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Equivalence of Lp stability and exponential stability for a class of nonlinear semigroups
Vol. 8. No. 7. pp. 805415.
0362-%5&W S3.09 + @I Q 19.8~ Pergamon Press Ltd.
1984
EQUIVALENCE OF L, STABILITY AND EXPONENTIAL STABILITY FOR A CLASS OF NONLINEAR SEMIGROUPS* AKIRA ICHIKAWA Shizuoka University, Hamamatsu 432, Japan
Faculty of Engineering,
(Received 20 January 1983; in final form 30 October 1983; received for publication 1.5November 1983) Key words and phrases: Liapunov’s theorem, stochastic evolution equations.
semigroups,
Liapunov stability, exponential
stability,
1. INTRODUCTION
Let S(t), t 5 0 be a strongly continuous semigroup of bounded linear operators on a Banach space Y and A its infinitesimal generator. Then S(t)yo corresponds to the differential equation dy/dt = Ay,
Y(0) =yo*
(1.1)
Pazy [12] has shown the following. THEOREM
1.1. The two statements
(Dl) i,‘IS(t)y/pdf
below are equivalent:
< a, yE Y for somep>
1.
(D2) IS(t)/ s Me-“, I 2 0 for some M 2 1 and a > 0. Here we denote by 1 1norms of vectors and operators. This result was originally proved by Datko [3] for Hilbert spaces and p = 2. In this case there is yet another equivalent statement: (D3) There exists a nonnegative operator P 5 0 in 9(Y) such that 2Re(Py,Ay)
= -Iyl’,
Y E z(A).
(1.2)
Above 9(Y) denotes the space of bounded linear operators on Y, 9(A) the domain of A and ( , ) is the inner product in Y. In finite dimensions this is the well-known Liapunov theorem and can be found for example in [5]. Zabczyk [16] relaxes (Dl) replacing IS(t)y lp by m( IS(t)y I), where m is a strictly increasing convex function with m(0) = 0. It is also equivalent to a stronger one:
Pl’)
XIS(f)YIPdt~KlyIP,
It is natural to ask if the equivalence system
yEYforsomepslandK>O. still holds for the nonlinear semigroup of a perturbed
dy/dt = AY +f(r),
y(0) =YO,
(1.3)
* This work was carried out while the author was visiting the Department of Mathematics, University of British Columbia and was supported by the Natural Sciences and Engineering Research Council of Canada under grant A8051. 805
A.
806
I~HIKXW.A
wherefis Lipschitz continuous withf(0) = 0. We consider solutions in the mild sense, namely, as solutions of the associated integral equation r(t) = S(f)Yo +
j0
‘s(r - u)f(y(u))
(1.4)
du.
For this equation one can show the following [14]. PROPOSITION1.1. For each y. E Y (1.4) possesses a unique strongly continuous y(r; ~0). Moreover, Iy(t; yo) 1s Nebtjyo/, y. E Y for some N 2 1 and b > 0.
solution
We can pose a similar question for stochastic evolution equations. Let Y and H be Hilbert spaces. We assume for simplicity that they are real and separable. Let (S2, 9. u) be a complete probability space and let w(t) be a Wiener process in H with incremental covariance operator W [l]. Let 9, = a(w(s), 0 c s c t), the u-algebra generated by w(s), 0 c s c f. Consider the linear stochastic evolution equation dy = Ay dt + G(y) dw(t),
y(0) =yo E Y,
(1.5)
where A is the infinitesimal generator of a strongly continuous semigroup S(r) on Y and G E 5?( Y, .2?(H, Y)). A stochastic process y(t) is said to be a mild solution of (1.5) if y(t) is 9!-adapted and satisfies the associated integral equation Y
(t> = S(f)Yo +
‘S(r- u)G(y(u))
dw(u).
(1.6)
The stochastic integral with respect to w(t) is defined as in [l, lo]. One can show [ 10, 111: PROPOSITION1.2. There exists a unique mild solution y(t; yo) to (1.5) which is continuous in the pth mean, p > 0. Moreover, for each p > 0 there exist positive numbers iV = N(p) and 6 = b(p) such that Ely(t; ~0) lp s N ebrlyOIP,
for anyyo E Y,
t 2 0.
A stochastic analog of Datko’s result is known [8, 7, 171. THEOREM 1.2. Let y(t;yo) equivalent: (Sl) j=
Ely(t;yo)
be the mild solution of (1.5). Then three statements
below are
I*dt <“o;
y. E Y for some M > 1 and a > 0; (S2) ily(r; ya)12 ~Me-arlyo12, (S3) There exists a nonnegative operator 0 s P E 3(Y) such that 2 (Py,Ay)
+ tF. G*(y)PG(y)W
= - ly I*,
Y E Q(A).
