing the conditions of stable systems can be designed according to the Lyapunov stability theory. By applying output feedback control at the boundary of systems with the domain-averaged measurement and boundary-valued measurement respectively, the distributed parameter systems can reach the steady states from the unstable states in a short time. Finally, the simulations verify the effectiveness of the designed controllers.