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6.2.2 Infinitehorizon problem and its solution
We have now prepared the ground for taking the limit as
and considering the infinitehorizon LQR problem with the cost

(6.28) 
Recalling the optimal cost (6.26)
and the optimal control (6.24)
for the finitehorizon case, and passing to the limit as
,
it is natural to guess that the infinitehorizon optimal cost and optimal control will be^{6.3}

(6.29) 
and

(6.30) 
where
is the matrix limit (6.27) which satisfies the ARE (6.28). Note that the quadratic cost (6.30) is independent of
and the linear feedback law (6.31) is timeinvariant, which is consistent with the problem formulation and with our earlier findings in Section 5.1.3. Still, optimality of (6.31) is far from obvious, and it is not even clear whether a control yielding a bounded cost exists. Strictly speaking, the use of the asterisks in (6.31) is not yet justified; at this point,
is simply the trajectory of the system under the action of the feedback law (6.31) with
.
We now show that the above guess is indeed correct. Consider the function
. Its derivative along
the trajectory
is
where the last equality follows from the ARE (6.28).
We can then calculate the portion of the corresponding cost over an arbitrary finite interval
to be
where the last inequality follows from the fact that
. Taking the limit
as
, we obtain

(6.31) 
In particular, we can now be sure that the infinitehorizon problem is well posed, because
gives a bounded cost.
On the other hand, consider another trajectory
with the same initial condition corresponding to an arbitrary control
. Since
is the finitehorizon optimal cost, we have for every finite
that
where the second inequality relies on the positive (semi)definiteness of
and
. Passing to the limit as
yields
Comparing this inequality with (6.32) and remembering that
was arbitrary (and could in particular be equal to
), we see that
hence
is the infinitehorizon optimal cost and
is an optimal control, as claimed.
Next: 6.2.3 Closedloop stability
Up: 6.2 Infinitehorizon LQR problem
Previous: 6.2.1 Existence and properties
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Daniel
20101220