Abstract: A semi-active suspension system with shape memory alloy(SMA) spring of vehicle is built, considering the impact of uneven tire materials and other internal random factors. The Hamilton function is expressed as one dimension diffusion process by using stochastic average method, by judging the modality of the singular boundary, the condition of the first-passage behavior is calculated, and the backward Kolmogorov equation for reliability function and probability density function of the first-passage time is established. On the basis, combined with the initial conditions and boundary conditions, the first-passage issue is derived, and the numerical simulation is done. The results show that the possibility of destruction of system stability gradually increases with time passing and the system eventually becomes unstable. When the initial state is away from the right border, its reliability function decreases very slowly. At the same time point, the former possibility becomes smaller. With the initial state being gradually away from the right border, the moment that stability of the system most likely to be damaged is delayed. The model of the controlled system is established by adding the control unit, and the optimal control strategy aiming at obtaining the maximal reliable function is derived by utilizing the dynamic programming principle. Numerical simulation of partial differential equations fitted for reliability function and probability density function of the first-passage time is carried out by using finite difference methods. Numerical results show that, with the increase of the constraining force of control, the possibility of the system state staying in the security domain increases, and the possibility of destruction of the system steady state decreases.

Key words: First-passage, Optimal control, Reliability function, Semi-active suspension, Shape memory alloy(SMA) spring, Stochastic stability, Bearing, High speed vehicle, Vibration, Waviness