(1.7)
This is also well-known in finite dimensions [15]. A nonlinear version of (1.5) may be the following: dy = [AY +f(y)l where
f : Y+
dt + G(Y) dw(t)>
Y and G : Y+ Z(H, Y) are nonlinear
Y(O) =YO,
and satisfy the Lipschitz
(1.8) condition
807
Equivalence of 15, stability and exponential stability for a class of nonlinear semigroups
together with f(0) = 0 and G(0) = 0. The associated integral equation is y(r) = S(~)YO + 6’s(f
- u)f(y(u))
du + 6’W
--4G(yW)
dw(u).
(1.9)
In this case proposition 1.2 still holds [lo, 111. The question is whether or not two statements below are equivalent for each p 2 1. (Sl’) 6’ Ely(r;yo)lPdr~KlyolP, (S2’) Ely(C
~0) Ip c M
YoEY.
e-"'lyOIP,
yoEYforsomeMalanda>O.
This equivalence is not available even in the linear case. The questions above are also important, as one may expect, from the point of view of Liapunov’s direct method. For there are some situations where one may easily find “Liapunov” functions which are not strictly positive but assure L, stability i.e., (Dl’) or (Sl’). To be more specific let P 2 0 satisfy (1.2). Suppose further that 2Re(Py,Ay
+f(y))~
y EZ(A)
-dlyl*,
forsomed>O.
Then applying the Liapunov argument to (1.3) and (Py, y). it is possible to obtain [9] (PY(CYo),Y(CYo)) ‘I2< N eAbr]yo] for some :V > 0 and 6 > 0, I
o~~y(t;yo).‘dtsK~yo~‘,
yoEYforsomeK>O,
where y(t; ~0) is the mild solution of (1.3). However, ]y(t;yo)]6Me-“‘jyo]
we cannot in general conclude that
forsomeMslanda>O.
In fact Pazy [12] has shown that if A is the generator of an analytic semigroup, (Py, y)@ cannot be equivalent to Iy]. We would have a similar situation for P, the solution of (1.7) and the system (1.8). Later we shall positively answer these questions. In fact we shall generalize the following result by Datko [4]. 1.3. Let s(t, s), 0 s s s t c 50 be a strongly continuous two-parameter semigroup of bounded linear operators on a Banach space Y. Then two statements below are equivalent.
THEOREM
(Gl) IX
lSCt,s)ylP df=sK(Y)
(G2) Ik(l, s) j < M e-a(r-s),
< ~0,
y EY,
t==ssO,
for somep 3 1.
t
If s(t, s) is generated by a bounded operator, this result is given in [2] for any p > 0. In Section 2 we extend theorem 1.3 with K(y) = KIy]J’, K > 0, but p > 0, to nonlinear semigroups satisfying an additional condition. We also give a modification to cover stochastic systems when p < 1. Then the equivalence of (Dl’) and (D2’) for (1.3), and that of (Sl’) and (S2’) for (1.8) follow from our general results. Discussions on this will be given in Section 3. We can apply our results to ordinary and stochastic nonlinear differential equations.
808
A. ICHIKAWA 2. THE
MAIN
RESULTS
Now we extend theorem 1.3 to a class of nonlinear semigroups. Let 2 be a Banach space and let r(t, s), f 2 s 3 0, be a family of nonlinear operators with domain Y, C Z with properties:
t as,
T(r, J)Ys C Y,,
t 30,
T(t, t)Y = Y? T(t, K) T(K,
S)
(2.1.1)
= qt,
S)
y E y,
on Y,,
(2.1.2) S SK
(2.1.3)
St,
T(. , s)y is continuous on [s, =) for each y E Y,
(2.1.4)
We shall use the following simple result. LEMMA 2.1.
Let
nL ~2t 6 (n + l)L
0 < r < 1, L > 0 and let n be implies e-” 6 rn S (l/r)e-“‘, a = -(log
a
nonnegative
integer.
Then
r)/L > 0.
Proof. Note that rR = enlog’and log r < 0.
The theorem below is a nonlinear version of theorem THEOREM 2.1. Let g be a positive continuous following:
Then two statements
function on [0, ~0). Suppose T(t, s) satisfy the t 3s.
Y E Y,,
(B) I T(r, S)Y / s g(t - 3) 1~19
1.3.
below are equivalent:
(Nl) ~x~~(r,s)ylPd~~KP~~lP,
YE
(N2) l;(r, s)y I s M e-“(‘-‘)ly
/,
ys,
tss~Oforsomep>OandK>O; t 2s ~0 forsomeM
y E y*,
11 anda >O.
Proof. It is enough to show that (Nl) implies (N2). We generalize the proof of [13, theorem 3 41 slightly. For any 0 < s < t and y E Y, we have 1T(t, s)y lp I’g-p(t *
- K) dK = /‘g -P(t - K) 1T(t, s)y 1’ du s =
G
=
‘g-p(t
I3
- K) IT(t,
‘g-P(t -
Is II
K)g”(t
K)T(K,S)y
-
K) 1T(K,
lpdK by (2.1) S)J’
jp
du by (B)
jT(K,S)ylPdK
s KPlylP by (Nl).
(2.2)
809
Equivalence of L, stability and exponential stability for a class of nonlinear semigroups
Let i > 0 be arbitrary but fixed. