<s>
This	O
article	O
describes	O
Lyapunov	B-Algorithm
optimization	I-Algorithm
for	O
dynamical	O
systems	O
.	O
</s>
<s>
Lyapunov	B-Algorithm
optimization	I-Algorithm
refers	O
to	O
the	O
use	O
of	O
a	O
Lyapunov	O
function	O
to	O
optimally	O
control	O
a	O
dynamical	O
system	O
.	O
</s>
<s>
Lyapunov	B-Algorithm
drift	I-Algorithm
is	O
central	O
to	O
the	O
study	O
of	O
optimal	O
control	O
in	O
queueing	O
networks	O
.	O
</s>
<s>
backpressure	B-Protocol
routing	I-Protocol
algorithm	O
for	O
network	O
stability	O
,	O
also	O
called	O
the	O
max-weight	O
algorithm	O
.	O
</s>
<s>
Adding	O
a	O
weighted	O
penalty	O
term	O
to	O
the	O
Lyapunov	B-Algorithm
drift	I-Algorithm
and	O
minimizing	O
the	O
sum	O
leads	O
to	O
the	O
drift-plus-penalty	B-Algorithm
algorithm	I-Algorithm
for	O
joint	O
network	O
stability	O
and	O
penalty	O
minimization	O
.	O
</s>
<s>
The	O
drift-plus-penalty	O
procedure	O
can	O
also	O
be	O
used	O
to	O
compute	O
solutions	O
to	O
convex	O
programs	O
and	O
linear	B-Algorithm
programs	I-Algorithm
.	O
</s>
<s>
Define	O
the	O
Lyapunov	B-Algorithm
drift	I-Algorithm
as	O
the	O
change	O
in	O
this	O
function	O
from	O
one	O
slot	O
to	O
the	O
next	O
:	O
</s>
<s>
where	O
and	O
are	O
arrivals	O
and	O
service	O
opportunities	O
,	O
respectively	O
,	O
in	O
queue	O
on	O
slot	O
This	O
equation	O
can	O
be	O
used	O
to	O
compute	O
a	O
bound	O
on	O
the	O
Lyapunov	B-Algorithm
drift	I-Algorithm
for	O
any	O
slot	O
t	O
:	O
</s>
<s>
1	O
)	O
leads	O
to	O
the	O
following	O
bound	O
on	O
the	O
conditional	O
expected	O
Lyapunov	B-Algorithm
drift	I-Algorithm
:	O
</s>
<s>
2	O
)	O
reduces	O
to	O
the	O
drift	O
condition	O
used	O
in	O
the	O
following	O
Lyapunov	B-Algorithm
drift	I-Algorithm
theorem	O
.	O
</s>
<s>
Theorem	O
(	O
Lyapunov	B-Algorithm
Drift	I-Algorithm
)	O
.	O
</s>
<s>
Suppose	O
there	O
are	O
constants	O
such	O
that	O
for	O
all	O
and	O
all	O
possible	O
vectors	O
the	O
conditional	O
Lyapunov	B-Algorithm
drift	I-Algorithm
satisfies	O
:	O
</s>
<s>
To	O
stabilize	O
the	O
network	O
while	O
minimizing	O
the	O
time	O
average	O
of	O
the	O
penalty	O
network	O
algorithms	O
can	O
be	O
designed	O
to	O
make	O
control	O
actions	O
that	O
greedily	O
minimize	O
a	O
bound	O
on	O
the	O
following	O
drift-plus-penalty	B-Algorithm
expression	I-Algorithm
on	O
each	O
slot	O
:	O
</s>
<s>
Choosing	O
reduces	O
to	O
minimizing	O
a	O
bound	O
on	O
the	O
drift	O
every	O
slot	O
and	O
,	O
for	O
routing	O
in	O
multi-hop	O
queueing	O
networks	O
,	O
reduces	O
to	O
the	O
backpressure	B-Protocol
routing	I-Protocol
algorithm	O
developed	O
by	O
Tassiulas	O
and	O
Ephremides	O
.	O
</s>
<s>
Using	O
and	O
defining	O
as	O
the	O
network	O
power	O
use	O
on	O
slot	O
leads	O
to	O
the	O
drift-plus-penalty	B-Algorithm
algorithm	I-Algorithm
for	O
minimizing	O
average	O
power	O
subject	O
to	O
network	O
stability	O
developed	O
by	O
Neely	O
.	O
</s>
<s>
Using	O
and	O
using	O
as	O
the	O
negative	O
of	O
an	O
admission	O
control	O
utility	O
metric	O
leads	O
to	O
the	O
drift-plus-penalty	B-Algorithm
algorithm	I-Algorithm
for	O
joint	O
flow	O
control	O
and	O
network	O
routing	O
developed	O
by	O
Neely	O
,	O
Modiano	O
,	O
and	O
Li	O
.	O
</s>
<s>
A	O
generalization	O
of	O
the	O
Lyapunov	B-Algorithm
drift	I-Algorithm
theorem	O
of	O
the	O
previous	O
section	O
is	O
important	O
in	O
this	O
context	O
.	O
</s>
<s>
Theorem	O
(	O
Lyapunov	B-Algorithm
Optimization	I-Algorithm
)	O
.	O
</s>
