Difference between revisions of "D'Onofrio model (binary variant)"
ClemensZeile (Talk | contribs) |
ClemensZeile (Talk | contribs) |
||
| Line 17: | Line 17: | ||
\displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex] | \displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex] | ||
\mbox{s.t.} & \dot{x}_0 & = & - \zeta x_0 \text{ln} \left( \frac{x_0}{x_1} \right) - \sum\limits_{i=1}^{4} w_i\;c_{1,i}\; F \; x_0 , \\ | \mbox{s.t.} & \dot{x}_0 & = & - \zeta x_0 \text{ln} \left( \frac{x_0}{x_1} \right) - \sum\limits_{i=1}^{4} w_i\;c_{1,i}\; F \; x_0 , \\ | ||
| − | & \dot{x}_1 & = & b x_0 - \mu x_1 - d x_0^{\frac{2}{3}}x_1 -\sum\limits_{i=1}^{4} w_i c_{0,i} G x_1 - \sum\limits_{i=1}^{4} w_i\;c_{1,i} \eta x_1, \\ | + | & \dot{x}_1 & = & b x_0 - \mu x_1 - d x_0^{\frac{2}{3}}x_1 -\sum\limits_{i=1}^{4} w_i c_{0,i} \; G x_1 - \sum\limits_{i=1}^{4} w_i\;c_{1,i} \; \eta x_1, \\ |
& \dot{x}_2 & = & \sum\limits_{i=1}^{4} w_i\;c_{0,i}, \\ | & \dot{x}_2 & = & \sum\limits_{i=1}^{4} w_i\;c_{0,i}, \\ | ||
& \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex] | & \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex] | ||
& x_2 & \leq & x_2^{max}, \\ | & x_2 & \leq & x_2^{max}, \\ | ||
& x_3 & \leq & x_3^{max},\\ | & x_3 & \leq & x_3^{max},\\ | ||
| − | & 1 &=& \sum\limits_{i=1}^{ | + | & 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\ |
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4. | & w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4. | ||
\end{array} | \end{array} | ||
| Line 30: | Line 30: | ||
== Parameters == | == Parameters == | ||
| − | + | The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are | |
| − | <math> | + | <math>(c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0) \\ |
| + | (c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_0^{max},u_0^{max},0). | ||
| + | </math> | ||
== Reference Solutions == | == Reference Solutions == | ||
Revision as of 15:24, 11 January 2018
| D'Onofrio model (binary variant) | |
|---|---|
| State dimension: | 1 |
| Differential states: | 4 |
| Discrete control functions: | 4 |
| Path constraints: | 2 |
This site describes a D'Onofrio model variant with four binary controls instead which of only two continuous controls. The continuous controls are replaced via the outer convexifacation method.
Mathematical formulation
For ![t \in [t_0, t_f]](https://mintoc.de/images/math/5/5/8/55823791d9100bcb5461801aff4f6edd.png)
the optimal control problem is given by
![\begin{array}{llcl}
\displaystyle \min_{x, u} & x_0(t_f) &+& \alpha \int_{t_0}^{t_f} u_0(t)^2 \text{d}t \\[1.5ex]
\mbox{s.t.} & \dot{x}_0 & = & - \zeta x_0 \text{ln} \left( \frac{x_0}{x_1} \right) - \sum\limits_{i=1}^{4} w_i\;c_{1,i}\; F \; x_0 , \\
& \dot{x}_1 & = & b x_0 - \mu x_1 - d x_0^{\frac{2}{3}}x_1 -\sum\limits_{i=1}^{4} w_i c_{0,i} \; G x_1 - \sum\limits_{i=1}^{4} w_i\;c_{1,i} \; \eta x_1, \\
& \dot{x}_2 & = & \sum\limits_{i=1}^{4} w_i\;c_{0,i}, \\
& \dot{x}_3 & = & \sum\limits_{i=1}^{4} w_i\;c_{1,i}, \\ [1.5ex]
& x_2 & \leq & x_2^{max}, \\
& x_3 & \leq & x_3^{max},\\
& 1 &=& \sum\limits_{i=1}^{4}w_i(t), \\
& w_i(t) &\in& \{0, 1\}, \quad i=1\ldots 4.
\end{array}](https://mintoc.de/images/math/7/8/6/786dadd722eb90f8af8959226181d037.png)
Parameters
The parameters and scenarios are as in D'Onofrio_chemotherapy_model, the new fixed parameters are
Failed to parse (syntax error): (c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0) \\ (c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_0^{max},u_0^{max},0).
Reference Solutions
If the problem is relaxed, i.e., we demand that 
be in the continuous interval ![[0, 1]](https://mintoc.de/images/math/c/c/f/ccfcd347d0bf65dc77afe01a3306a96b.png)
instead of the binary choice 
, the optimal solution can be determined by means of direct optimal control.
The optimal objective value of the relaxed problem with 
is 
. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is 
.
- Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and
.
Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and
. The relaxed controls were approximated by Combinatorial Integral Approximation.
Source Code
Model description is available in
