View source for Fuller's initial value problem

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-->This site describes a Fuller's problem variant with no terminal constraints and additional Mayer term for penalizing deviation from given reference values. 

== Mathematical formulation ==

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

<p>
<math>
\begin{array}{llcl}
 \displaystyle \min_{x, w} & \int_{0}^{1} x_0^2 & \; \mathrm{d} t + (x(t_f)-x_T)^2 \\[1.5ex]
 \mbox{s.t.} & \dot{x}_0 & = & x_1, \\
 & \dot{x}_1 & = & 1 - 2 \; w, \\[1.5ex]
 & x(0) &=& x_S, \\
 & w(t) &\in&  \{0, 1\}.
\end{array} 
</math>
</p>


== Parameters ==

We use <math>x_S = x_T = (0.01, 0)^T</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 means of direct optimal control. 

The optimal objective value of the relaxed problem with  <math> n_t=6000, \, n_u=150  </math> is <math>1.45412214e-05</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.40273813e-05</math>.  

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
 Image:FullerRelaxed 6000 40 1.png| Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=150</math>.
 Image:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
</gallery>








<!--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:Tracking objective]]
[[Category:Chattering]]
[[Category:Sensitivity-seeking arcs]]
[[Category:Population dynamics]]


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