Fuller's problem

Fuller's problem
State dimension:	1
Differential states:	2
Discrete control functions:	1
Interior point equalities:	4

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

$\begin{array}{ll}\underset{x,w}{\min }& \underset{0}{\overset{1}{\int }}{x}_0^2\mathrm{d}t\\ \text{s.t.}& {\dot{x}}_0(t)& =& x_1(t),\\ & {\dot{x}}_1(t)& =& 1-2w(t),\\ & x(0)& =& x_S,\\ & x(t_f)& =& x_T,\\ & w(t)& \in & \{0,1\}.\end{array}$

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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