ABSTRACT
R e m a r k 1 .1 .1 . Upon setting B i — 0, and replacing ( I + 7 ) w ith 7 for B 0 in (1.1.6), the a,bove model (1.1.1)-(1.1.4) becomes the abstract parabolic problem, whose quadratic cost optim al control problem has been recently resolved in full generality, see [B-D-DM], [DaP-I.l], [F.l], [L-T.1]-[L-T.5]. However, in the applications to boundary control problems for partia l differential equations which m otivate the present paper (see Section 4 ), the presence in the model (1 .1 .1)-(1.1.4) of the highly unbounded operator B \ which
satisfies (1.1.6) is an intrinsic property. Thus, this is a new feature of the present paper which introduces additional conceptual and technical difficulties over the theory for Bi = 0 of the aforementioned references. This is confirmed by the new features which appear in the resulting theory, see Section 1.4. Needless to say, in the applications which we have in mind, all of the above assumptions, (H .l) through (H.3), of the dynamics are autom atically satisfied; see a sample in Section 4, involving boundary control for p .d .e .’s which are second order in tim e and which have a high internal damping. □
R e m a r k 1 .1 .2 . Assum ption (H.3) will be used only in Section 3.6 (to show analyticity of the feedback semigroup on Z), and in Section 3.7 (to prove pointwise feedback synthesis of the optim al pair). □
We note th a t the semigroup eA\ originally defined on F , extends as a s.c., analytic semigroup, denoted by the same symbol eAt, in particular on the space
Z = [ V ( ( - A * r ) } ' (1.1.8)
(which will be the basic space for the solution zx (t) in (1.1.1)), the dual of A*)7) with respect to the pivot space F , endowed with the norm
\\x\\z = \\x\\[D((-A*)y)]' = \ \ A ~ ^ x \\y , x e Z, (1.1.9) where the fractional powers (—A*)0, O < 0 < l , o f ( —A*) are well defined by (H .l). It follows from (1.1.5) and (1.1.9) th a t H e ^ H ^ ) < M e ~ 8\ t > 0 , as well. Henceforth, x in (1.1.1) will be taken precisely in the space Z : x E Z. Regularity of the ¿-dynamics in (1.1.1) will be given in Section 1.3 below, along with additional comments.
