Abstract: The stochastic optimal control of a spherically symmetric piezoelectric shell under random boundary excitations is studied. The dynamic equilibrium equation, the kinetic equation, the charge conservative equation and the electrical and mechanical coupling constitutive equations of the piezoelectric shell are given. These equations and the performance parameters construct a stochastic optimal control problem. The electric potential integral is presented to convert the governing equations into a mechanical vibration equation. Using the displacement transformation and the Galerkin method, the partial differential vibration equation is converted into ordinary differential equations for optimal control. Based on the stochastic dynamical programming principle, the HJB equation is derived and then the optimal control electric potential for the piezoelectric shell is determined. The frequency-response function, response power spectrum density and correlation function matrices of the controlled shell system are presented for estimating the random response. Finally, numerical results are given to demonstrate the effectiveness of the stochastic optimal control of the piezoelectric shell.

Key words: vibration and wave, piezoelectric shell, random vibration, optimal control, random response

摘要： 分析研究球对称压电壳在边界随机激励下的最优控制问题。给出压电壳的机电动力学方程、应力和电位移表达式，建立其随机最优控制问题方程；通过电势积分转化为机械振动控制方程。通过位移变换和Galerkin法，导出关于模态位移的多自由度振动最优控制方程。根据随机动态规划原理，建立HJB方程，得到压电壳的最优控制电势；并给出受控壳系统的频响函数、响应谱密度和相关函数等表达式，以计算其随机响应。最后给出数值结果，显示压电壳的随机最优控制效果。

关键词: 振动与波, 压电壳, 随机振动, 最优控制, 随机响应