Evolutionary Coprocesses and Perturbations of Kato–stable Families of Operators

Stefano Bertoni^{∗}^{†}^{‡}

June 29, 1999

Abstract

In this paper we introduce and study a new object, a co–process U^{⊕}(t, s), associated to an evolutionary process U(t, s), 0 ≤ s ≤ t ≤ T, generated by a family {A(t)}_{0≤t≤T} of operators. Moreover, to prove a theorem of existence for the co–process, we obtain a theorem of Kato–stability for a family of operators with respect to perturbations given in a constant domain. ^{1}

1 Introduction

This study is part of a research to extend the variation–of–constants formula

for perturbed non–autonomous linear differential equations in Banach spaces,

following the lines used by Desch & Schappacher [2] for the autonomous case.

∗Institut für Mathematik, Karl–Franzens–Universität Graz, Austria.

†Dipartimento di Matematica, Università di Trento, Povo (TN), Italy.

‡e-mail: bertoni@science.unitn.it

1Key words: evolutionary process, non–autonomous linear abstract differential equation, co–process, family of generators of C_{0}–semigroups, Kato–stability, perturbation.

AMS Subject Classification: 34G10, 47D03, 47A55.

This paper was written and discussed while the author was at the Institut für Mathematik of the Karl–Franzens University of Graz (Austria), with the financial support of the CNR (Italy), while the basic ideas were developed in 1997, in the Department of Mathematics, University of Trento (Italy).

1.1 The Problem

The starting problem is the following differential equation:

u′(t) = A(t) [u(t) + f(t, u(t))],  (1)

where {A(t)}_{0≤t≤T} generates an evolutionary process in X, f: [0, T]×X → Z, and Z is an intermediate space between D(A(t)) and X.

To obtain a definition of integral solution for (1), we are studying on X the auxiliary problem

v′(t) = A^{⊕}(t) v(t),  (2)

where A^{⊕}(t) = (I + (A(t)^{−1})′) A(t) = A(t) + (A(t)^{−1})′A(t). The corresponding evolutionary process, {U^{⊕}(t, s) | 0 ≤ s ≤ t ≤ T }, is called the co–process associated to U(t, s): we have U^{⊕}(t, s)|_{D(A(s))} = A(t)^{−1}U(t, s) A(s).

If U^{⊕}(t, s) is well–defined on X and not only on D(A(s)), we can define the integral form of Pbm. (1) as

u(t) = U(t, s) x + A(t) ∫_{s}^{t} U^{⊕}(t, r) f(r, u(r)) dr  (3)

where the integral may belong to the domain D(A(t)) for suitable spaces Z extending the domain. For this class of problems we can prove an existence and uniqueness theorem via contraction arguments.

The property A(t) U^{⊕}(t, s) = U(t, s) A(s) is used to obtain (3). As equivalent alternative, you may study the problem via the change of variable v(t) = A(t)^{−1}u(t).

Essentially U^{⊕}(t, s) allows a transfer of the regularity conditions from f(t, u(t)) to A(t).

For this class of motivations, the co–process U^{⊕}(t, s) appears an interesting case of evolutionary process. The study of its properties and of the applications of this approach is actually in progress.

I proved already some results for the parabolic case with the interpolation spaces (Bertoni [1]), both for the co–process and the integral equation. In the present paper the hyperbolic case is instead treated.

We will show that, in the case of constant domain, the only request for the well–posedness of Pbm. (2) and the existence of U^{⊕}(t, s) is only a further degree of regularity for A(·) with respect to the classical theorem for A(t) and U(t, s) (see Tanabe [7], Thm. 7.4 and Engel–Nagel [3], Thm. 9.a.1, p. 461).

To this aim we will prove a theorem of perturbation of Kato–stable families of infinitesimal generators of C_{0}–semigroups, for perturbations in the constant domain D.

We thank, for the discussions and the suggestions, A. Pugliese (Univ. of Trento), W. Schappacher (Univ. of Graz) and in particular W. Desch (Univ. of Graz) for Example 1 in Sec. 3.3.

