*2.2. Scaled Problem Formulation*

The problem is formulated in terms of standard form described in Equations (2)–(8), where *x*(·), *v*(·) are the decision variables. The endpoint cost *E x tf* is also referred to as the Mayer cost. The running cost *F*(*x*(*t*), *u*(*t*)) is also referred to as the Lagrange cost (usually with the integral). The standard cost function *J*[*x*(·), *u*(·)] is also referred to as the Bolza cost as the sum of the Mayer cost and Lagrange cost. Endpoint constraints *e x tf* are equations that are selected to be zero when the endpoint is unity.

$$\mathbf{x}^T = [\mathbf{x}, \mathbf{v}], \quad \mathbf{u} = [\mathbf{u}] \tag{2}$$

$$\text{Minimize } \quad J[\mathbf{x}(\cdot), \mathbf{z}(\cdot), \mathbf{u}(\cdot)] = E\left(\mathbf{x}\left(t\_f\right)\right) + \int\_0^{t\_f} F(\mathbf{x}(t), \mathbf{u}(t))dt = \frac{1}{2} \int\_0^{t\_f} \mathbf{u}^2 dt \tag{3}$$

$$\text{Subject to} \quad \dot{\mathbf{x}}\_1 = f\_1(\mathbf{x}(t), \boldsymbol{\mu}(t)) = \boldsymbol{v} \tag{4}$$

$$
\dot{\mathbf{x}}\_2 = f\_2(\mathbf{x}(t), \boldsymbol{\mu}(t)) = \dot{\boldsymbol{\nu}} = \boldsymbol{\mu} \tag{5}
$$

$$(x\_0, v\_0) = (0, 0) \tag{6}$$

$$\left(\mathbf{x}\_{f-1}, \mathbf{v}\_{f}, \mathbf{t}\_{f-1}\right) = \left(\mathbf{0}, \mathbf{0}, \mathbf{0}\right) \tag{7}$$

$$e\left(\mathbf{x}\left(t\_f\right)\right) = 0\tag{8}$$

where

*J*[*x*(·), *u*(·)] cost function *x<sup>T</sup>* = [*x*, *v*] state vector of motion state *x* and rate *v* with initial condition *<sup>x</sup><sup>T</sup>* <sup>=</sup> [*x*, *<sup>v</sup>*] (*x*0, *<sup>v</sup>*0) and final conditions *xf*−1, *vf* , *tf*−<sup>1</sup> = (0, 0, 0) *u* = [*u*] decision vector *H* Hamiltonian operator corresponding to system total energy *λ<sup>T</sup>* adjoint operators, also called co-states (corresponding to each state) *υ<sup>T</sup>* endpoint costates *e x tf* endpoint constraints.
