Abstracts - Facultatea de Matematică

On a wrong solution to a trivial optimal control problem in
mathematical economics
S¸tefan MIRICĂ and Touffik BOUREMANI
Faculty of Mathematics, University of Bucharest
Academiei 14, 010014 Bucharest, ROMANIA
bouremani@yahoo.com, mirica@fmi.unibuc.ro
This work is intended, in the first place, as a warning to the increasing number of
authors that try to solve concrete optimal control problems without enough knowledge
and even basic mathematical abilities; secondly, our aim is to show that the Dynamic
Programming approach in [2] is much more efficient than the PMP approach, in the
study of this type of problems.
A tipical example is the one in the recent paper, [1], in which the authors believe
that they solved the problem of minimizing the cost functional
subject to:
� T
J(P (.), I(.)) := e −ρt [h(I(t)) + K(P (t))]dt
0
I ′ (t) = P (t) − D(t, I(t)), P (t) ≥ D(t, I(t)) > 0,
a.e. ([0, T ], I(0) = I0, I(T ) = IT that arise from some (not very convincing) “model in
mathematical economics”.
Applying a non-existent (in [3]) variant of Pontryagin’s Maximum Principle (PMP)
as a “necessary optimality condition”, the authors of [1] conclude in their Th.1 that
in the particular case in which: K(p) := 1
2 kp2 , h(I) := 1
2 hI2 , D(t, I) := d1(t) +
d2I, k, h, d2, d1(t) > 0, the “optimal trajectory” of the problem above is of the form:
I ∗ (t) := a1e m1t + a2e m2t + Q(t).
On the other hand, using the Dynamic Programming approach in [2] (adapted to
non-autonomous problems) we prove that the only optimal trajectories are the constant
functions, � I(t) ≡ I0, hence the solution in [1] is wrong and the problem above is rather
“trivial”. Moreover, we also identify the main scientific errors made by the authors
of [1], among which, the first one is the fact that they apply the “classical form” of
PMP to a problem defined by a differential inclusion since in this case the set of control
parameters is variable (depending on the time and the state).
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