2 Preliminaries

Consider a family {A(t) | t ∈ [0, T]} of closed linear operators on a Banach space X (T > 0). We denote by L_{s}(D, X) the Banach space L(D, X) equipped with the strong topology (otherwise the norm topology is meant).

2.1 Hypotheses on A(t)

Hypothesis 1 (Invertibility) For every 0 ≤ t ≤ T, 0 ∈ ρ(A(t)).

Hypothesis 2 (Constant Domain) D(A(t)) = D, for every 0 ≤ t ≤ T, where (D, | · |_{D}) is a Banach space, dense and continuously included in X.

Hypothesis 3 (Regularity) t ↦ A(t) ∈ C^{2}([0, T], L_{s}(D, X)).

Hyp. 1 implies that A(t)^{−1} is defined for every t ∈ [0, T]. Hyp. 2 is a simplifying approach. Hyp. 3 is essential and there exist counterexamples for lesser degree of regularity (see Ex. 1 in Sec. 3.3).

2.2 Regularity of A(t) and A(t)^{−1}

Assume the existence of a Banach space (Y, | · |_{Y}) ↪ D(A(t)) for every t ∈ [0, T].

For the uniform boundedness theorem, the following asserts hold:

Proposition 1 (Lipschitz Continuity) The strong Lipschitz continuity for A(·) is equivalent to the norm Lipschitz continuity, that is: A(·) ∈ Lip([0, T], L_{s}(Y, X)) if and only if A(·) ∈ Lip([0, T], L(Y, X)).

Corollary 2 If A(·) ∈ C^{1}([0, T], L_{s}(Y, X)), then A(·) ∈ C([0, T], L(Y, X)) ∩ BV([0, T], L(Y, X)).

From now on, assume Hyp. 2: Y = D.

Proposition 3 (Existence of the Inverse) If A(·) ∈ C([0, T], L(D, X)) and for a particular s ∈ [0, T] there exists A(s)^{−1}, then there exists ϵ > 0 such that A(t)^{−1} exists for every t ∈ [0, T], |t − s| < ϵ, and A(·)^{−1} ∈ C([0, T] ∩ (s − ϵ, s + ϵ), L(X, D)).

Proof: the existence and the continuity follow from the expansion

A(t)^{−1} = A(s)^{−1} ∑_{n=0}^{∞}(−1)^{n}{[A(t) − A(s)] A(s)^{−1}}^{n},  (4)

well–defined for ∥A(t) − A(s)∥_{L(D,X)} < \frac{1}{∥A(s)^{−1}∥}, verified for |t − s| small enough.

Now, assume also Hyp. 1, that implies the existence of A(t)^{−1} for every 0 ≤ t ≤ T.

Remember that a strongly continuous function is also uniformly bounded.

Theorem 4

a) A(·) ∈ C([0, T], L(D, X)) if and only if A(·)^{−1} ∈ C([0, T], L(X, D)).

b) A(·) ∈ Lip([0, T], L(D, X)) if and only if A(·)^{−1} ∈ Lip([0, T], L(X, D)).

c) A(·): [0, T] → L(D, X) is strong differentiable if and only if A(·)^{−1}: [0, T] → L(X, D) is strong differentiable. Moreover we have

\frac{d}{dt} (A(t)^{−1}x) = −A(t)^{−1}A′(t) A(t)^{−1}x,  (5)

where A′(t) is the strong derivative of A(t) in L(D, X).

d) A(·) ∈ C^{1}([0, T], L_{s}(D, X)), if and only if A(·)^{−1} ∈ C^{1}([0, T], L_{s}(X, D)).

e) A(·) ∈ BV([0, T], L(D, X)) and ∥A(t)^{−1}∥_{L(X,D)} is uniformly bounded for t ∈ [0, T], if and only if A(·)^{−1} ∈ BV([0, T], L(X, D)) and ∥A(t)∥_{L(D,X)} is uniformly bounded for t ∈ [0, T].

