First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems

• Published in: International journal of non-linear mechanics Vol. 38; no. 8; pp. 1133 - 1148
• Main Authors: Zhu, W.Q., Huang, Z.L., Deng, M.L.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.10.2003; Elsevier Science
• Subjects: Dynamical programming; Exact sciences and technology; First-passage failure; First-passage time; Fracture mechanics (crack, fatigue, damage...); Fracture mechanics, fatigue and cracks; Fundamental areas of phenomenology (including applications); Non-linear system; Physics; Reliability; Solid mechanics; Stochastic averaging; Stochastic excitation; Stochastic optimal control; Structural and continuum mechanics; First-passage time; Non-linear system; Stochastic excitation; First-passage failure; Stochastic optimal control; Dynamical programming; Reliability; Stochastic averaging; Averaging method; Itô equation; Probabilistic approach; Random excitation; Rupture; Modeling; Stochastic integral; First passage time; Hamiltonian mechanics; Hamiltonian system; Dynamic programming; Random vibration
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/S0020-7462(02)00058-6

An n degree-of-freedom Hamiltonian system with r (1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system. A partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi-partially integrable Hamiltonian system. In the present paper, the procedures for studying the first-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems are proposed. First, the stochastic averaging method for quasi-partially integrable Hamiltonian systems is briefly reviewed. Then, based on the averaged Itô equations, a backward Kolmogorov equation governing the conditional reliability function, a set of generalized Pontryagin equations governing the conditional moments of first-passage time and their boundary and initial conditions are established. After that, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and of maximization of mean first-passage time are formulated. The relationship between the backward Kolmogorov equation and the dynamical programming equation for reliability maximization, and that between the Pontryagin equation and the dynamical programming equation for maximization of mean first-passage time are discussed. Finally, an example is worked out to illustrate the proposed procedures and the effectiveness of feedback control in reducing first-passage failure.