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.

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[edit] Example

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

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

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

\Gamma(t)\,x_0 = x(t), 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

\Gamma(s + t) = \Gamma(s) \Gamma(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(B) induced by the norm on B, which amounts to check that

\lim_{t\to0^+} \|\Gamma(t)\,x_0 - x_0 \| = 0

for each x0 in D.

[edit] 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 := idB ,   (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 .

[edit] Infinitesimal generator

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

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

whenever the limit exists. The domain of A, D(A), is the set of x \in B for which this limit does exist.

Γ(t) may also be denoted by the symbol

Γ(t) = etA.

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

[edit] Stability

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

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

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

\|\Gamma(t)\| \leq Me^{wt}

for all t ≥ 0.

The semigroup is exponentially stable, i.e.

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

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 x \in B there is C > 0 such that

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


[edit] 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.
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