C0-semigroup

In mathematics, a C₀-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations.

Formally, a strongly continuous semigroup is a representation of the semigroup (R₊,+) on some Banach space X that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup.

Formal definition

A strongly continuous semigroup on a Banach space X is a map

T : R₊ → L(X)

such that

1. T(0) = I ,   (identity operator on X)
2. ∀ t,s ≥ 0 : T(t+s) = T(t) T(s)
3. ∀ x₀ ∈ X : || T(t) x₀ - x₀ || → 0 , as t ↓ 0.

The first two axioms are algebraic, and state that it is a representation of the semigroup (R₊,+); the last is topological, and states that the map T is continuous in the strong operator topology.

Elementary examples

Let A be a bounded operator on the Banach space X, then

$T(t)={\rm {e}}^{At}:=\sum _{k=0}^{\infty }{\frac {A^{k}}{k!}}t^{k}$

is a strongly continuous semigroup (it is even continuous in the uniform operator topology). Conversely, any uniformly continuous semigroup is necessarily of this form for some bounded linear operator A. In particular, if X is a finite-dimensional Banach space, then any strongly continuous semigroup is necessarily of this form for some linear operator A.

Infinitesimal generator

The infinitesimal generator A of a strongly continuous semigroup T is defined by

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

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

The strongly continuous semigroup T with generator A is often denoted by the symbol e^At. This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via the spectral theorem.

Stability

Exponential stability

The growth bound of a semigroup T is the constant

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

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

$\|T(t)\|\leq Me^{\omega t}$

for all t ≥ 0.

The semigroup is exponentially stable, i.e.

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

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 λI−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.