General Multiple-Phase Problem Solvable Using GPOPS-II
The general multiple-phase optimal control problem that can be solved by \(\mathbb{GPOPS-II}\) is given as follows. First, let \(p\in[1,\ldots,P]\) be the phase number where \(P\) as the total number of phases. The optimal control problem is to determine the state, \(\mathbf{y}^{(p)}(t)(t)\in\mathbb{R}^{n_y^{(p)}}\), the control, \(\mathbf{u}^{(p)}(t)(t)\in\mathbb{R}^{n_u^{(p)}}\), integrals, \(\mathbf{q}^{(p)}\in\mathbb{R}^{n_q^{(p)}}\), phase start times, \(t_0^{(p)}\in\mathbb{R}\), phase terminus times, \(t_f^{(p)}\in\mathbb{R}\), in all phases \(p\in[1,\ldots,P]\), along with the static parameters \(\mathbf{s}\in\mathbb{R}^{n_s}\), that minimize the objective functional
$$
J = \phi \left( \mathbf{e}^{(1)},\ldots,\mathbf{e}^{(P)}, \mathbf{s} \right),
$$
subject to the dynamic constraints
$$
\dot{\mathbf{y}}^{(p)}(t) = \mathbf{a}^{(p)}(\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t,\mathbf{s}), \quad (p=1,\ldots,P),
$$
the event constraints
$$ \label{general-event}
\mathbf{b}_{\min} \leq \mathbf{b} \left(\mathbf{e}^{(1)}, \ldots,\mathbf{e}^{(P)}, \mathbf{s} \right)\leq \mathbf{b}_{\max},
$$
where
$$\small
\mathbf{e}^{(p)}=\left[\mathbf{y}^{(p)}\left(t_0^{(p)}\right),t_0^{(p)},\mathbf{y}^{(p)}\left(t_f^{(p)}\right),t_f^{(p)},\mathbf{q}^{(p)}\right],\;(p=1,\ldots,P),$$
the inequality path constraints
$$
\mathbf{c}_{\min}^{(p)} \leq \mathbf{c}^{(p)}(\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t^{(p)},\mathbf{s})\leq \mathbf{c}_{\max}^{(p)}, \; (p=1,\ldots,P),
$$
the static parameter constraints
$$
\mathbf{s}_{\min} \leq \mathbf{s} \leq \mathbf{s}_{\max},
$$
and the integral constraints
$$
\mathbf{q}_{\min}^{(p)} \leq \mathbf{q} ^{(p)} \leq \mathbf{q}_{\max}^{(p)}, \; (p=1,\ldots,P),
$$
and the integrals in each phase are defined as
$$
q_i^{(p)} = \int_{t_0^{(p)}}^{t_f^{(p)}} g_i^{(p)} (\mathbf{y}^{(p)}(t),\mathbf{u}^{(p)}(t),t^{(p)},\mathbf{s})dt, \left[\begin{array}{c} i=1,\ldots n_q^{(p)},\\ p=1,\ldots,P \end{array} \right].
$$
It is important to note that the event constraints of Eq.~(\ref{general-event}) can contain any functions that relate information at the start and/or terminus of any phase (including relationships that include both static parameters and integrals) and that the phases themselves need not be sequential.
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