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C0-semigroupIn mathematics, a C0-semigroup is a continuous morphism from (R+,+) into a topological monoid, usually L(H), the algebra of linear continuous operators on some Hilbert space H. Thus, strictly speaking, not the C0-semigroup, but rather its image, is a semigroup. Example C0-semigroups occur for example in the context of initial value problems, -
where x and f take values in a Hilbert space H. If the solution of (CP) is unique (depending on f) for x0 in some given domain D ⊂ H, one has the "solution operator" defined by - , where x(t) is solution of (CP).
Thus one can view Γ as an "evolution operator", and it is clear that one should have - Γ(s+t)=Γ(s) Γ(t)
on the domain D. This is just the condition of a semigroup-morphism. Then one can study the conditions under which Γ is continuous for the topology on L(H) induced by the norm on H, which amounts to check that -
Formal definition All that follows concerns the following definition: A (strongly continuous) C0-semigroup on a Hilbert space H is a map - Γ : R+ → L(H)
such that - Γ(0) = I := idH , (identity operator on H)
- ∀ t,s ≥ 0 : Γ(t+s) = Γ(t) Γ(s)
- ∀ x0 ∈ H : || Γ(t) x0 - x0 || → 0 , as t → 0 .
Infinitesimal generator The infinitesimal generator A of a C0-semigroup Γ is defined by -
whenever the limit exists. The domain of A, D(A), is the set of x ∈ H for which this limit does exist. Stability The growth bound of a semigroup Γ (on a Hilbert space) is the constant - .
The semigroup is exponentially stable, i.e. -
iff its growth bound is negative. One has the following Theorem: A semigroup is exponentially stable iff for every there is such that - .
See also References - E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1957.
- R F Curtain, H J Zwart: An introduction to infinite dimensional linear systems theory. Springer Verlag, 1995.
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