The problem of optimally controlling a Wiener process until it leaves an interval (a; b) for the first time is considered in the case when the infinitesimal parameters of the process are random. When a = ��1, the exact optimal control is derived by solving the appropriate system of differential equations, whereas a very precise approximate solution in the form of a polynomial is obtained in the two-barrier case.
In this paper the basic methodology of the coupled response-degradation modelling of stochastic dynamical systems is presented along with the effective analysis of selected problems. First, the general formulation of the problems of stochastic dynamics coupled with the evolution of deterioration process is given. Then some specific degrading oscillatory systems under random excitation are analyzed with a special attention on the systems with fatigue-induced stiffness degradation. Both, the general discussion and the analysis of selected exemplary problems indicate how the reliability of deteriorating stochastic dynamical systems can be assessed.