C0 semigroup

The C₀- semigroup (C₀- はんぐん British: C₀-semigroup) in the field of the mathematics or the strong continuation one parameter semigroup is one generalization of the exponential function. I am given the solution of the differential equation that a linear fixed number coefficient in the バナッハ space is the way it goes by strong continuation semigroup so that the solution of the differential equation that it is the way it goes to assume the linear scalar fixed number a coefficient gives it in an exponential function. The differential equation in such バナッハ space appears, for example, in a field of a delay differential equation (English version) and the partial differential equation.

The strong continuation semigroup is expression of semigroup (R₊,+) on consecutive バナッハ space X in strong operator phase formally. Therefore, as for the strong continuation semigroup, it may be said that it is continuous expression of the very special semigroup rather than semigroup when I say closely.

Table of contents

Definition

バナッハ space ${\displaystyle X}$  The representation that meets the next property with the upper strong continuation semigroup ${\displaystyle T:\mathbb {R} _{+}\to L(X)}$  It is with a saw:

1. ${\displaystyle T(0)=I}$ ,${\displaystyle X}$  Upper 恒等作用素)
2. ${\displaystyle \forall t,s\geq 0:\ T(t+s)=T(t)T(s)}$ 
3. ${\displaystyle \forall x_{0}\in X:\ T(t)x_{0}-x_{0} \to 0}$ , as ${\displaystyle t\downarrow 0}$ .

Two axioms of the beginning are algebraic things,${\displaystyle T}$  But, it is semigroup${\displaystyle \mathbb {R} _{+},+}$ I mean that it is expression of). The last axiom is topologic and represents it ${\displaystyle T}$  But, I mean that it is continuation in strong operator phase.

Simple example

I assume A existence world operator on バナッハ space X. Then,

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

It is は 強連続半群 (I really continue in same operator phase (English version)). On the contrary, existence boarder-line type operator A that I can write in the upper form in any same consecutive semigroup by all means exists [¹]. Particularly, alignment operator A which I can write in the upper form in any strong continuation semigroup by all means exists if X is limited dimensional バナッハ space [²].

Infinitesimal generation operator

Infinitesimal generation operator A of strong continuation semigroup T

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

It is defined によって (when the limit of the right side exists). Domain D(A) of A is a meeting consisting of x ∈ X where there is such a limit. D(A) is a linear subspace, and A is linear on the domain [³]. A is not necessarily existence world, but it is shut, and the domain is dense again in X [⁴].

Strong continuation semigroup T with generation operator A is often expressed using sign e^At. This scale adapts to the scale for the function of the operator defined through a line exponential function and a Pan-function calculation (e.g., a spectrum theorem).

Issue of abstract Cauchy

I think about the issue of following abstract Cauchy:

${\displaystyle u'(t)=Au(t),~~~u(0)=x.}$ 

A assumes it a shut operator on バナッハ space X here and assumes it x ∈ X. With two concepts following in the solution of this problem:

• Continuous function u: which can differentiate it [0, ∞] the thing meeting the initial condition that satisfy u(t) ∈ D(A) for all t≥0 in → X and was given is called the classic solution of the upper Cauchy problem.
• Continuous function u: [0, ∞] in → X

${\displaystyle \int _{0}^{t}u(s)\,ds\in D(A){\text{ and }}A\int _{0}^{t}u(s)\,ds=u(t)-x}$ 

The thing satisfying を is called a soft solution of the issue of big Cauchy.

All classic solutions are soft solutions. A necessary and sufficient condition for a soft solution to be a classic solution is that it is continuous differentiability [⁵].

The next theorem relates to the abstract issue of Cauchy and relations of the strong continuation semigroup.

I assume theorem [⁶] A a shut operator on バナッハ space X. The following claims are the equivalent:

1. For all x ∈ X, there is merely one soft solution for the issue of abstract Cauchy.
2. Operator A generates a certain strong continuation semigroup.
3. The レゾルベント set of A is not empty, and, for all x ∈ D(A), there is an only horn classic solution for the issue of abstract Cauchy.

