Difference between revisions of "Bang-bang approximation of a traveling wave"

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The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints.
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<!-- and was introduced in <bibref>Hante2009</bibref>. -->
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|nd        = 2
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|nx        = 1
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|nw        = 1
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|nc        = 2
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}}
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The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
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and was introduced in <bib id="Hante2009" />.  
  
 
The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling  
 
The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling  
 
wave such that the <math>L^2</math>-distance between the traveling wave and the resulting flow is minimized.
 
wave such that the <math>L^2</math>-distance between the traveling wave and the resulting flow is minimized.
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== Mathematical formulation ==
 
== Mathematical formulation ==
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is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and  
 
is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and  
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<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) over the interval <math>[0,1]</math>.  
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<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of <math>q(\cdot)</math> over the interval <math>[0,1]</math>.  
 
Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the  
 
Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the  
 
characteristic equations.
 
characteristic equations.
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Systems biology
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== Reference solution ==
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For <math>c=0.0075</math> the best known solution is given by
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<math>
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q^*(t)=\chi_{[0,0.2]}(t)+\chi_{[0.4,0.6]}(t)+\chi_{[0.8,1]}(t),\quad 0\leq t\leq 1
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</math>
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where <math>\chi_{[a,b]}(t)</math> denotes the indicator function of the interval <math>[a,b]</math>.
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<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
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Image:sinPGsws.png| Solution <math>q^*(t)</math> obtained using a projected gradient method based on switching time sensitivities.
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Image:sinFinalTime.png| Corresponding plots of the differential states (blue) and the wave (red) at <math>t=1</math>.
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</gallery>
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== References ==
 
== References ==
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<bibreferences/>
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<biblist />
  
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
 
<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
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[[Category:PDE model]]
 
[[Category:PDE model]]
 
[[Category:Transport]]
 
[[Category:Transport]]
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[[Category:Tracking objective]]

Latest revision as of 16:02, 27 January 2016

Bang-bang approximation of a traveling wave
State dimension: 2
Differential states: 1
Discrete control functions: 1
Path constraints: 2


The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints and was introduced in [Hante2009]Address: Berlin, Heidelberg
Author: Hante, Falk M.; Leugering, G\"unter
Booktitle: HSCC '09: Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control
Pages: 209--222
Publisher: Springer-Verlag
Title: Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions
Year: 2009
Link to Google Scholar.

The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling wave such that the L^2-distance between the traveling wave and the resulting flow is minimized.


Mathematical formulation

\begin{array}{ll} \displaystyle \min_{x, q} & \displaystyle \int_0^1\int_0^1 |x(t,s)-x_d(t,s)|^2\,ds\,dt + c \bigvee_0^1 q(t)\,dt \\[1.5ex] \mbox{s.t.} & \displaystyle \frac{\partial}{\partial t}x(t,s)+\frac{\partial}{\partial s}x(t,s) = 0,\quad 0<s<1,~0<t<1\\[1.5ex] & \displaystyle x(t,0) = q(t),\quad 0<t<1 \\ & \displaystyle x(0,s) = x_d(0,s),\quad 0<s<1 \\ & \displaystyle q(t) \in \{0,1\},\quad 0<t<1 \\ \end{array}

where

x_d(t,s)=\frac12\sin(5\pi(t-s))+1,\quad 0\leq t \leq 1,~0\leq s\leq 1

is the traveling wave (oscillating between 0 and 1), c>0 is a (small) regularization parameter and \bigvee_0^1 q(t)\,dt denotes the variation of q(\cdot) over the interval [0,1]. Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the characteristic equations. Systems biology

Reference solution

For c=0.0075 the best known solution is given by

q^*(t)=\chi_{[0,0.2]}(t)+\chi_{[0.4,0.6]}(t)+\chi_{[0.8,1]}(t),\quad 0\leq t\leq 1

where \chi_{[a,b]}(t) denotes the indicator function of the interval [a,b].

  • Obtained solution plots

  • Solution q^*(t) obtained using a projected gradient method based on switching time sensitivities.

  • Corresponding plots of the differential states (blue) and the wave (red) at t=1.


References

There were no citations found in the article.

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