<s>
In	O
automata	B-Application
theory	I-Application
,	O
a	O
hybrid	B-Application
automaton	I-Application
(	O
plural	O
:	O
hybrid	B-Application
automata	I-Application
or	O
hybrid	B-Application
automatons	I-Application
)	O
is	O
a	O
mathematical	O
model	O
for	O
precisely	O
describing	O
hybrid	O
systems	O
,	O
for	O
instance	O
systems	O
in	O
which	O
digital	O
computational	O
processes	O
interact	O
with	O
analog	O
physical	O
processes	O
.	O
</s>
<s>
A	O
hybrid	B-Application
automaton	I-Application
is	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
with	O
a	O
finite	O
set	O
of	O
continuous	O
variables	O
whose	O
values	O
are	O
described	O
by	O
a	O
set	O
of	O
ordinary	O
differential	O
equations	O
.	O
</s>
<s>
In	O
general	O
,	O
hybrid	B-Application
automata	I-Application
have	O
been	O
used	O
to	O
model	O
and	O
analyze	O
a	O
variety	O
of	O
embedded	B-Architecture
systems	I-Architecture
including	O
vehicle	O
control	O
systems	O
,	O
air	B-Application
traffic	I-Application
control	I-Application
systems	I-Application
,	O
mobile	O
robots	O
,	O
and	O
processes	O
from	O
systems	O
biology	O
.	O
</s>
<s>
An	O
Alur	O
–	O
Henzinger	O
hybrid	B-Application
automaton	I-Application
comprises	O
the	O
following	O
components	O
:	O
</s>
<s>
Hybrid	B-Application
automata	I-Application
come	O
in	O
several	O
flavors	O
:	O
The	O
Alur	O
–	O
Henzinger	O
hybrid	B-Application
automaton	I-Application
is	O
a	O
popular	O
model	O
;	O
it	O
was	O
developed	O
primarily	O
for	O
algorithmic	O
analysis	O
of	O
hybrid	O
systems	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
The	O
model	B-Application
checking	I-Application
tool	O
is	O
based	O
on	O
this	O
model	O
.	O
</s>
<s>
Another	O
formalism	O
,	O
which	O
is	O
useful	O
to	O
model	O
implementations	O
of	O
hybrid	B-Application
automaton	I-Application
,	O
is	O
the	O
lazy	B-Application
linear	I-Application
hybrid	I-Application
automaton	I-Application
.	O
</s>
<s>
Given	O
the	O
expressiveness	O
of	O
hybrid	B-Application
automata	I-Application
it	O
is	O
not	O
surprising	O
that	O
simple	O
reachability	O
questions	O
are	O
undecidable	O
for	O
general	O
hybrid	B-Application
automata	I-Application
.	O
</s>
<s>
In	O
fact	O
,	O
a	O
straightforward	O
reduction	O
from	O
counter	B-Application
machines	I-Application
to	O
three	O
variables	O
hybrid	B-Application
automata	I-Application
(	O
two	O
variables	O
for	O
storing	O
counter	O
values	O
and	O
one	O
to	O
restrict	O
spending	O
a	O
unit-time	O
per	O
location	O
)	O
proves	O
the	O
undecidability	O
of	O
the	O
reachability	O
problem	O
for	O
hybrid	B-Application
automata	I-Application
.	O
</s>
<s>
A	O
sub-class	O
of	O
hybrid	B-Application
automata	I-Application
are	O
timed	B-Application
automata	I-Application
where	O
all	O
of	O
the	O
variables	O
grow	O
with	O
uniform	O
rate	O
(	O
i.e.	O
,	O
all	O
continuous	O
variables	O
have	O
derivative	O
1	O
)	O
.	O
</s>
<s>
Other	O
notable	O
decidable	O
subclasses	O
include	O
initialized	O
rectangular	O
hybrid	B-Application
automata	I-Application
,	O
one-dimensional	O
piecewise-constant	O
derivatives	O
(	O
PCD	O
)	O
systems	O
,	O
priced	O
timed	B-Application
automata	I-Application
,	O
and	O
constant-rate	O
multi-mode	O
systems	O
.	O
</s>
