| Summary: | In a recent work of A. Bensoussan and J. Turi Degenerate Dirichlet Problems
Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008, it
has been shown that the solution of a stochastic variational inequality modeling an
elasto-plastic oscillator excited by a white noise has a unique invariant probability
measure. The latter is useful for engineering in order to evaluate statistics of plastic
deformations for large times of a certain type of mechanical structure. However, in terms
of mathematics, not much is known about its regularity properties. From then on, an
interesting mathematical question is to determine them. Therefore, in order to investigate
this question, we introduce in this paper approximate solutions of the stochastic
variational inequality by a penalization method. The idea is simple: the inequality is
replaced by an equation with a nonlinear additional term depending on a parameter
n
penalizing the solution whenever it goes beyond a prespecified area. In this context, the
dynamics is smoother. In a first part, we show that the penalized process converges
towards the original solution of the aforementioned inequality on any finite time interval
as n goes to
∞. Then, in a second part,
we justify that for each n it has a unique invariant probability measure.
Finally, we provide numerical experiments and we give an empirical convergence rate of the
sequence of measures related to the penalized process.
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