246
the we have
II y*(T,· : u~) - yd(.) IIc(IT\w)s; E.
Moreover such a control u~ can be found as a fixed point of the application A : C([O, T] x
(IT - w)) -+ P(C([O, T] x (IT - w)) defined by
A(z) = {Y(',' : u) solution of (PL) with a = fN(z) and u satisfying (53), (54)}.
z
From estimate (56) of Lemma 1 we deduce that if u~ is a fixed point of A it must satisfy
II u~ IIc([o,T]xoDG) S; C
with C (independent of N) given in (56). Then, by the maximum priciple we conclude
that if N 2' : No is large enough then the function y*(t, x : u~) satisfies
ly*(t, x : u~)l S; N If (t, x) E [0, T] x (IT - w)
and so, in fact, y*(O,' : u~) satisfies the requirements of the statement of Theorem 10 .
•
In order to complete the proof of Theorem 7 we need to use some other auxiliary
results.
Lemma 5 (Diaz and Fursikov [1994])
Let U e E C([O, T] x ow) fixed. There exists ve E C([O, T] x w) such that the solution
y(t : Vel of
satisfies
1
~t - fly + >'IW-1y = Ve
y = Ue
Y(O,·) = Yo(')
in (0, T) x w
on (0, T) x ow
on w
II y(T : Vel - Yd Ilc(W)S; E.
We would need to regularize the matching between the functions iJ and y given in
Theorem 10 and Lemma 4 respectively.
Lemma 6
Let We be an open regular subset of w such that d( We, ow) S; E. Then there exists
y* E C([O, T] x IT) n C 2 ((0, T) x n) such that y* = y on [0, T] x We and y* = i J on
[0, T] x (IT - w).
The proof of this result uses standard regularization techniques and the details are
left to the reader. The last technical result is consequence of the continuous dependence
of the solutions of problem (Pp) with respect to different initial data.