Robust Averaged Control of Vibrations for the Bernoulli-Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli-Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting which is so general as to include different types of random variables and second order random fields as sources of uncertainty. The second order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. Since the stochastic direct and adjoint systems are uncoupled, the set of decoupled deterministic problems that arises from the stochastic collocation method are solved in parallel which reduces the computational burden and permits to solve problems with a relative large number of terms in the truncated Karhunen-Loève expansion of a random field. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.