First-passage failure of quasi-integrable Hamiltonian systems under time-delayed feedback control

A procedure for studying the first-passage failure of quasi-integrable Hamiltonian systems under time-delayed feedback control is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems under time-delayed feedback control is firstly introduced. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.