We will prove these enunciates only in one sense, because the proof of the converse is symmetrical. Proof: a) it is immediate for Prop. 3;

b) If A(t) is Lipschitz continuous, substitute ϵ by L |t − s| in the proof of Prop. 3 and obtain the local Lipschitz continuity of A(t)^{−1}. Hence the global one on [0, T] follows by the uniformity of the lengths of the intervals.

c) For Cor. 2, the strong differentiablity implies the norm continuity, so we are can prove differentiability of A(t)^{−1} using (4) and the previous point a):

|\frac{A(t)^{−1}x − A(s)^{−1}x}{t − s} + A(s)^{−1} \frac{A(t) − A(s)}{t − s} A(s)^{−1}x|_{D}

= | A(s)^{−1} \frac{A(t) − A(s)}{t − s} A(s)^{−1} ∑_{n=1}^{∞}(−1)^{n}\{[A(t) − A(s)] A(s)^{−1}\}^{n} x|_{D}

= o(|t − s|)

and the strong differentiability of A(t)^{−1} follows from that one of A(t). Moreover, Eq. (5) holds.

d) Strong continuity of A′(t) on L(D, X) implies Lipschitzianity of A(t) and strong continuity of (d/dt)A(t)^{−1} on L(X, D), so that d) holds.

e) Define K_{A} := sup_{0≤t≤T} ∥A(t)^{−1}∥_{L(X,D)}. For every t, s ∈ [0, T], from A(t)^{−1} − A(s)^{1} = −A(t)^{−1}(A(t) − A(s)) A(s)^{−1},

there follows

V_{0}^{T} [A^{−1}(·)]_{L(X,D)} ≤ K_{A}^{2} · V_{0}^{T} [A(·)]_{L(D,X)},  (6)

where V_{0}^{T} [f]_{Z} denotes the total variation of f: [0, T] → Z, for Z normed space.

2.3 Evolutionary Processes and Equations

Let X be a Banach space. Recall some definitions (see [3, 4]; [6], Chp. 4: [7], Chp. 7):

Definition 1 (Evolutionary Process) A family {U(t, s) | 0 ≤ s ≤ t ≤ T } of bounded linear operators is a (continuous) evolutionary process in X if a) U(t, t) = I and U(t, r) U(r, s) = U(t, s) for 0 ≤ s ≤ r ≤ t ≤ T and

b) {(t, s) | 0 ≤ s ≤ t ≤ T } ∋ (t, s) ↦ U(t, s) is strongly continuous.

Consider the nonautonomous linear abstract Cauchy problem

\{ \begin{matrix}u′(t) = A(t) u(t), & for t ≥ s, t, s ∈ [0, T], \\ u(s) = x, & x ∈ X, \end{matrix}  (7)

where {A(t) | t ∈ [0, T]} is a family of closed linear operators in X.

A (strict) solution for Pbm. (7) is a function u(·; s, x) = u ∈ C^{1}([s, T], X) such that u(t) ∈ D(A(t)), u′(t) = A(t) u(t) for t ≥ s and u(s) = x.

Definition 2 Pbm. (7) is well–posed if there exists an evolutionary process U(t, s) such that U(t, s) x is a solution, for every x ∈ D(A(s)). Then we say that U(t, s) is generated by the family {A(t) | 0 ≤ s ≤ t ≤ T }.

Consider now the family {A(t) | 0 ≤ s ≤ t ≤ T }.

Definition 3 (Kato–stability) A family {A(t) | t ∈ [0, T]} of generators of C_{0}–semigroups is Kato–stable, with stability constants M ≥ 1, ω ∈ ℝ, if ρ(A(t)) ⊇ (ω, ∞), t ∈ [0, T] and for every finite sequence 0 ≤ t_{1} ≤ t_{2} ≤ … ≤ t_{k} ≤ T, k = 1, 2, …, λ > ω

∥ ∏_{j=1}^{n}(λ − A(t_{j}))^{−1}∥ ≤ \frac{M}{(λ − ω)^{n}} ,  (8)

where ∏_{j=1}^{n}(λ − A(t_{j}))^{−1} = (λ − A(t_{k}))^{−1} … (λ − A(t_{1}))^{−1}.

