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6.2.2 Infinite-horizon problem and its solution

We have now prepared the ground for taking the limit as and considering the infinite-horizon LQR problem with the cost

 (6.28)

Recalling the optimal cost (6.26) and the optimal control (6.24) for the finite-horizon case, and passing to the limit as , it is natural to guess that the infinite-horizon optimal cost and optimal control will be6.3

 (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 time-invariant, 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 infinite-horizon 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 finite-horizon 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 infinite-horizon optimal cost and is an optimal control, as claimed.

Next: 6.2.3 Closed-loop stability Up: 6.2 Infinite-horizon LQR problem Previous: 6.2.1 Existence and properties   Contents   Index
Daniel 2010-12-20