Robbins - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T10:41:44Z

Reference Solutions

← Older revision		Revision as of 10:41, 28 November 2025
Line 43:		Line 43:
	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="500px" heights="~~500px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Robbins.png\| States and discretized control for a local optimum.		Image:Robbins.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Reference Solutions */

2025-11-28T10:41:30Z

Reference Solutions

← Older revision		Revision as of 10:41, 28 November 2025
Line 43:		Line 43:
	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="500px" perrow="1">
	Image:Robbins.png\| States and discretized control for a local optimum.		Image:Robbins.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2025-11-24T15:10:46Z

Mathematical formulation

← Older revision		Revision as of 15:10, 24 November 2025
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	\displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) dt \\		\displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_1}(t) & = & x_2(t)),\\		\quad \dot{x_1}(t) & = & x_2(t),\\
	\quad \dot{x_2}(t) & = & x_3(t), \\		\quad \dot{x_2}(t) & = & x_3(t), \\
	\quad \dot{x_3}(t) & = & u(t), \\		\quad \dot{x_3}(t) & = & u(t), \\

RobertLampel: /* Mathematical formulation */

2025-11-24T12:08:33Z

Mathematical formulation

← Older revision		Revision as of 12:08, 24 November 2025
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) \\		\displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x_1}(t) & = & x_2(t)),\\		\quad \dot{x_1}(t) & = & x_2(t)),\\

RobertLampel: /* References */

2025-11-24T12:08:09Z

References

← Older revision		Revision as of 12:08, 24 November 2025
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	== References ==		== References ==
	<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/~~jackson~~.md<br>		<span id="OCPjl">[1]</span> Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/robbins.md<br>


	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:ODE model]]		[[Category:ODE model]]

RobertLampel: /* Reference Solutions */

2025-11-24T12:07:14Z

Reference Solutions

← Older revision		Revision as of 12:07, 24 November 2025
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	<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">		<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
	Image:~~Jackson~~.png\| States and discretized control for a local optimum.		Image:Robbins.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Mathematical formulation */

2025-11-24T12:03:55Z

Mathematical formulation

← Older revision		Revision as of 12:03, 24 November 2025
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	\quad \dot{x_2}(t) & = & x_3(t), \\		\quad \dot{x_2}(t) & = & x_3(t), \\
	\quad \dot{x_3}(t) & = & u(t), \\		\quad \dot{x_3}(t) & = & u(t), \\
	\quad x_1(t) & \geq & 0 \ \quad \forall t \in [0, T] \\		\quad x_1(t) & \geq & 0 \ \quad & \forall t \in [0, T], \\
	\quad x(0) & = & (1, -2, 0)^T, \\		\quad x(0) & = & (1, -2, 0)^T, \\
	\quad x(T) & = & (0, 0, 0)^T \\		\quad x(T) & = & (0, 0, 0)^T \\

RobertLampel: /* Parameters */

2025-11-24T12:03:07Z

Parameters

← Older revision		Revision as of 12:03, 24 November 2025
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	! Symbol !! Value !! Description		! Symbol !! Value !! Description
	\|-		\|-
	\| align=center \| <math>~~k_1~~</math> \|\| align=right \| 1 \|\| ~~Interaction between <math>x_1</math> and <math>x_2</math>~~		\| align=center \| <math>\alpha</math> \|\| align=right \| 3 \|\| Weight on state
	\|-		\|-
	\| align=center \| <math>~~k_2~~</math> \|\| align=right \| 10 \|\| ~~Interaction between <math>x_1</math> and <math>x_2</math>~~		\| align=center \| <math>\beta</math> \|\| align=right \| 0 \|\| Weight on squared state
	\|-		\|-
	\| align=center \| <math>~~k_3~~</math> \|\| align=right \| 1 \|\| ~~Growth of <math>x_3</math> under complementary~~ control		\| align=center \| <math>\gamma</math> \|\| align=right \| 0.5 \|\| Weight on squared control
	\|-		\|-
	\| align=center \| <math>~~t_\mathrm{f}~~</math> \|\| align=right \| 1 \|\| ~~Horizon of the control problem~~		\| align=center \| <math>T</math> \|\| align=right \| 10 \|\| Final time
	\|}		\|}

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''Robbins problem''' is a classical benchmark in optimal control. This description is taken from [1]. == Mathematical formulation ==
$\begin{array}{lll} \displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) \\ \text{subject to} \\ \quad \dot{x_1}(t) & = & x_2(t)),\\ \quad \dot{x_2}(t) & = & x_3(t), \\ \quad \dot{x_3}(t) & = & u(t),..."$

2025-11-24T12:01:45Z

Created page with "{{Dimensions |nd = 1 |nx = 3 |nw = 1 }} The '''Robbins problem''' is a classical benchmark in optimal control. This description is taken from [1]. == Mathematical formulation == <math> \begin{array}{lll} \displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) \\ \text{subject to} \\ \quad \dot{x_1}(t) & = & x_2(t)),\\ \quad \dot{x_2}(t) & = & x_3(t), \\ \quad \dot{x_3}(t) & = & u(t),..."

New page

{{Dimensions
|nd = 1
|nx = 3
|nw = 1
}}

The '''Robbins problem''' is a classical benchmark in optimal control. This description is taken from [[#OCPjl | [1]]].

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^T (\alpha \cdot x_1(t) + \beta \cdot x_1(t)^2 + \gamma \cdot u(t)^2) \\
\text{subject to} \\
\quad \dot{x_1}(t) & = & x_2(t)),\\
\quad \dot{x_2}(t) & = & x_3(t), \\
\quad \dot{x_3}(t) & = & u(t), \\
\quad x_1(t) & \geq & 0 \ \quad \forall t \in [0, T] \\
\quad x(0) & = & (1, -2, 0)^T, \\
\quad x(T) & = & (0, 0, 0)^T \\
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>k_1</math> || align=right | 1 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_2</math> || align=right | 10 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_3</math> || align=right | 1 || Growth of <math>x_3</math> under complementary control
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

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

== Miscellaneous and Further Reading ==
This formulation and a detailed description can be found in [[#OCPjl|[1]]].

== References ==
[1] Caillau, J.-B., Cots, O., Gergaud, J., & Martinon, P. OptimalControlProblems.jl: a collection of optimal control problems with ODE's in Julia. https://github.com/control-toolbox/OptimalControlProblems.jl/blob/main/ext/Descriptions/jackson.md 

[[Category:MIOCP]]
[[Category:ODE model]]