An equivalent definition is obtained by substituting (8) by

∥ ∏_{j=1}^{n} e^{s_{j}A(t_{j})}∥ ≤ M e^{ω ∑_{j=1}^{n} s_{j}},

s_{j} ≥ 0, j = 1, 2, …, k; see [7], Prop. 7.3.

3 The Evolutionary Co–process

3.1 Co–process and Co–generators

Let U(t, s) be an evolutionary process generated by a family {A(t) | 0 ≤ t ≤ T } of generators of C_{0}–semigroups. Assume Hyp. 1.

Definition 4 (Co–process) Define co–process, associated to {U(t, s) | 0 ≤ s ≤ t ≤ T }, an evolutionary process {U^{⊕}(t, s) | 0 ≤ s ≤ t ≤ T } in X, such that, for 0 ≤ s ≤ t ≤ T,

U^{⊕}(t, s)|_{D(A(s))} = A(t)^{−1}U(t, s) A(s).  (9)

From now on assume Hypotheses 1, 2 and suppose A(·) ∈ C^{1}([0, T], L_{s}(D, X)).

Definition 5 (Co–generators) Define family of co–generators associated to the family {A(t) | 0 ≤ t ≤ T }, the family of operators {A^{⊕}(t) | 0 ≤ t ≤ T }, where

A^{⊕}(t) = (I + \frac{d}{dt} (A(t)^{−1})) A(t),  (10)

with constant domain D(A^{⊕}(t)) = D(A(t)) = D.

Remark 1 For A(t) = A = constant, we have A^{⊕}(t) = A and U^{⊕}(t, s) = U(t, s) = e^{(t−s)A} for every 0 ≤ s ≤ t ≤ T.

Proposition 5 Let {A(t) | 0 ≤ t ≤ T } be a family of generators of C_{0}–semigroups. Then the operators A^{⊕}(t) are well-defined and form a family of generators of C_{0}–semigroup on X, with domain D, for 0 ≤ t ≤ T.

Proof: it follows from Theorem (A) in Desch and Schappacher [2].

It is immediate to see the following assertion: Proposition 6 Given the process U(t, s), 0 ≤ s ≤ t ≤ T, generated by the family {A(t) | 0 ≤ t ≤ T }, the co–process U^{⊕}(t, s) exists (and it is unique) if and only if there exists a constant M ≥ 0, such that, for 0 ≤ s ≤ t ≤ T,

∥A(t)^{−1} U(t, s) A(s) y∥ ≤ M ∥y∥, for every y ∈ D.  (11)

Moreover, there exists a constant M_{0}, such that, for 0 ≤ s ≤ t ≤ T, we have:

∥U^{⊕}(t, s)∥_{L(D)} ≤ M_{0} ∥U(t, s)∥_{L(X)}.  (12)

Proposition 7 Assume that Eq. (7) is well-posed and generates the evolutionary process U(t, s), the corresponding co–process U^{⊕}(t, s) exists on X and Eq. (2), v′(t) = A^{⊕}(t) v(t), is well–posed.

Then the evolutionary process associated to Eq. (2) is precisely the co–process U^{⊕}(t, s), that is generated by {A^{⊕}(t)}_{0≤t≤T}.

Proof: take u(t) = U(t, s) u(s) and consider the change of variable:

v(t) = A(t)^{−1} u(t),  (13)

so that v(t) = A(t)^{−1}U(t, s) u(s) = U^{⊕}(t, s) v(s), and v(s) ∈ D. The evolutionary process, solution of Eq. (2), coincides with U^{⊕}(t, s) on a dense set.

3.2 Theorems of Existence and Perturbation

Theorem 8 (Existence of the co–process) Let Hyp. 1–3 hold. If the family {A(t)}_{0≤t≤T} of generators is Kato–stable on X, then the co–family {A^{⊕}(t)}_{0≤t≤T} is Kato–stable on D and on X, and it generates on D and X an evolutionary process {U^{⊕}(t, s) | 0 ≤ s ≤ t ≤ T }, that solves Problem (2).

This assertion follows after Thm. 10.

Proposition 9 (Kato–stability down–up) Let {A(t)}_{0≤t≤T} be a family of generators of C_{0}–semigroups in X, and let Hyp. 2 hold. Then {A(t)|_{D}}_{0≤t≤T} is a family of generators of C_{0}–semigroups in D.

