Acta Mathematica et Informatica Universitatis

Acta Mathematica et Informatica Universitatis Ostraviensis

Danilo Rastović A note on stability properties of integrated semigroups Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 3 (1995), No. 1, 61--(65)

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Acta Mathematica et lnformatica

Universitatis Ostravieiisis 3(1995)61-65

61

A note on stability properties of integrated semigroups DANILO RASTOVIČ

A b s t r a c t . Asymptotic stability of a certain class of integrated semigroups is discussed by means of Lyapunov functionals. 1991 M a t h e m a t i c s S u b j e c t

1

Classification:

Introduction

We consider t h e a b s t r a c t C a u c h y p r o b l e m , ( A C P ) : u = Au(t),

t > 0;

ti(0) = x

in a Hilbert space H a n d discuss t h e a s y m p t o t i c stability of t h e i n t e g r a t e d semi­ group S(t) associated with ( A C P ) . D e f i n i t i o n 1. Let S(t) be a n o n d e g e n e r a t e N-times i n t e g r a t e d s e m i g r o u p on H such t h a t \S(t)\ < M exp(u;l) for t > 0, some w £ R a n d some M > 1. If a closed linear o p e r a t o r A h a s t h e resolvent H(lx; A) a n d satisfies />oo

R(f.i; A) — fjtN

/ Jo

t h e n A is called t h e generator

exp (—p,t)S(t)xdt

for /t > LJ a n d x £ H,

of 5 ( i ) .

We need t h e following p r o p o s i t i o n s . P r o p o s i t i o n 1. (F. Neubrander) By a solution of (ACP) is meant an H-valued, strongly continuous function u(-) : [0, + o o ) —> H which satisfies the evolution equation (DE) for t > 0 and the initial condition (IC). If A is the generator of a nondegenerate, exponentially bounded N-times integrated semigroup S(t) on H, then (ACP) is well-posed in the sense that there exist constants M and u such that for each x £ D(AN+1) there exists a unique solution u(-) such that \u(t)\ < M exp (u)t)\x\N, where \X\N denotes the graph norm of the space D(AN) defined by \x\N = |a?| + | i x | + ... +

\ANx\.

62

D. Rastovic

Proposition 2. (S. Oharu) (a) The space D(AN) is a Banach space under the graph norm \ • \N- We write Y for the Banach space, namely, Y =

\D[A

o]

(b) Since A is the generator of a nondegenerate, exponentially bounded N-times integrated semigroup S(t) on H, A has a nonemty resolvent set p(A). (c) If D(A) is dense in H, D(A ) is dense in D(AJ) for every pair of nonnegati ve integers j and k with j < k. In particular, Y is dense in H.

2

On stability properties of integrated semigroups

Given a (Co) semigroup T(t) on a Banach space X, a continuous functional V : X —> R is called a Lyapunov functional for T(t), if V(0) = 0 and V(x) = l i m s u p r ^ ( T ^ ) * ) - V(x)) < 0

for x E X.

£10

Moreover, a continuous functional V : X —•> R is said to be quadratic if there exists a bounded linear operator B from X into its dual space X* such that (Bx)(y) = (By)(x)

for x, y £ X

and V(x) = (Bx)(x) for x E X. If a Lyapunov functional V for T(t) is quadratic, we say that V is a quadratic Lyapunov functional for T(t). It is the main feature of the argument of this paper to use quadratic Lyapunov functionals to discuss asymptotic stability of integrated semigroups. Definition 2. We say that a (Co-)-semigroup T(t) on a Banach space X is asymptotically stable, if T(t)x -> 0 in X as t —> oo for each x G N, and that an N-times integrated semigroup S(t) on H is asymptotically stable, if (N\/tN)S(t)x —> 0 in H as t —> oo for each x E H. In what follows, we put the following conditions on the operator A.: (A) The operator A has a dense domain D(A) in H and is the generator of a nondegenerate, exponentially bounded N-times integrated semigroup S(t) on H. (B) There exists a bounded linear operator B from the Banach space Y into H with the three properties below: (-92/1, 2/2)// = (£"2, 2/1 ) # = (2/1, 5y2)Lr (Hy, y)# > 7 |7/| 2

for yu y2 e Y,

for g G y and some 7 > 0,

(Bl)

(B2)

A note on stability properties . . .

Re(By,

(A - uI)y)H

< 0

for y eY

63

and some to G R. N

An inner p r o d u c t (•, -)x is defined on the subspace D(A ) (yu V2)x = (Byu In fact, (y, y)x

— (By, Also

N

y2 G D(A ,

and (g, y)x

(y2, yx)x

> 0 for y G D(AN)

and y = 0 by (B2). For yu (.Vi, .y2>x °y ( B 1 ) -

y-2)n

(B3)

z)x

= 0 implies \y\ = 0 yx)H

= (Byu

+ a2(y2)

z)x

y 2) H

=

for

yu

of H

One can then define a n o r m | • \x on D(AN) by \y\x = (y, y ) x • Let X be a completion of D(A ) with respect to the n o r m | • | x - T h e n the inner p r o d u c t (•, -)x is n a t u r a l l y induced on X and X becomes a Hilbert space. From (B2) it follows t h a t X can be regarded as a dense subspace of H a n d is continuously embedded in II. On the other h a n d , B is a b o u n d e d linear o p e r a t o r from Y into /I, and so the Banach space Y is continuously embedded in X. Therefore X is an intermediate Hilbert space in the sense t h a t Y X ^> II. T h e part A of A in X is defined by D(A)

