Bryson Denham - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:47:05Z

Reference Solutions

← Older revision		Revision as of 13:47, 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:Bryson-Denham.png\| States and discretized control for a local optimum.		Image:Bryson-Denham.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Miscellaneous and Further Reading */

2025-08-20T09:48:34Z

Miscellaneous and Further Reading

← Older revision		Revision as of 09:48, 20 August 2025
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	== Miscellaneous and Further Reading ==		== Miscellaneous and Further Reading ==
	The Bryson-Denham problem is a variation of the double integrator problem [[#BH75\|[1]]]. This formulation ~~detailed description~~ can be found in [[#openmdao\|[2]]].		The Bryson-Denham problem is a variation of the double integrator problem [[#BH75\|[1]]]. This formulation can be found in [[#openmdao\|[2]]].

	== References ==		== References ==

RobertLampel: /* Mathematical formulation */

2025-08-20T09:10:01Z

Mathematical formulation

← Older revision		Revision as of 09:10, 20 August 2025
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	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot u(t)^2 dt \\		\displaystyle \min_{w} && \int_0^{1} \frac{1}{2} \cdot w(t)^2 dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & v(t),\\		\quad \dot{x}(t) & = & v(t),\\
	\quad \dot{v}(t) & = & u(t), \\		\quad \dot{v}(t) & = & w(t), \\
	\quad x(0) &=& 0, \\		\quad x(0) &=& 0, \\
	\quad v(0) &=& 1, \\		\quad v(0) &=& 1, \\

RobertLampel: /* Mathematical formulation */

2025-08-20T09:09:38Z

Mathematical formulation

← Older revision		Revision as of 09:09, 20 August 2025
Line 13:		Line 13:
	<math>		<math>
	\begin{array}{lll}		\begin{array}{lll}
	\displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot w(t)^2 dt \\		\displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot u(t)^2 dt \\
	\text{subject to} \\		\text{subject to} \\
	\quad \dot{x}(t) & = & v(t),\\		\quad \dot{x}(t) & = & v(t),\\

RobertLampel: /* References */

2025-08-20T09:07:30Z

References

← Older revision		Revision as of 09:07, 20 August 2025
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	[[Category:MIOCP]]		[[Category:MIOCP]]
	[[Category:~~Sensitivity~~-~~seeking~~ arcs]]		[[Category:Path-constrained arcs]]

RobertLampel: /* Reference Solutions */

2025-08-20T09:06:11Z

Reference Solutions

← Older revision		Revision as of 09:06, 20 August 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:~~Rao_Mease~~.png\| States and discretized control for a local optimum.		Image:Bryson-Denham.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 }} The '''Bryson-Denham problem''' is a two-dimensional toy ODE model. It aims to minimize a quadratic Lagrange term. The optimal integer control functions exhibits a singular arc. == Mathematical formulation ==
$\begin{array}{lll} \displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot w(t)^2 dt \\ \text{subject to} \\ \quad \dot{x}(t)..."$

2025-08-20T09:02:22Z

Created page with "{{Dimensions |nd = 1 |nx = 2 |nw = 1 }} The '''Bryson-Denham problem''' is a two-dimensional toy ODE model. It aims to minimize a quadratic Lagrange term. The optimal integer control functions exhibits a singular arc. == Mathematical formulation == <math> \begin{array}{lll} \displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot w(t)^2 dt \\ \text{subject to} \\ \quad \dot{x}(t)..."

New page

{{Dimensions
|nd = 1
|nx = 2
|nw = 1
}}

The '''Bryson-Denham problem''' is a two-dimensional toy [[:Category:ODE model|ODE model]]. It aims to minimize a quadratic Lagrange term.

The optimal integer control functions exhibits a [[:Category:Sensitivity-seeking arcs|singular arc]].

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{u} && \int_0^{1} \frac{1}{2} \cdot w(t)^2 dt \\
\text{subject to} \\
\quad \dot{x}(t) & = & v(t),\\
\quad \dot{v}(t) & = & u(t), \\
\quad x(0) &=& 0, \\
\quad v(0) &=& 1, \\
\quad x(1) &=& 0, \\
\quad v(1) &=& -1, \\
\quad x(t) &\leq& \frac{1}{9}
\end{array}
</math>


== Reference Solutions ==

Here is one local solution to the above control problem.

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

== Miscellaneous and Further Reading ==
The Bryson-Denham problem is a variation of the double integrator problem [[#BH75|[1]]]. This formulation detailed description can be found in [[#openmdao|[2]]].

== References ==
[1] Arthur E Bryson and Yu-Chi Ho. Applied Optimal Control: Optimization, Estimation and Control. CRC Press, 1975. 
[2] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/bryson_denham/bryson_denham.html 

[[Category:MIOCP]]
[[Category:Sensitivity-seeking arcs]]