ABSTRACT
For the terms on the right of (2.3.55) (first and second are similar), we note that, by (1.2.2) and (1.3.18),
R i - A T i - A y - ^ e ^ i - A ^ l x + B lU°x(0)] e i - PC([0, oo]; Y), (2.3.56)
while by (1.3.16) and (2.3.2) already proved in part (i), du°
R B i - f e 1_/,C'([0,oo]; r ) , (2.3.57)
while, finally, by (2.3.2) and (2.3.27), du°
RL o ^ e 1_2pC([0,oo]; y ). (2.3.58)
Then, (2.3.56)-(2.3.58), used in (2.3.55), yield(2.3.4) as desired. The analysis is similar for obtaining (2.3.3). Theorem 2.3.1 is proved. □
R em ark 2.3.1. We note that under assumption (H.3), it is possible to prove that the (/-valued and Z-valued functions ux and zx(t) are analytic for t > 0. To this end, we use arguments akin to those used in the regular parbolic problems in [L-T.4] and [L-T.2, Chapter 2], by making use of the compactness assumption (H.3) = (1.1.7). However, in Section 3.6, we shall provide a different proof of this result. Indeed, we shall first prove that [zx(t) — B iux(t)] is a s.c. analytic semigroup on the natural space Z = [D((—A*)7)]' for t > 0, by an argument which again uses the compactness assumption (H.3) and which, this time, is focused, however, on its generator. Then, as a result, the property of analyticity is first transferred to the U-valued function ux(t) using the feedback expression (3.2.2) [or (3.3.23)], and finally to the Z-valued function m - □
In this section we obtain the pointwise feedback synthesis of the optimal pair {u°(t), through a “gain operator” which has, however, a more complicated form than the one which arises in the regular theory [DaP-I.l], [F.l], [L-T. 1]-[L-T.5]; see e.g., the Appendix for a finite-dimensional precursor rooted in the work of D. Lukes of 1967, as well as the recent infinite-dimensional version with bounded control operator [L-W.l], which became available to us after the main results of this paper had already been obtained. The gain operator is still expressed [albeit in a more complicated ex pression] in terms of an operator P which is the unique (within a class) self-adjoint solution of an algebraic Riccati equation (A.R.E.). In line with [L-T.1]-[L-T.5], we first define P in terms of the problem data, and subsequently we show that P satisfies an A.R.E. There are, however, two main technical and conceptual additional difficulties of the present problem for (1.1.1) over the regular parabolic case of, say, the references quoted above, (i) First, the “traditional” trick at the level of (3.1.1) (to extract a
pointwise expression fails now, since z® does not possess the semigroup property. This difficulty is resolved by Lemma 3.1.3 (and Theorem 3.1.2), which points out new in trinsic property of the problem, and which eventually will lead (in Theorem 3.3.1) to assert that it is [z®(t) — Biux(t)] that describes now a s.c. semigroup on Z. Second, the pointwise synthesis of the optimal pair presents a new obstacle in asserting a nonobvious bounded invertibility of an operator. This difficulty is resolved in Section 3.7, where for this purpose, we make use of the compactness assumption (H.3). Finally, the uniqueness of the self-adjoint solution of the A.R.E. also presents additional difficulties.
