Accessibility

in control theory

A state of a control system has the accessibility property if the positive orbit of this state has non-empty interior in the phase space of the system, and it has the strong accessibility property if for some $T>0$ the set of states being attainable from it at time $T$ has non-empty interior. For example, for the control system defined in the $(x,y)$-plane by the two fields of admissible velocities $(1,0)$, $(1,f(x,y))$, where $f(x,y)=0$ when $x\geq0$ and $f(x,y)=\exp(1/x)$ when $x<0$, all points of the set $x<0$ have the (strong) accessibility property but no point of the set $x\geq0$ has it.

The control system itself has the (strong) accessibility property if all of its states have this property. The accessibility property is typical for control systems. Namely, every control system defined on a smooth manifold by a pair of smooth admissible vector fields has the strong accessibility property if this pair belongs to some open everywhere-dense subset of the space of pairs in an appropriate topology (for example, in the fine $C^\infty$-topology).

Classical references for the notion of accessibility are [a1], [a2], [a3], [a4].

#### References[edit]

[a1] |  C. Lobry, "Dynamical polysystems and control theory" D.Q. Mayne (ed.) R.W. Brockett (ed.) , Geometric Methods in System Theory. Proc. NATO Advanced Study Institute, London, August 27–September 7, 1973 , D. Reidel (1973) pp. 1–42

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[a2] |  V. Jurjevic, "Certain controllability properties of analytic control systems" SIAM J. Control , 10 : 2 (1972) pp. 354–360
[a3] |  H.J. Sussmann, V. Jurjevic, "Controllability of nonlinear systems" J. Diff. Eq. , 12 (1972) pp. 95–116
[a4] |  H. Hermes, "On local and global controllability" SIAM J. Control , 12 : 2 (1974) pp. 252–261