In mathematical analysis, 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.
X
T:R+\toL(X)
L(X)
X
T(0)=I
X
\forallt,s\ge0: T(t+s)=T(t)T(s)
\forallx0\inX: \|T(t)x0-x0\|\to0
t\downarrow0
T
{(R+,+)}
T
The infinitesimal generator A of a strongly continuous semigroup T is defined by
Ax=\limt\downarrow0
| 1t(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; D(A) is a linear subspace and A is linear on this domain.[1] The operator A is closed, although not necessarily bounded, and the domain is dense in X.[2]
The strongly continuous semigroup T with generator A is often denoted by the symbol
eAt
\exp(At)
A uniformly continuous semigroup is a strongly continuous semigroup T such that
| \lim | |
| t\to0+ |
\|T(t)-I\|=0
holds. In this case, the infinitesimal generator A of T is bounded and we have
l{D}(A)=X
and
T(t)=eAt
| ||||
| :=\sum | ||||
| k=0 |
tk.
Conversely, any bounded operator
A\colonX\toX
is the infinitesimal generator of a uniformly continuous semigroup given by
T(t):=eAt
Thus, a linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator. If X is a finite-dimensional Banach space, then any strongly continuous semigroup is a uniformly continuous semigroup. For a strongly continuous semigroup which is not a uniformly continuous semigroup the infinitesimal generator A is not bounded. In this case,
eAt
Consider the Banach space
C0(R):=\{f:R → Ccontinuous: \forall\epsilon>0~\existsc>0suchthat\vertf(x)\vert\leq\epsilon~ \forallx\inR\setminus[-c,c]\}
\Vertf\Vert:=supx\in\vertf(x)\vert
q:R → C
sups\inRe(q(s))<infin
Mqf:=q ⋅ f
D(Mq):=\{f\inC0(R):q ⋅ f\inC0(R)\}
(Tq(t))t\geq
Tq(t)f:=eqtf.
Mq
q
Mq
C0(R)
q
Let
Cub(R)
R
(Tl(t))t\geq
Tl(t)f(s):=f(s+t), s,t\inR
Its generator is the derivative
Af:=f'
D(A):=\{f\inCub(R):fdifferentiablewithf'\inCub(R)\}
Consider the abstract Cauchy problem:
u'(t)=Au(t),~~~u(0)=x,
where A is a closed operator on a Banach space X and x∈X. There are two concepts of solution of this problem:
Theorem:[3] Let A be a closed operator on a Banach space X. The following assertions are equivalent:
When these assertions hold, the solution of the Cauchy problem is given by u(t ) = T(t )x with T the strongly continuous semigroup generated by A.
In connection with Cauchy problems, usually a linear operator A is given and the question is whether this is the generator of a strongly continuous semigroup. Theorems which answer this question are called generation theorems. A complete characterization of operators that generate exponentially bounded strongly continuous semigroups is given by the Hille–Yosida theorem. Of more practical importance are however the much easier to verify conditions given by the Lumer–Phillips theorem.
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]].
See main article: analytic semigroup.
A C0-semigroup Γ(t), t ≥ 0, is called a quasicontraction semigroup if there is a constant ω such that ||Γ(t)|| ≤ exp(ωt) for all t ≥ 0. Γ(t) is called a contraction semigroup if ||Γ(t)|| ≤ 1 for all t ≥ 0.[4]
A strongly continuous semigroup T is called eventually differentiable if there exists a such that (equivalently: for all and T is immediately differentiable if for all .
Every analytic semigroup is immediately differentiable.
An equivalent characterization in terms of Cauchy problems is the following: the strongly continuous semigroup generated by A is eventually differentiable if and only if there exists a such that for all the solution u of the abstract Cauchy problem is differentiable on . The semigroup is immediately differentiable if t1 can be chosen to be zero.
A strongly continuous semigroup T is called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operator (equivalently[5] 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.
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, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous.[6]
The growth bound of a semigroup T is the constant
\omega0=inft>0
| 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)\|\leqMe\omega
for all t ≥ 0.
The following are equivalent:[7]
\|T(t)\|\leqM{\rme}-\omega,
\limt\toinfty\|T(t)\|=0
\|T(t0)\|<1
| infty(C | |
| H | |
| +;L(X)) |
The spectral bound of an operator A is the constant
s(A):=\sup\{{\rmRe}λ:λ\in\sigma(A)\}
The growth bound of a semigroup and the spectral bound of its generator are related by[9] s(A) ≤ ω0(T ). There are examples[10] 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.[11] This gives another equivalent characterization of exponential stability for these semigroups:
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.
A strongly continuous semigroup T is called strongly stable or asymptotically stable if for all x ∈ X:
\limt\toinfty\|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
\|T(t)\|\leqM
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.