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Let $H$ be an Hamiltonian and denote $\vec{H}$ the associated Hamiltonian vector field. I am interested in solving numerically the following problem $$ \dot z(t) = \vec{H}(z(s),t_1,\ldots, t_p) \qquad \text{ for } t\in [0,T] \quad a.e. $$ subject to $$ \Gamma(z(0),z(T))=0 $$ and $$ \int_0^T v_i\ F_i(z(t),t_1,\dots,t_p)\ ds \le 0 $$ for all $v_i\in K(t_1,\dots,t_p)$, where $K(t_1,\dots,t_p)$ is an interval of $\mathbb{R}$ (typically $\mathbb{R}^+,\mathbb{R}^-, \mathbb{R}$ ...) and $t_1,\dots,t_p$ are parameters to be computed.

I would like to get some references about this kind of problems.

I found this reference article DVI but my problem is not exactly a differential variational inequalities problem since I have parameters instead of functions. I would call it "Differential parameterized inequalities" but I didn't find any references.

What I tried numerically is to implement a primal dual interior point method where I don't minimize anything but it does not work properly.

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