When these claims are concluded, the solution of the issue of Cauchy is given by u(t) = T(t)x. But T is strong continuation semigroup generated by A.

Generation theorem

When mostly I was given a certain linear operator A, in conjunction with the issue of Cauchy, it is in a problem it a generation bare となるかどうかという point of the strong continuation semigroup. The theorem becoming the answer to this problem is called a generation theorem. One perfect characterization about the operator which generated strong continuation semigroup was given by a theorem of fin - Yoshida. In addition, I was given the condition that it was easy to confirm it by a theorem of roux mer - Philips more practically while being important.

Special kind of the semigroup

The same consecutive semigroup

If t → T(t) is consecutive mapping from [0, ∞] to L(X), strong continuation semigroup T is told to be the same continuation.

It is generation bare は of the same consecutive semigroup, existence world operator [¹].

Analysis semigroup

The details refer to "analysis semigroup"

Reduction semigroup

The details refer to "reduction semigroup"

The semigroup that can differentiate it

A certain t₀ where T(t₀)X ⊂ D(A) (as the condition that or is it and the equivalent for all t≥t₀ T(t)X ⊂ D(A)) is established in strong continuation semigroup T > If 0 exists, I am called when I can differentiate it for an end. In addition, it is all t > I am called when I can differentiate it promptly if T(t)X ⊂ D(A) is established for 0.

All analysis semigroup can differentiate it promptly.

The characterization that is one equivalent in the issue of Cauchy is the following thing: A necessary and sufficient condition for strong continuation semigroup generated by A to be able to differentiate it for an end is that t1≥0 which there is which becomes when solution u of the issue of abstract Cauchy can differentiate it on (t₁, ∞) for all x ∈ X exists. It is if I can choose t₁ to become zero when such a semigroup can differentiate it promptly.

Compact semigroup

A certain t₀ where T(t₀) becomes the compact operator in strong continuation semigroup T > If 0 exists; for an end called the compact (^[7] that this condition is the equivalent with T(t) being compact for all t≥t₀). It is all t > If T(t) is a compact operator, I am called for 0 when such a semigroup is compact promptly.

Norm consecutive semigroup

If t0≥0 which there is where t → T(t) becomes the consecutive representation from (t₀, ∞) to L(X) exists, the strong continuation semigroup is called it for an end when it is norm continuation. I am called if I can choose t₀ as zero when such a semigroup is norm continuation promptly.

For the semigroup which is norm continuation, in t → T(t), it should be noted consecutive things in t = 0 promptly (the semigroup becomes the same continuation if it is continuation).

Analysis semigroup, the semigroup which can differentiate it (for an end), semigroup compact (for an end) are all norm consecutive semigroup for an end [⁸].

Stability

Index stability

The growth upper limit of semigroup T is the fixed number

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

It is defined によって. This number

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

But, because is given as the lower limit of real number ω which there is fixed number M (≥1) established for all t≥0; such; call it, and do.

All the conditions to give next are the equivalent [⁹]:

1. For all t≥0 ${\displaystyle T(t) \leq M{\rm {e}}^{-\omega t}}$  But, established M,ω>0 exists.
2. Growth upper limit ω ₀ <0 is minus number.
3. The semigroup converges to zero in same operator phase (English version). In other words,${\displaystyle \lim _{t\to \infty} T(t) =0}$  となる.
4. ${\displaystyle T(t_{0}) <1}$  であるようなある t₀ > 0 exists.
5. t₁ where a spectrum radius of T(t₁) becomes smaller than 1 closely > 0 exists.
6. For all x ∈ X ${\displaystyle \int _{0}^{\infty} T(t)x ^{p}\,dt<\infty}$  となるような p ∈ [1, ∞] exists.
7. For all p ∈ [1, ∞] and x ∈ X,${\displaystyle \int _{0}^{\infty} T(t)x ^{p}\,dt<\infty}$  But, it is established.

It is said that the semigroup meeting the condition that is these equivalent is index stability or the same stability (in the related documents, one of three conditions of the upper beginning is often treated as a definition). It is known that a condition of L^p is index stability and the equivalent as a theorem of ダツコ - ペジー.

