Application of pseudospectral method in stochastic optimal control of nonlinear structural systems

This paper presents the results of numerical study on the optimal control strategy of nonlinear stochastic systems. The systems under investigation are mechanical oscillators and a damping device. The numerical approach to obtain the optimal control strategy involves solving a nonlinear partial differential equation - the Hamilton-Jacobi-Bellman equation. Since civil engineering structural systems usually exhibit nonlinear hysteretic behavior under extreme loading conditions, the potential application of the obtained control strategy could provide an optimal feedback control law to reduce the system response under the random excitations (such as earthquakes, wind load and sea waves). Several numerical examples are presented to verify optimality and demonstrate the efficacy of the proposed optimal control solution. First, a linear oscillator is used to verify that the obtained solution is indeed the optimal solution by comparing it to the closed form solution. Then the proposed method is applied to several nonlinear systems. In each case, optimality is demonstrated by comparing the system responses and costs under optimal control with those obtained using linearized optimal control.