Difference between revisions of "Fuller's initial value problem"

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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="4">
 
<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: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>.
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  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.
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  Image:FullerRelaxed 6000 40 2.png| Optimal relaxed controls.
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Image:FullerCIA 6000 40 1.png| Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=150</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
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Image:FullerCIA 6000 40 2.png| Optimal binary controls.
 
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Revision as of 17:28, 8 January 2018

Fuller's initial value problem
State dimension: 1
Differential states: 2
Discrete control functions: 1
Interior point equalities: 2

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

\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}


Parameters

We use x_S = x_T = (0.01, 0)^T.


Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, the optimal solution can be determined by means of direct optimal control.

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

  • Reference solution plots

  • Optimal relaxed states determined by an direct approach with ampl_mintoc (Radau collocation) and n_t=6000, \, n_u=150.

  • Optimal relaxed controls.

  • Optimal differential states trajectories of binary controls determined by an direct approach (Radau collocation) with ampl_mintoc and n_t=6000, \, n_u=150. The relaxed controls were approximated by Combinatorial Integral Approximation.

  • Optimal binary controls.

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