View source for Annihilation of calcium oscillations with PLC activation inhibition

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This control problem is closely related to [[Annihilation of calcium oscillations]]. The only difference is an additional control function, the inhibition of PLC activation. We state only the differences in this article.

== Mathematical formulation ==

For <math>t \in [t_0, t_f]</math> almost everywhere the mixed-integer optimal control problem is given by

<math>
\begin{array}{llcl}
 \displaystyle \min_{x, w, w^{\mathrm{max}}} & & & {\int_{t_0}^{t_f} || x(\tau) - \tilde{x} ||_2^2  + p_1 w_1(\tau) + p_2 w_2(\tau) \; \mathrm{d}\tau} \\[1.5ex]
 \mbox{s.t.} & \dot{x}_0 & = & k_1 + k_2 x_0 - \frac{k_3 x_0 x_1}{x_0 + K_4} -  \frac{k_5 x_0 x_2}{x_0 + K_6} \\
& \dot{x}_1 & = & (1 - w_2) k_7 x_0 - \frac{k_8 x_1}{x_1 + K_9} \\
& \dot{x}_2 & = & \frac{k_{10} x_1 x_2 x_3}{x_3 + K_{11}} + k_{12} x_1 + k_{13} x_0 - \frac{k_{14} x_2}{w_1 \cdot x_2 + K_{15}} - \frac{k_{16} x_2}{x_2 + K_{17}} + \frac{x_3}{10} \\
& \dot{x}_3 & = & - \frac{k_{10} x_1 x_2 x_3}{x_3 + K_{11}} + \frac{k_{16} x_2}{x_2 + K_{17}} - \frac{x_3}{10} \\[1.5ex]
 & x(0) &=& (0.03966, 1.09799, 0.00142, 1.65431)^T, \\
 & x(t) & \ge & 0.0, \\
 & w_1(t) &\in&  \{1, w^{\mathrm{max}}\}, \\
 & w_2(t) &\in&  \{0, 1\}, \\
 & w^{\mathrm{max}} & \ge & 1.1, \\
 & w^{\mathrm{max}} & \le & 1.3.
\end{array} 
</math>

Note that we write <math>w_1(t)</math> instead of <math>w(t)</math> and have an additional control function <math>w_2(t)</math>. The regularization parameters are set to <math>p_1 = p_2 = 100</math>. 

== Reference Solutions ==

A solution for this problem is described in <bib id="Lebiedz2005" />. A local minimum that is actually slightly worse than the solution provided for only one control, is shown in the next plots.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:calcium2Controls.png| Optimal stimuli determined by mixed-integer optimal control.
 Image:calcium2States.png| Corresponding differential states.
</gallery>

== Variants ==

[[Annihilation of calcium oscillations| Only one control function]].

== References ==
<biblist />

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