Goddart's rocket problem - Revision history

RobertLampel: /* Parameters */

2025-10-10T12:45:56Z

Parameters

2025-10-10T12:45:43Z

Variables

← Older revision		Revision as of 12:45, 10 October 2025
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	<dd>		<dd>
	<math>		<math>
	D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).		D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= \exp(-k\cdot (r-r_0)).
	</math>		</math>
	</dd>		</dd>

2025-10-10T12:45:26Z

Mathematical formulation

2025-10-10T12:45:12Z

Mathematical formulation

← Older revision		Revision as of 12:45, 10 October 2025
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	\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]		\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]
	\mbox{s.t.} & \dot{r} & = & v, \\		\mbox{s.t.} & \dot{r} & = & v, \\
	& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]		& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{\mathrm{max}}u-D(r,v)) \\[1.5ex]
	& \dot{m} & = & -b ~~T_{max}~~ u, \\		& \dot{m} & = & -b u, \\
	& u(t) &\in& [0,1] \\		& u(t) &\in& [0,1] \\
	& r(0) &=& r_0, \\		& r(0) &=& r_0, \\
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	& r(T) &=& r_T, \\		& r(T) &=& r_T, \\
	& D(r,v)&\le& C \\		& D(r,v)&\le& C \\
	& T \, free		& T \, \text{free}
	\end{array}		\end{array}
	</math>		</math>

2016-02-22T16:08:53Z

Mathematical formulation

← Older revision		Revision as of 16:08, 22 February 2016
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	\begin{array}{llcll}		\begin{array}{llcll}
	\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]		\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]
	\mbox{s.t.} & \dot{r}~~(t)~~ & = & v, \\		\mbox{s.t.} & \dot{r} & = & v, \\
	& \dot{v}~~(t)~~ & = & -\frac{1}{r~~(t)~~^2} + \frac{1}{m~~(t)~~} (T_{max}u~~(t)~~-D(r,v)) \\[1.5ex]		& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
	& \dot{m}~~(t)~~ & = & -b T_{max} u~~(t)~~, \\		& \dot{m} & = & -b T_{max} u, \\
	& u(t) &\in& [0,1] \\		& u(t) &\in& [0,1] \\
	& r(0) &=& r_0, \\		& r(0) &=& r_0, \\
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	& m(0) &=& m_0, \\		& m(0) &=& m_0, \\
	& r(T) &=& r_T, \\		& r(T) &=& r_T, \\
	& D(r~~(t)~~,v~~(t)~~)&\le& C \\		& D(r,v)&\le& C \\
	& T \, free		& T \, free
	\end{array}		\end{array}

2016-01-28T09:33:08Z

Source Code

← Older revision		Revision as of 09:33, 28 January 2016
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	* [[:Category: Bocop \| Bocop code]] at [[Goddart's rocket problem (Bocop)]]		* [[:Category: Bocop \| Bocop code]] at [[Goddart's rocket problem (Bocop)]]
			* [[:Category: AMPL/TACO \| AMPL/TACO code]] at [[Goddart's rocket problem (TACO)]]

	== References ==		== References ==

2016-01-27T20:06:54Z

2016-01-27T16:03:50Z

2016-01-27T16:03:04Z

← Older revision		Revision as of 16:03, 27 January 2016
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			{{Dimensions
			\|nd = 1
			\|nx = 3
			\|nw = 1
			\|nre = 3
			}}

	In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.		In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.
	~~[[Category:MIOCP]]~~
	== Variables ==		== Variables ==
	The state variables <math>r,v,m</math> describe the altitude(radius), speed and mass respectively.		The state variables <math>r,v,m</math> describe the altitude(radius), speed and mass respectively.
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	The Problem can be found in the [http://bocop.org/ BOCOP User Guide].		The Problem can be found in the [http://bocop.org/ BOCOP User Guide].

			[[Category:MIOCP]]
	[[Category:Aeronautics]]		[[Category:Aeronautics]]
	[[Category:Minimum energy]]		[[Category:Minimum energy]]

2016-01-27T13:08:00Z

← Older revision		Revision as of 13:08, 27 January 2016
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	Image:Goddartmass.png\| Mass m over time.		Image:Goddartmass.png\| Mass m over time.
	</gallery>		</gallery>

			== Source Code ==
			Model descriptions are available in:

			* [[:Category: Bocop \| Bocop code]] at [[Goddart's rocket problem (Bocop)]]


	== References ==		== References ==