Feedback minimization of the first-passage failure of a hysteretic system under random excitations

The stochastic optimal bounded control of a hysteretic system for minimizing its first-passage failure is presented. The hysteretic system subjected to random excitation is firstly replaced by an equivalent nonlinear non-hysteretic system. The controlled non-hysteretic system is reduced to a one-dimensional controlled diffusion process by using the stochastic averaging of the energy envelope method. The dynamical programming equations and their associated boundary and final-time conditions for the problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for the maximum reliability problem and the mean first-passage time problem are finalized and solved numerically. Finally, numerical results are worked out to illustrate the application and effectiveness of the proposed method.