# Differential parameterized inequalities

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.