Approximate Analytical Solutions of Reliability for Controlled Single First-Integral Nonlinear Stochastic Systems

We study the reliability of stochastically excited and controlled single first-integral systems and present an approximate analytical solution for the first-passage rate (FPR). By introducing the stochastic averaging method (SAM), we reduce the dimension of the original system to an averaged one-dimensional controlled Ito differential equation. We then modify the classic Laplace integral method (LIM) and apply it to deal with the arduous integrals in the expression of reliability function. The procedure of acquiring the analytical solution for the reliability is illuminated in detail as well. In addition, we provide two controlled single first-integral nonlinear vibration systems, namely, the classical bistable model and the two coupled nonlinear oscillators, as examples. By comparing the results obtained from the modified Laplace integral method (MLIM) to Monte Carlo simulations (MCS), we verify the effectiveness and exactness of the proposed procedure. We identified two properties in the obtained analytical solution: One is that the solutions are independent of the initial system state. The other is that they are only effective in the high passage threshold range. Finally, a reasonable explanation has been given to explain these two properties.