C0-semigroup
Encyclopedia
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 equation
s, strongly continuous semigroup
s provide solutions of linear constant coefficient ordinary differential equations in Banach space
s. Such differential equations in Banach spaces arise from e.g. delay differential equation
s and partial differential equation
s.
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.
X is a map
such that
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
.
on the Banach space
X, then
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.
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. The operator A is closed
, although not necessarily bounded
, and the domain is dense in X.
The strongly continuous semigroup T with generator A is often denoted by the symbol eAt. This notation is compatible with the notation for matrix exponential
s, and for functions of an operator defined via functional calculus
(for example, via the spectral theorem
).
:
where A is a closed operator
on a Banach space
X and x∈X. There are two concepts of solution of this problem:
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable.
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem 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.
.
The generator of a uniformly continuous semigroup is a bounded operator
.
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 t1 ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t1, ∞). The semigroup is immediately differentiable if t1 can be chosen to be zero.
(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.
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.
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
for all t ≥ 0.
The following are equivalent:
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 Lp 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
. This is called the Gearhart-Pruss theorem.
The spectral bound of an operator A is the constant
,
with the convention that s(A) = −∞ if 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:
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.
.
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s, strongly continuous semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
s provide solutions of linear constant coefficient ordinary differential equations in Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. Such differential equations in Banach spaces arise from e.g. delay differential equation
Delay differential equation
In mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
s and partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s.
Formally, a strongly continuous semigroup is a representation of the semigroup (R+,+) on some Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X that is continuous in the strong operator topology
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
. 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 spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X is a map
- T : R+ → L(X)
such that
- T(0) = I , (identity operator on X)
- ∀ t,s ≥ 0 : T(t + s) = T(t) T(s)
- ∀ 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
Strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the weakest locally convex topology on the set of bounded operators on a Hilbert space such that the evaluation map sending an operator T to the real number \|Tx\| is continuous for each vector...
.
Elementary examples
Let A be a bounded operatorBounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
on the Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X, then
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
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. The operator A is closed
Closed operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
, although not necessarily bounded
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
, and the domain is dense in X.
The strongly continuous semigroup T with generator A is often denoted by the symbol eAt. This notation is compatible with the notation for matrix exponential
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
s, and for functions of an operator defined via functional calculus
Functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch of the field of functional analysis, connected with spectral theory. In mathematics, a functional calculus is a theory allowing one to apply mathematical...
(for example, via the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
).
Abstract Cauchy problems
Consider the abstract Cauchy problemCauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
:
where A is a closed operator
Closed operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
on a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X and x∈X. There are two concepts of solution of this problem:
- a continuously differentiable function u:[0,∞)→X is called a classical solution of the Cauchy problem if u(t) ∈ D(A) for all t ≥ 0 and it satisfies the initial value problem,
- a continuous function u:[0,∞) → X is called a mild solution of the Cauchy problem if
Any classical solution is a mild solution. A mild solution is a classical solution if and only if it is continuously differentiable.
The following theorem connects abstract Cauchy problems and strongly continuous semigroups.
Theorem Let A be a closed operator on a Banach space X. The following assertions are equivalent:
- for all x∈X there exists a unique mild solution of the abstract Cauchy problem,
- the operator A generates a strongly continuous semigroup,
- the resolvent setResolvent setIn linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.-Definitions:...
of A is nonempty and for all x ∈ D(A) there exists a unique classical solution of the Cauchy problem.
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.
Generation theorems
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 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 theoremLumer-Phillips theorem
In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.-Statement of the...
.
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
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
.
Differentiable semigroups
A strongly continuous semigroup T is called eventually differentiable there exists a t0 > 0 such that T(t0)X⊂D(A) (equivalently: T(t)X ⊂ D(A) for all t ≥ t0) and T is immediately differentiable if T(t)X ⊂ D(A) for all t > 0.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 t1 ≥ 0 such that for all x ∈ X the solution u of the abstract Cauchy problem is differentiable on (t1, ∞). The semigroup is immediately differentiable if t1 can be chosen to be zero.
Compact semigroups
A strongly continuous semigroup T is called eventually compact if there exists a t0 > 0 such that T(t0) is a compact operatorCompact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
(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, (eventually) differentiable semigroups and (eventually) compact semigroups are all eventually norm continuous.
Exponential stability
The growth bound of a semigroup T is the constant
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
for all t ≥ 0.
The following are equivalent:
- There exist M,ω>0 such that for all t ≥ 0:
- The growth bound is negative: ω0 < 0,
- The semigroup converges to zero in the uniform operator topology:
, - There exists a t0 > 0 such that
, - There exists a t1 > 0 such that the spectral radiusSpectral radiusIn mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...
of T(t1) is strictly smaller than 1, - There exists a p ∈ [1, ∞) such that for all x∈X:
, - For all p ∈ [1, ∞) and all x ∈ X:
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 Lp conditions are equivalent to exponential stability is called the Datko-Pazy theorem.
In case X is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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
Hardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
. This is called the Gearhart-Pruss theorem.The spectral bound of an operator A is the constant
,with the convention that s(A) = −∞ if the spectrum
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
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:.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 is bounded: there exists a M ≥ 1 such that
, - A has not residual spectrum on the imaginary axis, and
- 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.
See also
- Hille–Yosida theorem
- Lumer–Phillips theorem
- Analytic semigroupAnalytic semigroupIn mathematics, an analytic semigroup is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value...
- Contraction semigroup
- Matrix exponentialMatrix exponentialIn mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
- Strongly continuous family of operators