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.
Annihilation of calcium oscillations with PLC activation inhibition
State dimension: 1
Differential states: 4
Discrete control functions: 2
Interior point equalities: 4

Contents

Mathematical formulation

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

\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}

Note that we write w1(t) instead of w(t) and have an additional control function w2(t). The regularization parameters are set to p1 = p2 = 100.

Reference Solutions

A solution for this problem is described in [1]. A local minimum that is actually slightly worse than the solution provided for only one control, is shown in the next plots.

Reference solution plots

Optimal stimuli determined by mixed-integer optimal control.

Corresponding differential states.

Variants

Only one control function.

References

  1. ↑ Lebiedz, D., Sager, S., Bock, H. G., & Lebiedz, P. (2005). Annihilation of limit cycle oscillations by identification of critical phase resetting stimuli via mixed-integer optimal control methods. Physical Review Letters, 95, 108303. Bib
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