Bang-bang approximation of a traveling wave

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

$\begin{array}{ll}\underset{x,q}{\min }& \underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}|x(t,s)-x_d(t,s){|}^{2}dsdt+c\underset{0}{\overset{1}{\bigvee }}q(t)dt\\ \text{s.t.}& \frac{\partial }{\partial t}x(t,s)+\frac{\partial }{\partial s}x(t,s)=0,0<s<1,0<t<1\\ & x(t,0)=q(t),0<t<1\\ & x(0,s)=x_d(0,s),0<s<1\\ & q(t)\in \{0,1\},0<t<1\\ \end{array}$

where

$x_d(t,s)=\frac{1}{2}\sin (5\pi (t-s))+1,0\le t\le 1,0\le s\le 1$

is the traveling wave (oscillating between 0 and 1), $c>0$ is a (small) regularization parameter and $\underset{0}{\overset{1}{\bigvee }}q(t)dt$ 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

$q^{*}(t)={\chi }_{[0,0.2]}(t)+{\chi }_{[0.4,0.6]}(t)+{\chi }_{[0.8,1]}(t),0\le t\le 1$

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

References

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