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 T 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.

Special classes of semigroups

Uniformly continuous semigroups

The strongly continuous semigroup T is called uniformly continuous if the map t→T(t) is continuous from [0,∞) to L(X).

The generator of a uniformly continuous semigroup is a bounded operator.

Analytic semigroups

Main article: analytic semigroup

Differentiable semigroups

A strongly continuous semigroup T is called eventually differentiable if there exists a t₀≥0 such that for all x∈X the map t→T(t)x is differentiable on (t₀,∞). The semigroup is called immediately differentiable if t₀ can be chosen to be zero.

Equivalently: T is eventually differentiable if there exists a t₁>0 such that T(t₁)X⊂D(A) (equivalently: for all t≥t₁) and T is immediately differentiable if T(t)X⊂D(A) for all t>0.

Every analytic semigroup is immediately differentiable.

Compact semigroups

A strongly continuous semigroup T is called eventually compact if there exists a t₀>0 such that T(t₀) is a compact operator (equivalently if T(t) is a compact operator for all t≥t₀) . The semigroup is called immediately compact if T(t) is a compact operator for all t>0.

Norm continuous semigroups

A strongly continuous semigroup is called eventually norm continuous if there exists a t₀≥0 such that the map t→T(t) is continuous from (t₀,∞) to L(X). The semigroup is called immediately norm continuous if t₀ can be chosen to be zero.

Note that for an immediately norm continuous semigroup the map t→T(t) may not be continuous in t=0 (that would make the semigroup uniformly continuous).

Analytic semigroups, differentiable semigroups and compact semigroups are all eventually differentiable.

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 ω such that there exists a constant M (≥ 1) with

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

for all t ≥ 0.

The following are equivalent:

1. There exist M,ω>0 such that for all t≥0: $\|T(t)\|\leq M{\rm {e}}^{-\omega t}$,
2. The growth bound is negative: ω₀<0,
3. The semigroup converges to zero in the uniform operator topology: $\lim _{t\to \infty }\|T(t)\|=0$,
4. There exists a t₀>0 such that $\|T(t_{0})\|<1$,
5. There exists a t₁>0 such that the spectral radius of T(t₁) is strictly smaller than 1,
6. There exists a p∈[1,∞) such that for all x∈X: $\int _{0}^{\infty }\|T(t)x\|^{p}\,dt<\infty$,
7. For all p∈[1,∞) and all x∈X: $\int _{0}^{\infty }\|T(t)x\|^{p}\,dt<\infty$.

A semigroup that satisfies these equivalent conditions is called exponentially stable or uniformly stable (either of the first three of the above statements is taken as the definition in certain parts of the literature). That the L^p conditions are equivalent to exponential stability is called the Datko-Pazy theorem.

In case X is a Hilbert space there is another condition that is equivalent to exponential stability in terms of the resolvent operator of the generator: all λ with positive real part belong to the resolvent set of A and the resolvent operator is uniformly bounded on the right half plane, i.e. (λI-A)^-1 belongs to the Hardy space $H^{\infty }(\mathbb {C} _{+};L(X))$. This is called the Gearhart-Pruss theorem.

The spectral bound of an operator A is the constant

$s(A):=\sup\{{\rm {Re}}\lambda :\lambda \in \sigma (A)\}$,

with the convention that s(A)=-∞ of the spectrum of A is empty.

The growth bound of a semigroup and the spectral bound of its generator are related by: s(A)≤ω₀(T). There are examples where s(A)<ω₀(T). If s(A)=ω₀(T), then T is said to satisfy the spectral determined growth condition. Eventually norm-continuous semigroups satisfy the spectral determined growth condition. This gives another equivalent characterization of exponential stability for these semigroups:

• An eventually norm-continuous semigroup is exponentially stable if and only if s(A)<0.

Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually norm-continuous so that the spectral determined growth condition holds in particular for those semigroups.

Strong stability

A strongly continuous semigroup T is called strongly stable or asymptotically stable if for all x∈X: $\lim _{t\to \infty }\|T(t)x\|=0$.

Exponential stability implies strong stability, but the converse is not generally true if X is infinite-dimensional (it is true for X finite-dimensional).

The following sufficient condition for strong stability is called the Arendt-Batty-Lyubich-Phong theorem: Assume that

1. T is bounded: there exists a M≥1 such that $\|T(t)\|\leq M$,
2. A has not residual spectrum on the imaginary axis, and
3. The spectrum of A located on the imaginary axis is countable.

Then T is strongly stable.

If X is reflexive then the conditions simplify: if T is bounded, A has no eigenvalues on the imaginary axis and the spectrum of A located on the imaginary axis is countable, then T is strongly stable.

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.
• Engel, Klaus-Jochen; Nagel, Rainer (2000), One-parameter semigroups for linear evolution equations, Springer
• Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser
• Staffans, Olof (2005), Well-posed linear systems, Cambridge University Press
• Luo, Zheng-Hua; Guo, Bao-Zhu; Morgul, Omer (1999), Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer
• Arendt, Wolfgang; Batty, Charles (1988), Tauberian theorems and stability of one-parameter semigroups, Transactions of the American mathematical society
• Lyubich, Yu; Phong, Vu Quoc (1988), Asymptotic stability of linear differential equations in Banach spaces, Studia Mathematica