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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