mintOC - User contributions [en]

Bang-bang approximation of a traveling wave

2010-08-24T09:27:35Z

FalkHante:

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
and was introduced in <bibref>Hante2009</bibref>.

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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<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
characteristic equations.

== Reference solution ==

For <math>c=0.0075</math> the best known solution is given by

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

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

<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
Image:sinPGsws.png| Solution <math>q^*(t)</math> obtained using a projected gradient method based on switching time sensitivities.
Image:sinFinalTime.png| Corresponding plots of the differential states (blue) and the wave (red) at <math>t=1</math>.
</gallery>

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]

Bang-bang approximation of a traveling wave

2010-08-24T09:24:15Z

FalkHante: Added figures

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
and was introduced in <bibref>Hante2009</bibref>.

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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<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
characteristic equations.

== Reference solution ==

For <math>c=0.0075</math> the best known solution is given by

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

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

<gallery caption="Obtained solution plots" widths="240px" heights="167px" perrow="3">
Image:sinPGsws.png| Solution obtained using a projected gradient method based on switching time sensitivities.
Image:sinFinalTime.png| Corresponding final time plot of the differential states (blue) and the wave (red).
</gallery>

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]

File:SinFinalTime.png

2010-08-24T09:17:22Z

FalkHante:

File:SinPGsws.png

2010-08-24T09:16:54Z

FalkHante:

Bang-bang approximation of a traveling wave

2010-08-16T12:31:15Z

FalkHante: typo

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
and was introduced in <bibref>Hante2009</bibref>.

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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) 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
characteristic equations.

== Reference solution ==

For <math>c=0.0075</math> the best known solution is given by

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

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

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]

Bang-bang approximation of a traveling wave

2010-08-16T12:29:12Z

FalkHante: Added reference

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints
and was introduced in <bibref>Hante2009</bibref>.

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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) 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
characteristic equations.

== Reference Solution ==

For <math>c=0.0075</math> the best known solution is given by

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

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

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]

Bang-bang approximation of a traveling wave

2010-08-16T12:20:45Z

FalkHante: Added reference solution

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints.


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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) 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
characteristic equations.

== Reference Solution ==

For <math>c=0.0075</math> the best known solution is given by

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

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

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]

Bang-bang approximation of a traveling wave

2010-08-16T12:06:10Z

FalkHante: Created page with 'The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints. <!-- and was introduced in <bibref>Hante2009</bibref>....'

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints.


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.

== Mathematical formulation ==

<math>
\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}
</math>

where

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

is the traveling wave (oscillating between 0 and 1), <math>c>0</math> is a (small) regularization parameter and
<math>\bigvee_0^1 q(t)\,dt</math> denotes the variation of q(\cdot) 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
characteristic equations.

== References ==
<bibreferences/>


[[Category:MIOCP]]
[[Category:PDE model]]
[[Category:Transport]]