Optimal Control Problem (OCP):

Find the states \(y : [t_0 , t_f] \rightarrow \mathbb{R}^{n_y}\), the parameter vector \(p \in \mathbb{R}^{n_p}\) and the controls \(u : [t_0 , t_f] \rightarrow \mathbb{R}^{n_u}\), such that $$\begin{align*} \varphi(y(t_0), y(t_f), p) \end{align*}$$ is minimal, subject to the nonlinear constraints $$\begin{align*} y'(t) &= F(t, y(t), u(t), p) && \\ \psi_{\min} &\leq \psi(y(t_0), y(t_f), p) \leq \psi_{\max}&& \\ v_{\min} &\leq v(t, y(t), u(t), p) \leq v_{\max} && \end{align*}$$ with \(t \in \left[t_0, t_f\right]\) and the box conditions $$\begin{align*} y_{\min} &\leq y(t) \leq y_{\max}&& t \in \left[t_0, t_f\right]\\ u_{\min} &\leq u(t) \leq u_{\max}&& t \in \left[t_0, t_f\right]\\ p_{\min} &\leq p \leq p_{\max}\text{.}&& \end{align*}$$ Herein, the functions $$\begin{align*} \varphi&: \mathbb{R}^{n_y} \times \mathbb{R}^{n_y} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R} \\ F&: \left[t_0, t_f\right] \times \mathbb{R}^{n_y} \times \mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_y} \\ \psi&: \mathbb{R}^{n_y} \times \mathbb{R}^{n_y} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_\psi} \\ v&: \left[t_0, t_f\right] \times \mathbb{R}^{n_y} \times \mathbb{R}^{n_u} \times \mathbb{R}^{n_p} \rightarrow \mathbb{R}^{n_v} \text{.} \end{align*}$$ are user-defined functions.

Nonlinear Optimazation Problem (NLP):

Find the vector \(x \in \mathbb{R}^n\) to minimize $$\begin{align*} f(x) \end{align*}$$ subject to the nonlinear constraints $$\begin{align*} h(x) &= 0 \\ g(x) &\leq 0 \end{align*}$$ with \(f: \mathbb{R}^n \rightarrow \mathbb{R}\), \(h: \mathbb{R}^n \rightarrow \mathbb{R}^p\) and \(g: \mathbb{R}^n \rightarrow \mathbb{R}^q\).