Mountain Car - Revision history

RobertLampel at 08:27, 27 January 2026

2026-01-27T08:27:31Z

← Older revision		Revision as of 08:27, 27 January 2026
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	The '''Mountain Car problem''' proposes a vehicle stuck in a "well". It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.		The '''Mountain Car problem''' proposes a vehicle stuck in a "well". It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.

	The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].		The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990 [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].
	Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].		Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].

RobertLampel at 08:26, 27 January 2026

2026-01-27T08:26:59Z

← Older revision		Revision as of 08:26, 27 January 2026
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	}}		}}

	The '''Mountain Car problem''' proposes a vehicle stuck in a ~~“well~~.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.		The '''Mountain Car problem''' proposes a vehicle stuck in a "well". It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.

	The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].		The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].

RobertLampel: /* Reference Solutions */

2025-11-28T13:46:33Z

Reference Solutions

← Older revision		Revision as of 13:46, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Mountain_Car.png\| States and discretized control for a local optimum.		Image:Mountain_Car.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel at 06:31, 4 September 2025

2025-09-04T06:31:45Z

← Older revision		Revision as of 06:31, 4 September 2025
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	The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].		The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].
	Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].		Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].

	~~The optimal integer control functions exhibits a [[:Category:Bang bang\|bang bang]] structure.~~

	== Mathematical formulation ==		== Mathematical formulation ==

RobertLampel: /* References */

2025-08-20T14:59:42Z

References

← Older revision		Revision as of 14:59, 20 August 2025
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	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:~~Path-constrained arcs~~]]		[[Category:Bang bang]]

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-20T14:58:48Z

Miscellaneous and Further Reading

← Older revision		Revision as of 14:58, 20 August 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	This formulation detailed description can be found in [[#openmdao\|[1]]].		This formulation and a detailed description can be found in [[#openmdao\|[1]]].

	== References ==		== References ==

RobertLampel at 14:58, 20 August 2025

2025-08-20T14:58:19Z

← Older revision		Revision as of 14:58, 20 August 2025
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	The '''Mountain Car problem''' proposes a vehicle stuck in a “well.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.		The '''Mountain Car problem''' proposes a vehicle stuck in a “well.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.

	The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [Moo90]. The implementation here is taken from [[]] and based on that given by Melnikov, Makmal, and Briegel [MMB14].		The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [[#Moo90 \| [2]]]. The implementation here is taken from [[#openmdao \| [1]]] and based on that given by Melnikov, Makmal, and Briegel [[#MMB14 \| [3]]].
	Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].		Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].

RobertLampel: /* References */

2025-08-20T14:57:07Z

References

← Older revision		Revision as of 14:57, 20 August 2025
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	== References ==		== References ==
	<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html<br>		<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html<br>
	<span id="Moo90">[1]</span> Andrew William Moore: Efficient memory-based learning for robot control<br>		<span id="Moo90">[2]</span> Andrew William Moore. Efficient memory-based learning for robot control. Technical Report UCAM-CL-TR-209, University of Cambridge, Computer Laboratory, November 1990. URL: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-209.pdf, doi:10.48456/tr-209.<br>
			<span id="MMB14">[3]</span> Alexey A Melnikov, Adi Makmal, and Hans J Briegel. Projective simulation applied to the grid-world and the mountain-car problem. arXiv preprint arXiv:1405.5459, 2014.<br>

	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Path-constrained arcs]]		[[Category:Path-constrained arcs]]

RobertLampel: /* References */

2025-08-20T14:55:47Z

References

← Older revision		Revision as of 14:55, 20 August 2025
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	== References ==		== References ==
	<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html<br>		<span id="openmdao">[1]</span> Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/mountain_car/mountain_car.html<br>
			<span id="Moo90">[1]</span> Andrew William Moore: Efficient memory-based learning for robot control<br>

	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:Path-constrained arcs]]		[[Category:Path-constrained arcs]]

RobertLampel at 14:53, 20 August 2025

2025-08-20T14:53:34Z

← Older revision		Revision as of 14:53, 20 August 2025
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	The '''Mountain Car problem''' proposes a vehicle stuck in a “well.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.		The '''Mountain Car problem''' proposes a vehicle stuck in a “well.” It lacks the power to directly climb out of the well, but instead must accelerate repeatedly forwards and backwards until it has achieved the energy necessary to exit the well.

	The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [Moo90]. The implementation here is taken from [] and based on that given by Melnikov, Makmal, and Briegel [MMB14].		The problem is a popular machine learning test case. It first appeared in the PhD thesis of Andrew Moore in 1990. [Moo90]. The implementation here is taken from [[]] and based on that given by Melnikov, Makmal, and Briegel [MMB14].
	Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].		Its dynamics are given by a two-dimensional [[:Category:ODE model\|ODE model]].