Now assume that

a) there exists μ ∈ ρ(A(t)) for every t ∈ [0, T],

b) t ↦ A(t) ∈ BV([0, T], L(D, X)) and ∥(μ I − A(t))^{−1}∥_{L(X,D)} is uniformly bounded for t ∈ [0, T].

Under these assumptions, if {A(t)|_{D}}_{0≤t≤T} results Kato–stable on D, then {A(t)}_{0≤t≤T} is Kato–stable on X.

Proof: analogous to Prop. 7.5 in Tanabe [7]; we use A_{μ}(t) = μ I − A(t) as isomorphisms between D and X, for 0 ≤ t ≤ T. From Prop. 7.2 in [7], {A(t)|_{D}}_{t} is a family of generators of C_{0}–semigroups; for the Kato–stability in D, (λ − A(t))^{−1} = A_{μ}(t)^{−1}(λ − A(t)|_{D})^{−1} A_{μ}(t) for large enough real λ.

Moreover, set P_{j} := [A_{μ}(t_{j})^{−1} − A_{μ}(t_{j−1})^{−1}] A_{μ}(t_{j−1}) and

K_{A} := max{ sup_{0≤t≤T} ∥A_{μ}(t)∥_{L(D,X)}, sup_{0≤t≤T} ∥A_{μ}(t)^{−1}∥_{L(X,D)}} < ∞  (14)

for assumption b). Let M_{D} and ω_{D} be the stability constants of {A(t)|_{D}}_{0≤t≤T}. Then

∥ ∏_{j=1}^{k}(λ I − A(t))^{−1}∥_{L(X)} = ∥ ∏_{j=1}^{k} A_{μ}(t) (λ I − A(t))^{−1}A_{μ}(t)^{−1}∥_{L(X)}

≤ K_{A}^{2} · ∥[ ∏_{j=2}^{k}(λ I − A(t_{j}))^{−1}(I + P_{j})] (λ I − A(t_{1}))^{−1}∥_{L(D)}

≤ K_{A}^{2} M_{D} e^{M_{D} K_{A} V} \frac{1}{(λ − ω_{D})^{k}} ,  (15)

where V is the total variation of t ↦ A_{μ}(t)^{−1} ∈ BV([0, T], L(X, D)) for Thm. 4, e) (for the last passage, see again the proof of Prop. 7.5 in [7]). Hence {A(t)}_{t} is Kato–stable in X.

Theorem 10 (Kato–stability for perturbations in L(D)) Let {A(t)}_{0≤t≤T} be a Kato–stable family of generators of C_{0}–semigroups in X. Assume that Hyp. 2 holds and

a) ∃ μ ∈ ℝ, such that, for every t ∈ [0, T]: μ ∈ ρ(A(t)) and

sup_{0≤t≤T} ∥(μ I − A(t))^{−1}∥_{L(X,D)} < ∞.  (16)

b) t ↦ A(t) ∈ BV([0, T], L(D, X)),

c) t ↦ B(t) ∈ BV([0, T], L(D)).

Then {A(t) + B(t)}_{0≤t≤T} is a Kato–stable family of generators of C_{0}–semigroups in X.

Remark 2 For the Kato–stability of {A(t)}_{t}, Ass. a) is equivalent to a1) ∃ μ_{0} ∈ ℝ, ∃ L > 0, such that, for every λ > μ_{0}:

sup_{0≤t≤T} ∥(λ I − A(t))^{−1}∥_{L(X,D)} ≤ L < ∞.  (17)

Proof: let M, ω be the stability constants of {A(t)}_{t}, A_{μ}(t) := (μ I − A(t)), K_{A} be given in (14) and K_{B} := sup_{0≤t≤T} ∥B(t)∥_{L(D)}.

From Prop. 7.5 in Tanabe [7], with Y = D and {A_{μ}(t)}_{t} as family of isomorphisms, we obtain the Kato–stability of {A(t)|_{D}}_{t} in D.

