ABSTRACT
L em m a 3 .2 Let operator B in (1.1) be invertible with B ~ l G C(H). Let ( s , X ) G [0,T] x E and be an admissible trajectory-control pair at (5, X ) . Then, for any ( s , X ) G [5 , T] x E and any r > 0, there exists a trajectory-control pair {Yr)ur } which is admissible at (5, X ) and such that, for all t > s ,
u:(s) = - B * V x w e(s i Y ; (s ) ) (2 .11)
u n « - n o n * < [ii* - n s ) i i w + ƒ (3.1)
and
/ T / v rTA (s+r)( \ur(t)\2 — |w(£)|2) dt < C j \u(t)\2dt + C [ i | x 0 - ÿ 0(s ) |2 + I | | X - F ( s ) | |^
Lr° t
(3.2)
where C = C (||B ||, | |5 l \\,T). Moreover, if s + r < T, then
Yr(t) = Y (t ) ,u T(t) = u(t) Vi G [s + r , T ]
Since B is assumed to be invertible, without loss of generality we may take B = I here and in the sequel. Moreover, we drop subscripts in all norms occurring in proofs. P ro o f - Fix (s,X) G [s,T] x E , r > 0, and define, for any t G [s,T],
Yr(t) = Ar(t)Y(t) + [1 - Ar(i)]Y(i), (3.3) where Y (t) = e^~^AX and Ar : R —*· R is given by
Ar(i) = 0 t < s
3 ( Î71) 2 - 2 ( t 2) 3 s < t < s + r 1 s + r < t
We note that 0 < Ar < 1, A'(s) = 0 = A'T(s + r ), and
K l < “ , Wl < (3-4)
Since y ( i ) , y ( i ) G E, by convexity we have that Yr {t) G E. Moreover, it is easy to see that Yr is a trajectory of (1.11). In fact, denoting by yr the first component of Yr) by construction we have that
?/''(*) 4-Ayr {t) = ur (t), s < t < T yr (s) = x 0 l/r(s) = x i
where Ur (t) = Ar(t)u(t) + 2\'r (t)[tf(t) - y'(t)\ + K ( t ) [y ( t ) - y(t)]. (3.5)
So, yr is a trajectory of (1.1). We now proceed to show (3.1). Since Yr(t) — Y ( t ) = ( l — Xr( t ) ) ( Y (t ) — Y (¿)), it suffices
to consider the difference Y ( t ) — Y ( t ) . Passing to the first component, we have
(:V ~ V)" + A ( y - y ) = u , t e [5, T] ( y - y ) ( s ) = x o - y ( s ) (:y - y) '(s) = x i - y'(s)
Taking the scalar product of the above equation with (y — y ) f and integrating over [s, t] we obtain
\ [|(v - y)'(t)I2 + \VA(y - y)(t)\2] = + \ ' / A ( x o - y { s ) ) \ 2} - < % { y - y ) ' > da (3.6)
< ^ [l*i - y'(s)!2 + \VA{xo - y(s))|2 + ƒ \u\2da + ƒ |(y - y)'\2da
Hence, by GronwalPs Lemma we obtain (3.1).
