geometric minimal action method using optimal control

[oameye]

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I would like to compute the maximum likelihood path (MLP) of an underdamped stochastic ordinary system. Suppose we SDE of the form:

x^{\prime}(t)=f(x(t))+\varepsilon \eta(t) .

with $\eta$ additive Wiener Process. The geometric minimal action method states that in the limit of $\varepsilon\ll1$ the MLP is given by optimizing the following action integral:

\hat{S}_T(x)=\int_0^1\left\{\left|x^{\prime}\right||f(x)|-(x^{\prime}, f(x))\right\} \ d s

where $\left(\cdot,\cdot\right)$ represents the dot product. Is there a way to encode this in OptimalControl.jl?

A typical example would be a double well problem such as the Maier Stein system given as:

\begin{align}
    \dot{u} &= u-u^3 - 10*u*v^2\\
    \dot{v} &= -(1+u^2)*v
\end{align}

where one wants to know the optimal path between the two attractor (-1, 0) and (1, 0).