Free boundary value problems and hjb equations for the stochastic optimal control of elasto-plastic oscillators

We consider the optimal stopping and optimal control problems related to stochastic variational inequalities modeling elasto-plastic oscillators subject to random forcing. We formally derive the corresponding free boundary value problems and Hamilton-Jacobi-Bellman equations which belong to a class of nonlinear partial of differential equations with nonlocal Dirichlet boundary conditions. Then, we focus on solving numerically these equations by employing a combination of Howard’s algorithm and the numerical approach [A backward Kolmogorov equation approach to compute means, moments and correlations of non-smooth stochastic dynamical systems; Mertz, Stadler, Wylie; 2017] for this type of boundary conditions. Numerical experiments are given

Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise

International audienceThe risk of failure of mechanical structures under random forcing is an important concern in earthquake engineering. For a class of simple structures that can be modeled by an elasto-plastic oscillator, the risk of failure can be expressed in terms of the probability that, on a certain interval of time, the plastic deformation goes beyond a threshold related to a failure zone. In this note, asymptotic formulae for the risk of failure of an elasto-perfectly-plastic oscillator excited by a white noise are proposed. Our approach exploits the long cycle (repeating pattern) property of the aforementioned oscillator as introduced in A. Bensoussan, L. Mertz, S.C.P. Yam, Long cycle behaviour of the plastic deformation of an elasto-perfectly-plastic oscillator with noise, C. R. Acad. Sci. Paris Ser. I, 2012. We show that asymptotically the plastic deformation behaves like a Wiener process for which analytical formulae are available. Our result is a consequence of the Anscombe–Donsker Invariance Principle. Numerical experiments and comments are carried out

Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise

The risk of failure of mechanical structures under random forcing is an important concern in earthquake engineering. For a class of simple structures that can be modeled by an elasto-plastic oscillator, the risk of failure can be expressed in terms of the probability that, on a certain interval of time, the plastic deformation goes beyond a threshold related to a failure zone. In this note, asymptotic formulae for the risk of failure of an elasto-perfectly-plastic oscillator excited by a white noise are proposed. Our approach exploits the long cycle (repeating pattern) property of the aforementioned oscillator as introduced in A. Bensoussan, L. Mertz, S.C.P. Yam, Long cycle behaviour of the plastic deformation of an elasto-perfectly-plastic oscillator with noise, C. R. Acad. Sci. Paris Ser. I, 2012. We show that asymptotically the plastic deformation behaves like a Wiener process for which analytical formulae are available. Our result is a consequence of the Anscombe–Donsker Invariance Principle. Numerical experiments and comments are carried out