<s>
The	O
value	B-Algorithm
function	I-Algorithm
of	O
an	O
optimization	O
problem	O
gives	O
the	O
value	O
attained	O
by	O
the	O
objective	O
function	O
at	O
a	O
solution	O
,	O
while	O
only	O
depending	O
on	O
the	O
parameters	O
of	O
the	O
problem	O
.	O
</s>
<s>
In	O
a	O
controlled	O
dynamical	O
system	O
,	O
the	O
value	B-Algorithm
function	I-Algorithm
represents	O
the	O
optimal	O
payoff	O
of	O
the	O
system	O
over	O
the	O
interval	O
[	O
t	O
,	O
t1 ]	O
when	O
started	O
at	O
the	O
time-t	O
state	O
variable	O
x(t )	O
=	O
x	O
.	O
</s>
<s>
If	O
the	O
objective	O
function	O
represents	O
some	O
cost	O
that	O
is	O
to	O
be	O
minimized	O
,	O
the	O
value	B-Algorithm
function	I-Algorithm
can	O
be	O
interpreted	O
as	O
the	O
cost	O
to	O
finish	O
the	O
optimal	O
program	O
,	O
and	O
is	O
thus	O
referred	O
to	O
as	O
"	O
cost-to-go	O
function.	O
"	O
</s>
<s>
In	O
an	O
economic	O
context	O
,	O
where	O
the	O
objective	O
function	O
usually	O
represents	O
utility	O
,	O
the	O
value	B-Algorithm
function	I-Algorithm
is	O
conceptually	O
equivalent	O
to	O
the	O
indirect	O
utility	O
function	O
.	O
</s>
<s>
In	O
a	O
problem	O
of	O
optimal	O
control	O
,	O
the	O
value	B-Algorithm
function	I-Algorithm
is	O
defined	O
as	O
the	O
supremum	O
of	O
the	O
objective	O
function	O
taken	O
over	O
the	O
set	O
of	O
admissible	O
controls	O
.	O
</s>
<s>
If	O
the	O
value	B-Algorithm
function	I-Algorithm
happens	O
to	O
be	O
continuously	O
differentiable	O
,	O
this	O
gives	O
rise	O
to	O
an	O
important	O
partial	O
differential	O
equation	O
known	O
as	O
Hamilton	O
–	O
Jacobi	O
–	O
Bellman	O
equation	O
,	O
</s>
<s>
The	O
value	B-Algorithm
function	I-Algorithm
is	O
the	O
unique	O
viscosity	O
solution	O
to	O
the	O
Hamilton	O
–	O
Jacobi	O
–	O
Bellman	O
equation	O
.	O
</s>
<s>
In	O
an	O
online	B-Algorithm
closed-loop	O
approximate	O
optimal	O
control	O
,	O
the	O
value	B-Algorithm
function	I-Algorithm
is	O
also	O
a	O
Lyapunov	O
function	O
that	O
establishes	O
global	O
asymptotic	O
stability	O
of	O
the	O
closed-loop	O
system	O
.	O
</s>