Then {A(t)|_{D} + B(t)}_{t} is a Kato–stable family of generators of C_{0}–semigroups in D (Prop. 7.4 in [7]). But {A(t) + B(t)}_{t} is a family of generators of C_{0}–semigroups in X; moreover, for λ large enough, then

(λ I − (A(t) + B(t)))^{−1} = A_{λ}(t)^{−1}

+A_{λ}(t)^{−1}B(t) A_{μ}(t)^{−1} ∑_{j=0}^{∞} (A_{λ}(t)^{−1}A(t) B(t) A_{μ}(t)^{−1})^{j} A(t) A_{λ}(t)^{−1}

(Desch & Schappacher [2]), and ∥(λ I − (A(t) + B(t)))^{−1}∥_{L(X,D)} is uniformly bounded in t ∈ [0, T] for the uniformly boundedness of ∥A_{μ}(t)^{−1}∥_{L(X,D)} and Remark 2.

From Prop. 9, {A(t) + B(t)}_{t} is a Kato–stable family of generators of C_{0}–semigroups.

Proof of Thm. 8: it follows immediately applying Thm. 10 and Theorem 7.3 in Tanabe [7] to the family {A^{⊕}(t)}_{t}, with A^{⊕}(t) = A(t) + B(t) and B(t) = (A(t)^{−1})′ A(t).

Corollary 11 If Hypotheses of Thm. 8 hold and M, ω are the stability constants of {A(t)}_{0≤t≤T}, then the stability constants M^{⊕}, ω^{⊕} of {A^{⊕}(t)}_{0≤t≤T} can be evaluated by

M^{⊕} ≤ M K_{A}^{4} e^{M K_{A} [V_{A}+K_{A}^{2} V_{A_{−1}} e^{M K_{A} V_{A}}]},  (18)
ω^{⊕} ≤ ω + M K_{B} K_{A}^{2} e^{M K_{A} V_{A}},  (19)

where K_{A} is given by Eq. (14) with μ = 0, V_{A} = ∫_{0}^{T} ∥A′(t)∥_{L(D,X)} dt,

V_{A_{−1}} = ∫_{0}^{T} ∥(A^{−1}(t))′∥_{L(X,D)} dt ≤ K_{A}^{2} V_{A},

K_{B} = sup_{0≤t≤T} ∥(A(t)^{−1})′ A(t)∥_{L(D)} ≤ K_{A} sup_{0≤t≤T} ∥A′(t)∥_{L(D,X)}.

Finally, ∥U(t, s)∥_{L(X)} ≤ M e^{ω (t−s)} and ∥U^{⊕}(t, s)∥_{L(X)} ≤ M^{⊕} e^{ω^{⊕}(t−s)}, for 0 ≤ s ≤ t ≤ T.

Proof: it follows from Eq. (15), Prop. 7.4 and Eq. (7.22) in [7].

3.3 An Example

Example 1 Take X = Y × Y, where Y is a Banach space. let B be the generator of a C_{0}–semigroup on Y, with 1 ∈ ρ(B), C(·) ∈ C^{1}([0, T], L(Y)) and

A(t) = ( \begin{matrix}−I & C(t) B \\ 0 & −(I − B) \end{matrix} ) .  (20)

For x = ( \begin{matrix}y_{1} \\ y_{2} \end{matrix} ) ∈ D(A(t)) = Y × D(B), we obtain:

U(t, s) x = e^{−(t−s)} ( \begin{matrix}y_{1} & ∫_{s}^{t} C(r) B e^{B(r−s)}y_{2} dr \\ 0 & e^{B(t−s)}y_{2} \end{matrix} ) ,

and, integrating by part,

U(t, s) = e^{−(t−s)} ( \begin{matrix}I & C(t) e^{B(t−s)} − C(s) − ∫_{s}^{t} C′(r) e^{B(r−s)}dr \\ 0 & e^{B(t−s)} \end{matrix} ) , (21)

defined on the whole space X = Y × Y.

