Annihilation of calcium oscillations with PLC activation inhibition

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

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

$\begin{array}{llcl}\underset{x,w,w^{\mathrm{m}\mathrm{a}\mathrm{x}}}{\min }& & & \underset{t_0}{\overset{t_f}{\int }}||x(\tau )-\tilde{x}|{|}_2^2+p_1{w}_1(\tau )+p_2{w}_2(\tau )\mathrm{d}\tau \\ \text{s.t.}& {\dot{x}}_0& =& k_1+k_2{x}_0-\frac{k_3{x}_0{x}_1}{x_0+K_4}-\frac{k_5{x}_0{x}_2}{x_0+K_6}\\ & {\dot{x}}_1& =& (1-w_2)k_7{x}_0-\frac{k_8{x}_1}{x_1+K_9}\\ & {\dot{x}}_2& =& \frac{k_{10}{x}_1{x}_2{x}_3}{x_3+K_{11}}+k_{12}{x}_1+k_{13}{x}_0-\frac{k_{14}{x}_2}{w_1\cdot x_2+K_{15}}-\frac{k_{16}{x}_2}{x_2+K_{17}}+\frac{x_3}{10}\\ & {\dot{x}}_3& =& -\frac{k_{10}{x}_1{x}_2{x}_3}{x_3+K_{11}}+\frac{k_{16}{x}_2}{x_2+K_{17}}-\frac{x_3}{10}\\ & x(0)& =& (0.03966,1.09799,0.00142,1.65431{)}^T,\\ & x(t)& \ge & 0.0,\\ & w(t)& \in & \{0,w^{\mathrm{m}\mathrm{a}\mathrm{x}}\},\\ & w^{\mathrm{m}\mathrm{a}\mathrm{x}}& \ge & 1.1,\\ & w^{\mathrm{m}\mathrm{a}\mathrm{x}}& \le & 1.3.\end{array}$

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

Reference Solutions

A solution for this problem is described in <bibref>Lebiedz2005</bibref>.

Variants

Only one control function.

References

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