View source for D'Onofrio model (binary variant)

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

<p>
<math>
\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} 
</math>
</p>

== Parameters ==
The parameters and scenarios are as in [[D'Onofrio_chemotherapy_model]], the new fixed parameters are

<math>(c_{0,1},c_{0,2},c_{0,3},c_{0,4})=(u_0^{max},u_0^{max},0,0), \qquad
(c_{1,1},c_{1,2},c_{1,3},c_{1,4})=(0,u_1^{max},u_1^{max},0).
</math>

== Reference Solutions ==

If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by  using a direct method such as collocation or Bock's direct multiple shooting method. 


The optimal objective value of scenario 2  of the relaxed problem with  <math> n_t=6000, \, n_u=100  </math> is <math>19.3561387</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>169.45773</math>.  The binary control solution was evaluated in the MIOCP by using a Merit function with additional Mayer term 
<math> 100 \max\limits_{t\in[0,1]}\{0,x_2(t)-x_2^{max}\}+1000 \max\limits_{t\in[0,1]}\{0,x_3(t)-x_3^{max}\}  </math>.


<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 Image:DonofrioRelaxed 6000 60 1.png| Optimal relaxed differential states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=100</math>.
 Image:DonofrioCIA 6000 60 1.png| Optimal binary differential states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>

== Source Code ==

Model description is available in
* [[:Category:AMPL | AMPL code]] at [[D'Onofrio model, binary variant (AMPL)]]




<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->
[[Category:MIOCP]]
[[Category:ODE model]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]



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