Bang-bang approximation of a traveling wave

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Revision as of 13:20, 16 August 2010 by FalkHante (Talk | contribs) (Added reference solution)

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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 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

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