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 and was introduced in [1].

The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling wave such that the L2-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 χ[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

  1. ↑ Hante, F. M., & Leugering, G. (2009). Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions. Paper presented at HSCC '09: Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control, Berlin, Heidelberg. Bib
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