ABSTRACT
It is well-known tha t dynamic programming techniques allow to show that the value function v of deterministic optimal control problems satisfies a nonlinear first order P D E of the form
(HJ) F(x,viDv) = 0 in ft C It"
in the global viscosity sense (the first results in this direction can be found in [14]). By this we mean tha t the following conditions (i), (ii) holds for any (p E Cx(ft):
v is upper semicontinuos and
F(x, v(x)i D<p(x)) < 0 at any local maximum x of v — <p
t; is lower semicontinuos and
F(xi v(x), D<p(x)) > 0 at any local minimum x of v — ip]
of course, the validity of (i) and (ii) implies the continuity of v. The asymptotic problems tha t we are going to study in the next sections can be
casted in the following framework. Let e > 0 be a parameter describing some perturbation of a given optimal control
problem and v£ E C(ft) be the corresponding value function satisfying the HamiltonJacobi equation
(HJ) £ F£(z ,</ ,L>i;£) = 0 in ft
in the viscosity sense.
