A general numerical solution to optimal nonlinear stochastic structural control problem

Civil engineering structural systems exhibit hysteretic behavior when under extreme loading conditions as well as when energy dissipation devices are employed. To investigate the optimal control strategy for reducing the system response under random excitations (earthquakes, wind gust or sea waves), a general control solution is proposed in this paper. The approach considers the solution of the Hamilton-Jacobi-Bellman equation for general nonlinear stochastic systems, under the assumption that the evolution of the state of the stochastic system can be described by a Markov diffusion process. Several numerical examples are provided to verify the efficacy of the optimal control solution obtained from the proposed method. 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 including Van der Pol and Duffing oscillators and a Bouc-Wen system. In each case, optimality is demonstrated by comparing the system responses and costs under optimal control with the ones obtained using linearized optimal control.