C0-semigroup

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In mathematics, a C₀-semigroup, also known as a (strongly continuous) one-parameter semigroup, is a continuous morphism from (R₊,+) into a topological monoid, usually L(B), the algebra of linear continuous operators on some Banach space B.

Thus, strictly speaking, not the C₀-semigroup, but rather its image, is a semigroup. C₀-semigroups generalize one-parameter groups.

Example

C₀-semigroups occur for example in the context of initial value problems of the form,

where x and f take values in a Banach space B.

If the solution of (CP) is unique (depending on f) for x₀ in some given domain D ⊂ B, one has the "solution operator" defined by

where x(t) is the solution of (CP).

Thus one can view Γ as an "evolution operator", and it is clear that one should have

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(B) induced by the norm on B, which amounts to check that

for each x₀ in D.

Formal definition

All that follows concerns the following definition:

A (strongly continuous) C₀-semigroup on a Banach space B is a map

Γ : R₊ → L(B)

such that

1. Γ(0) = I := id_B ,   (identity operator on B)
2. ∀ t,s ≥ 0 : Γ(t+s) = Γ(t) Γ(s)
3. ∀ x₀ ∈ B : || 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 for which this limit does exist.

Γ(t) may also be denoted by the symbol

Γ(t) = e^tA.

This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via the spectral theorem.

Stability

The growth bound of a semigroup Γ (on a Banach space) is the constant

It is so called as this number is also the infimum of all real numbers w such that there exists a constant M (≥ 1) with

for all t ≥ 0.

The semigroup is exponentially stable, i.e.

if and only if its growth bound is negative.

One has the following:

Theorem: A semigroup is exponentially stable if and only if for every there is C > 0 such that

References

• E Hille, R S Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975.
• R F Curtain, H J Zwart: An introduction to infinite dimensional linear systems theory. Springer Verlag, 1995.
• E.B. Davies: One-parameter semigroups (L.M.S. monographs), Academic Press, 1980, ISBN 0-12-206280-9.