When X is Hilbert space, the following different condition about the generation natural レゾルベント operator also becomes index stability and the equivalent of the semigroup: All complex number λ with an equilateral real part belongs to レゾルベント set of A, and, as for the レゾルベント operator, it is in the right half plane in same existence world. In other words, (λI-A)^-1 is Hardy space ${\displaystyle H^{\infty }(\mathbb {C} _{+};L(X))}$  に belongs [¹⁰]. This is called a theorem of gear heart - pulse.

The spectrum upper limit of operator A is the fixed number

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

It is defined として. But it is a spectrum of A ${\displaystyle \sigma (A)}$  But, I assume it s(A) = - ∞ when it is empty.

In the growth upper limit and the spectrum upper limit of the semigroup, I have a relationship called s(A)≤ω₀(T) [¹¹]. s(A) The example becoming <ω0(T) is seen by some documents [¹²]. T is said to meet a spectrum decision growth condition (spectral determined growth condition) if it is s(A) =ω0(T). The norm consecutive semigroup meets a spectrum decision growth condition for an end [¹³]. From this, the condition that is index stability and the equivalent of those semigroup is provided again:

• The necessary and sufficient condition for norm consecutive semigroup to be index stability for an end is s(A) It is <0.

Because it is norm continuation, compact semigroup, the semigroup which can differentiate it for an end, analysis semigroup and the same consecutive semigroup meet a spectrum decision growth condition for an end for an end.

Strong stability

Strong continuation semigroup T for all x ∈ X ${\displaystyle \lim _{t\to \infty} T(t)x =0}$  But, or strong stability is called if established if asymptotically stable.

The index stability means strong stability, but the reverse is not generally satisfied when X is an infinite dimension (if X is a limited dimension, the reverse is established).

The sufficient condition for strong stability to speak next called the theorem of the アレンド - Bhatti - リュビッヒ - phone [¹⁴]:

1. T is existence world. A certain M≥1 exists and ${\displaystyle T(t) \leq M}$  But, it is managed.
2. A does not have a surplus spectrum (English version) on the empty axis.
3. The spectrum of A located on the empty axis is a countable unit.

であるなら, T are strong stability.

These conditions are simplified as follows if X is recursive: T is strong stability if T is existence world, and A does not have eigenvalue on the empty axis, and the spectrum of A on the empty axis is a countable unit.

Allied item

• Theorem of fin - Yoshida
• Theorem of roux mer - Philips
• Analysis semigroup
• Reduction semigroup
• Line exponential function
• Strong; the family of consecutive operators

Explanatory note

1. ^ ^{a b} Engel and Nagel Theorem I.3.7
2. ^ Engel and Nagel Theorem I.2.9
3. ^ Partington (2004) page 23
4. ^ Partington (2004) page 24
5. ^ Arendt et. al. Proposition 3.1.2
6. ^ Arendt et. al. Theorem 3.1.12
7. ^ Engel and Nagel Lemma II.4.22
8. ^ Engel and Nagel (diagram II.4.26)
9. ^ Engel and Nagel Section V.1.b
10. ^ Engel and Nagel Theorem V.1.11
11. ^ Engel and Nagel Proposition IV2.2
12. ^ Engel and Nagel Section IV.2.7, Luo et. al. Example 3.6
13. ^ Engel and Nagel Corollary 4.3.11
14. ^ Arendt and Batty, Lyubich and Phong

References

• Hille, E.; Phillips, R. S. (1975). Functional Analysis and Semi-Groups. American Mathematical Society. 
• Curtain, R. F.; Zwart, H. J. (1995). An introduction to infinite dimensional linear systems theory. Springer Verlag. 
• Davies, E. B. (1980). One-parameter semigroups. L.M.S. monographs. Academic Press. 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 
• Partington, Jonathan R. (2004), Linear operators and linear systems, London Mathematical Society Student Texts, Cambridge University Press, ISBN 0-521-54619-2 

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