On Y × D(B^{2}), U^{⊕}(t, s) = A(t)^{−1}U(t, s) A(s) is given by

U^{⊕}(t, s) = e^{−(t−s)} ·

· ( \begin{matrix}I & −C(s) B + C(t) B e^{B(t−s)} + ∫_{s}^{t} C(r) e^{B(r−s)}B (I − B) dr \\ 0 & e^{B(t−s)} \end{matrix} ) .

You can observe that here integrating two times by part needs C(·) ∈ C^{2}([0, T], L(Y)), i. e. Hyp. 3. Assume this condition and obtain:

U^{⊕}(t, s) = e^{−(t−s)} ·

· ( \begin{matrix}I & (C + C′)(t) e^{B(t−s)} − (C + C′)(s) − ∫_{s}^{t}(C + C′)′(r) e^{B(r−s)} dr \\ 0 & e^{B(t−s)} \end{matrix} ) .

This is exactly the expression of the evolutionary process generated by

A^{⊕}(t) = ( \begin{matrix}−I & (C(t) + C′(t)) B \\ 0 & −(I − B) \end{matrix} ) .  (22)

4 Intrinsical Sense of the Co–process

The co–process appears interesting to define a generalized solution for Eq. (1). But U^{⊕}(t, s) is also related with the global behaviour of A(·).

We will show that U(t, s) = U^{⊕}(t, s) if and only the generator A(t) is constant. But, also if the derivation of the co–process from a process is unique, the converse generally does not hold.

We will obtain also an estimate of the difference U(t, s) − U^{⊕}(t, s), in relation with A(·).

4.1 Constant case

Theorem 12 Assume Hyp. 1–3, and let {A(t)}_{0≤t≤T} a Kato–stable family of infinitesimal generators of C_{0}–semigroups. Then A(·) = constant on [0, T] if and only if U(t, s) = U^{⊕}(t, s) for 0 ≤ s ≤ t ≤ T.

Proof: If A(·) is constant, the result is immediate. Vice versa, assume that U(t, s) = U^{⊕}(t, s): hence A(t) U(t, s) x = U(t, s) A(s) x for each x ∈ D that is

(\frac{∂}{∂t} + \frac{∂}{∂s}) U(t, s) x = 0,

therefore U(t, s) x = C(t − s, x), that is, for the linearity and the uniform boundedness, U(t, s) = C(t − s), hence A(t) = constant.

In fact, U(t, s) x = C(t − s) x implies \frac{∂}{∂t} U(t, s) x = \frac{∂}{∂t} U(t − s, 0) x = A(t − s) U(t − s, 0) x = A(t − s) U(t, s) x ≡ A(t) U(t, s) x, for each x ∈ D and 0 ≤ s ≤ t ≤ T. For t = s, you obtain A(t) = A(0) for every 0 ≤ t ≤ T.

4.2 Non–Uniqueness of the Inversion

The derivation of the co–process U^{⊕}(t, s) from U(t, s), in the hypotheses of Thm. 8, is unique and well-defined, but the converse, in general, is not true.

More precisely, suppose to have two co–processes U^{⊕}(t, s) and V^{⊕}(t, s), obtained respectively by two evolutionary processes U(t, s) and V (t, s), whose families of generators fit Thm 8. Then, in general,

U^{⊕}(t, s) = V^{⊕}(t, s) ⇏ U(t, s) = V (t, s) or A(t) = B(t).

Example 2 (Counterexample) A^{⊕}(·) = 0 ⇔ U^{⊕}(t, s) = I but

A(t) = −(t I + C)^{−1} and U_{C}(t, s) = (t I + C)^{−1}(s I + C),

where C is a suitable operator in L(X).

In particular, for C = λ I, λ > 0, we obtain:

U_{λ}(t, s) = \frac{s + λ}{t + λ}, for 0 ≤ s ≤ t ≤ T.

In this case we obtain an infinity of processes U(t, s) with the same U^{⊕}(t, s).

Remark 3 This lack of uniqueness may contribute to explain some difficulties to apply techniques of extrapolation spaces to U^{⊕}(t, s) (see [3], Chp. II.5). Another fact is that D(A) = D(B) does not imply X_{−1}(A) ≡ X_{−1}(B), but only that they are isomorphic.

