Fuller's problem

Fuller's problem
State dimension:	1
Differential states:	3
Discrete control functions:	1
Interior point equalities:	3

The first control problem with an optimal chattering solution was given by <bibref>Fuller1963</bibref>. An optimal trajectory does exist for all initial and terminal values in a vicinity of the origin. As Fuller showed, this optimal trajectory contains a bang-bang control function that switches infinitely often.

The mathematical equations form a small-scale ODE model. The interior point equality conditions fix initial and terminal values of the differential states.

Mathematical formulation

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

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

Parameters

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

Source Code

The differential equations in C code:

Miscellaneous and further reading

An extensive analytical investigation of this problem and a discussion of the ubiquity of Fuller's problem can be found in <bibref>Zelikin1994</bibref>, a recent investigation of chattering controls in relay feedback systems in <bibref>Johansson2002</bibref>.

References

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