C0-semigroup

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

Thus, strictly speaking, not the C₀-semigroup, but rather its image, is a semigroup. A C₀-semigroup is a type of one-parameter semigroup, a generalization of a one-parameter group.

Example

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

${\frac {\mathrm {d} x}{\mathrm {d} t}}=f(x);~x(0)=x_{0}~,\qquad {\rm {(CP)}}$

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

$\lim _{t\to 0^{+}}\|\Gamma (t)\,x_{0}-x_{0}\|=0$

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 .

The first two axioms are algebraic, and state that it is a map of semigroups; the last is topological, and states that the map is continuous in the strong operator topology.

Infinitesimal generator

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

$A\,x=\lim _{t\to 0}{\frac {1}{t}}\,(\Gamma (t)-I)\,x$

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

$\Gamma (t)$ may also be denoted by the symbol

$\Gamma (t)=e^{tA}.$

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

$\omega =\lim _{t\to 0}{\frac {1}{t}}\log \|\Gamma (t)\|.$

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.

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

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

$\int _{\mathbb {R} _{+}}{\|\Gamma (t)\,x\|}^{2}\mathrm {d} t<C.$

Non-example

Consider the Banach space BUC[0, ∞) of bounded uniformly continuous complex-valued functions of the interval [0, ∞). Let

$(1)\quad [T_{t}\varphi ](x)=\varphi (x+t),\quad x,t\in [0,\infty ).$

Then {T_t} is a strongly continuous one parameter semigroup. In this case the operators T_t have norm at most 1. The strong continuity property follows from the fact that the functions in the space BUC[0, ∞) are uniformly continuous. In fact, the family of translation operators defined by (1) on the larger space BC[0, ∞) of bounded continuous complex-valued functions on [0, ∞) is a one-parameter semigroup but fails to be strongly continuous.

Connection with resolvent operator and Laplace transform

The operator-valued function R(λ, A) defined as the inverse of &lambda I-A is called the resolvent of A. The resolvent is related to the semigroup {T_t} _{t ∈ [0,∞)} generated by A as follows:

Theorem. Suppose A generates {T_t} _{t ∈ [0,∞)}. Then

$\operatorname {R} (\lambda ,A)\varphi =\int _{0}^{\infty }e^{-\lambda t}T_{t}\varphi \ dt,\quad \forall \varphi \in E.$

The above integral is defined as a Bochner integral.

By definitions of the infinitesimal generator and the semigroup we have that

${\frac {d}{dt}}T_{t}\varphi =\lim _{h\to 0}{\frac {T_{t+h}\varphi -T_{t}\varphi }{h}}=AT_{t}\varphi .$

If we formally do the Laplace transform of the semigroup

${\mathcal {L}}T_{t}\varphi =\int _{0}^{\infty }e^{-\lambda t}T_{t}\varphi dt$

we get by integration by parts

${\mathcal {L}}{\frac {d}{dt}}T_{t}\varphi =\lambda {\mathcal {L}}T_{t}\varphi -T_{0}\varphi .$

When applied to the differential equation above we get

$\lambda {\mathcal {L}}T_{t}\varphi -\varphi =A{\mathcal {L}}T_{t}\varphi$

or

${\mathcal {L}}T_{t}\varphi =(\lambda I-A)^{-1}\varphi =\operatorname {R} (\lambda ,A)\varphi .$

If A and $\operatorname {R} (\lambda ,A)$ satisfy the conditions of Hille–Yosida theorem, then we can invert the Laplace transform and A generates the semigroup

$T_{t}\varphi ={\mathcal {L}}^{-1}\operatorname {R} (\lambda ,A)\varphi .$

See also

• Strongly continuous family of operators

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.