Approximate Analytical Solution Of First-Passage Time Of Controlled Strongly Non-linear Stochastic Systems

The first-passage failure of strongly nonlinear stochastic systems is one of the most important and difficult topics in stochastic dynamics, which has a natural connection with the failure and destruction of the systems. Finding its solution, especially the semi-analytical solution, always attracts the researchers’ attentions. In the present thesis, the controlled quasi-nonintegrable Hamiltonian system, controlled viscoelastic system and controlled nonlinear ship rolling system are considered and the first-passage failure is investigated, as well as the predator-prey ecosystem. For the controlled system, the stochastic averaging method of quasi-nonintegrable Hamiltonian system or the stochastic averaging method based on the generalized harmonic functions is applied to reduce the original stochastic systems to the lower dimensional systems with respect to the system energy or amplitude. Based on the dynamical programming principle and the performance index regarding the first passage, the optimal bounded control is derived. Substituting the optimal control yields the fully averaged stochastic differential equation and it can be studied by two procedures. One is that, based on its Fokker-Planck-Kolmogorov equation in the form of probability current and with the application of the modified Laplace integral method, the approximate analytical solution of the conditional reliability function is obtained, as well as the probability density of the first-passage time and its moments. The other is that, the corresponding backward Kolmogorov equation is numerically solved to obtain the results of the first-passage problem. The approximate analytical results of some illustrated examples are compared with the numerical results from directly solving the backward Kolmogorov equation and the Monte Carlo simulation results of the original systems. It is found that the approximate analytical solution is very accurate in the situation of high threshold and long first-passage time and it can be easily applied to investigate the first-passage failure of the practical stochastic systems. Finally, the first passage of the predator-prey ecosystem under random excitation is investigated by using the splitting method. It is first observed that the conditional probability density function of the first-passage time shows obviously periodic variation due to the threshold variation along the boundary of the rectangle domain and the quasi-periodic nature of the population densities.