Remark 4 We have in general A(t) U^{⊕}(t, s) = U_{A(·)}(t, s) A(s), where the same co–process U^{⊕}(t, s) works for an entire class of evolutionary processes, {U_{A(·)}(t, s)}_{0≤s≤t≤T}. Then U^{⊕}(t, s) appears like a sort of commutator for this class.

4.3 Variation and Estimates

Proposition 13 (Estimate) Let Hypotheses of Thm. 8 hold, with stability constants M and ω. Then there exist K_{1}, K_{2} > 0, depending on T, M, ω, sup_{s≤t≤T} ∥A(t)∥_{L(D,X)} and sup_{s≤t≤T} ∥A′(t)∥_{L(D,X)}, such that, for 0 ≤ s ≤ t ≤ T,

∥U^{⊕}(t, s) − U(t, s)∥_{L(D)} ≤ K_{1} ∫_{s}^{t} ∥A′(r)∥_{L(D,X)} dr,  (23)
∥U^{⊕}(t, s) − U(t, s)∥_{L(X)} ≤ K_{2} [∥A′(s)∥_{L(D,X)}
+ ∫_{s}^{t} ∥A′(r)∥_{L(D,X)} dr + ∫_{s}^{t} ∥A′′(r)∥_{L(D,X)} dr].  (24)

Proof: fixed any x ∈ D, Eq. (23) is given by

U^{⊕}(t, s) x − U(t, s) x = ∫_{s}^{t} U^{⊕}(t, r) (A(r)^{−1})′ A(r) U(r, s) x dr.  (25)

Applying Corollary 11, Eq. (23) follows.

Hence, integrating by part ∫_{s}^{t} U^{⊕}(t, r) (A(r)^{−1})′ \frac{∂}{∂r}[U(r, s)] x dr in Eq. (25), obtain an expression that can be extended by density for every x ∈ X , that is

U^{⊕}(t, s) − U(t, s) = (A(t)^{−1})′ U(t, s) − U^{⊕}(t, s) (A(s)^{−1})′ −
− ∫_{s}^{t} U^{⊕}(t, r) [A(r) (A(r)^{−1})′ + (A(r)^{−1})′ A(r) (A(r)^{−1})′
+(A(r)^{−1})′′] · U(r, s) dr.  (26)

Example 3 Take Example 1 of Sec. 3.3, with ∥e^{tB}∥_{L(Y)} ≤ M e^{ωt}, C ∈

C^{2}([0, T], L(Y)). Then A′(t) = ( \begin{matrix}0 & C′(t) B \\ 0 & 0 \end{matrix} ) ,

U(t, s) − U^{⊕}(t, s) =

= e^{−(t−s)} ( \begin{matrix}0 & −C′(t)e^{B(t−s)} + C′(s) + ∫_{s}^{t} C′′(r) e^{B(r−s)} dr \\ 0 & 0 \end{matrix} ) ,

∥U^{⊕}(t, s) − U(t, s)∥_{L(X)} ≤ (1 + k) ∥C′(s)∥_{L(Y)} + 2k ∫_{s}^{t} ∥C′′(r)∥_{L(Y)} dr, where k = M e^{|ω|(T −s)}.

References

[1] S. Bertoni A Functional Approach to Non-Autonomous Abstract Differential Equations: the Parabolic Case, Research Report No. 98–CNA– 005, preprint, Carnegie Mellon University of Pitttsburgh, Pittsburgh (PA, USA, 1998).

[2] W. Desch, W. Schappacher, Some Generation Results for Perturbed Semigroups, in Semigroup theory and applications, P. Clement, S. Invernizzi, E. Mitidieri and I. Vrabie (eds.), Dekker (1989), pp. 125–152.

[3] Klaus–Jochen Engel, Rainer Nagel & others, One–parameter Semigroups for Evolution Equations, to be published.

[4] G. Nickel, Evolution Semigroups for Nonautonomous Cauchy Problems, preprint, Tübingen, (1996).

[5] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York–Berlin (1983).

[6] H. Tanabe, Equations of Evolutions, Pitman, New York Inc., 1979.

[7] H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, Inc., 1997.