![](bg6.png)
It follows from Lemma 3.2 that the solution of (28) is
a lower bound n
1
(t).
Lemmas 3.3 and 3.4 complete the proof of The-
orem 3.1. One can add that if at some point in time
t = t
s
, the optimal trajectory will go left or right from
the value n
1
(t
s
) = n
∗
1
, then we can split the problem
into two pieces (namely from t = t
0
to t = t
s
and
from t = t
s
to t = t
f
) and build new upper or lower
bounds respectively from the initial point n
12
(t
s
) and
n
1
(t
s
) = n
∗
1
.
4 CONCLUSIONS
The analytical solution for the optimalperimeter feed-
back control with the maximum throughput criterion
in an urban region has been derived and described.
The modified Krotov-Bellman sufficient conditions of
optimality have been utilized for the proof of optimal-
ity. The resulting optimal control policy is oriented to
keep the state variable, i.e. the total number of the
moving vehicles in the region, as close as possible to
the critical accumulation, n
∗
1
, where the MFD value
is maximized. Though this optimal solution is in-
tuitively expected here it is rigorously proven. The
numerical simulations and comparison with existing
practices will be done in consequent papers.
ACKNOWLEDGEMENTS
This research was supported by Carl E. Schustak en-
ergy research and development fund.
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