Optimal Control of Partial Differential Equations

46 CHAPTER 4. EVOLUTION EQUATIONS
Definition 4.4.1 Let (e tA )t≥0 be a strongly continuous semigroup on Z, with infinitesimal
generator A. The semigroup (e tA )t≥0 is analytic if there exists a sector
with 0 < δ < π,
such that Σa,
π
2 2 +δ ⊂ ρ(A), and
Σa, π
2 +δ = {λ ∈ C | |arg(λ − a)| < π
+ δ}
2
(λI − A) −1 ≤ M
|λ − a|
for all λ ∈ Σa, π
2 +δ.
It can be proved that the semigroup (e tA )t≥0 satisfies the conditions of definition 4.4.1 if and
only if (e tA )t≥0 can be extended to a function λ ↦→ e λA , where e λA ∈ L(Z), analytic in the
sector
Σa,δ = {λ ∈ C | |arg(λ − a)| < δ},
and strongly continuous in
{λ ∈ C | |arg(λ − a)| ≤ δ}.
Such a result can be find in a slightly different form in [2, Chapter 1, Theorem 2.1]. A theorem
very useful for studying regularity of solutions to evolution equations is stated below.
Theorem 4.4.1 ([18, Chapter 2, Theorem 6.13]) Let (etA )t≥0 be a continuous semigroup with
infinitesimal generator A. Suppose that (4.4.8) is satisfied for some c > 0. Then etAZ ⊂
D((−A) α ), (−A) αetA ∈ L(Z) for all t > 0, and, for all 0 ≤ α ≤ 1, there exists k > 0 and
C(α) such that
(−A) α e tA L(Z) ≤ C(α)t −α e −kt
for all t ≥ 0. (4.4.9)
A very simple criterion of analitycity is known in the case of real Hilbert spaces.
Theorem 4.4.2 ([2, Chapter 1, Proposition 2.11]) If A is a selfadjoint operator on a real
Hilbert space Z, and if
(Az, z) ≤ 0 for all z ∈ D(A),
then A generates an analytic semigroup of contractions on Z.