= {x e D(A) n X; Ax G X},

Ax = i _

for „ G D ( A )

and has the following properties: P r o p o s i t i o n 3 . (a) The operator A is a densely defined, closed linear operator in the Hilbert space X. (b) The subspace D(AN+l) is a core of the operator A in X. (c) Let to be the constant appearing in (B3). Then A — u is dissipative in X. (d) The range condition holds in the sense that R(i — XA) = X for A > 0. From Proposition 3 and the Lurner-Phillips theorem it follows t h a t A generates a (C r o)-semigroup T(t) on A" such t h a t | T ( l ) | < exp (ut) for t > 0 and the constant UJ G R appearing in condition (B3). If UJ > 0, the semigroup T(t) is said to be quasi-contractive; if u < 0, it is said to be a contradiction s e m i g r o u p . Now the relationship between the (Co)-semigroup T(t) on X and the original integrated semigroup S(t) on H m a y be stated as follows: P r o p o s i t i o n 4 . Let T(t) be a (Co)-semigroup on X generated (a) For x G X and t > 0, S(t) maps X into itself and S(t)x

-

\ dliv_i / Jo Jo

• • • dti

I Jo

T(s)xds

where the integral is taken in the sense of Bochner

by A.

for t > 0 and x G K, and in the Hilbert space

X.

64

D.Rastovic

(b) For each t > 0, let S(t) be the restriction of S(t) to the Hilbert space X. Then S(t), t > 0, form a nodegenerate, N-times integrated semigroup on X and A is its generator. Moreover, S(t) is exponentially bounded in the sense that \S(t)x\x

< (N\)~HN

exp(u>t)\x\x

fort

> 0 and x E X.

Applying P r o p o s i t i o n s 3 a n d 4, we o b t a i n the following result concerning a s y m p ­ totic stability of S(t). P r o p o s i t i o n 5. Let to be the constant appearing in (B3) and assume that to < 0. (a) The Co-semigroup T(t) on the Hilbert space X is exponantially stable. N (b) Suppose that sup{r N\\S(t)x\H : t > 0} < -f-oo for x E H. Then S(t) is asymptotically stable on the Hilbert space H. P R O O F : (a) T h e result of Walker [3] implies t h a t t h e search for a q u a d r a t i c Lyapunov functional reduces to a search for a b o u n d e d linear o p e r a t o r B : X —> X* such t h a t (Bx)(y) = (By)(x) for all x, y E X a n d Re(Bx)(Ax) < 0 for all x E D(A), where A is the infitesimal generator. Now, the result follows from Walker's t h e o r e m T . 4 . 1 . [3]. (b) We consider the integral (c - A)N / 0 diN_x f*"'1 • •dt1 f*1 T(s)R(c, A)N xds, for some c > u), t > 0. According to t h e results of T h i e m e for N — 1 a n d Nicaise [6] for generally N, S(t) will be N-times i n t e g r a t e d s e m i g r o u p on X, and S(t) will be N-times integrated s e m i g r o u p on H. These results follow if we apply the t r a n s f o r m a t i o n s between integrated semigroups and C-semigroups with N C — R(c, A) as it is done in T a n a k a , M i y a d e r a [1] and M i y a d e r a [7]. Under the condition sup {t~ N N\\S(t)x\n '• t > 0} < + c o for x E H, from t h e P r o p o s i t i o n 4 follows t h a t S(t) is a s y m p t o t i c a l l y stable on t h e Hilbert space H. •

References [1] Tanaka, N., Miyadera I., Some remarks on C-sєniгgroups and Integrated Proc.Japan Acad. 6 3 (1987), 139-142.

Semigroups,

[2] Oharu S., Senгigroups of Linear Opєrators гn a Banach Space, Publ. RIMS Kyoto Univ. 7 (1971/72), 205-260. [3] Walker, J.A., On thє Application of Lyapunov Direct Method to Linєar Systems, Journcil of Math. Anal. AppL 53 (1976), 187-220. [4] Neubrandeг, F., Intєgrated Semigroups and thєir Applications ProЫem, Pac. J . M a t h . 1,135 (1986), 111-155. [5] Tanaka, N., On thє єxponentially (1987), 107-117.

Boundєd

C-sєnúgroups,

Dynamical

to the Abstract

Tokyo J . M a t h .

[6] Nicaise, S., Thє Hillє-Yosida and Trottєr-Kato thєorems for Intєgratєd Journal of Math. Anal. Appl. 180 (1993), 303-316.

Cauchy

1,10

Semigroups.

[7] Miyadera, I., On the Generators of Proc. Japan Acad. 6 2 A (1986), 239-242. [8] Okazawa, N., A Generation Math. J. 26 (1974), 39-51.

Exponentially

Theorem for Semigroups

Bounded

C-semigroups,

of Growth Order o;, Tohoku

[9] Thieme, H.R., Integrated Semigroups and Integrated Solution to Abstract Problems, Journal of Math. Anal. Appl. 152,2 (1990), 416-447. Address: Danilo Rastovic, Higher Technical School, University of Zagreb Konavljanska 2, Zagreb, Croatia

Cauchy