C0-semigroup

In 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,

<math>\frac{\mathrm dx}{\mathrm dt} = f(x,t) ;~ x(0) = x_0 ~,\qquad\rm(CP)<math>

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

<math> \Gamma(t)\,x_0 = x(t) <math> , 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

<math> \lim_{t\to0} \|\Gamma(t)\,x_0 – x_0 \| = 0 <math>

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

1. Γ(0) = I := id_H ,   (identity operator on H)
2. ∀ t,s ≥ 0 : Γ(t+s) = Γ(t) Γ(s)
3. ∀ x₀ ∈ H : || Γ(t) x₀ – x₀ || → 0 , as t → 0 .

Infinitesimal generator

The infinitesimal generator A of a C₀-semigroup Γ is defined by

<math> A\,x = \lim_{t\to0} \frac1t\,(\Gamma(t)- I)\,x <math>

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

<math> \omega = \lim_{t\to0} \frac1t \log \| \Gamma(t) \| <math> .

The semigroup is exponentially stable, i.e.

<math> \exists K,a > 0,~ \forall t\ge0: \| \Gamma(t) \| \le K\,e^{- a\,t} <math>

iff its growth bound is negative.

One has the following

Theorem: A semigroup is exponentially stable iff for every <math>x \in H<math> there is <math>C < 0<math> such that

<math>\int_{\mathbb R_+} {\|\Gamma(t)\,x\|}^2\mathrm dt < C <math> .

See also

• Schrödinger semigroups

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.