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

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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.

Revision as of 14:47, 16 August 2010

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 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.


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].


References

<bibreferences/>

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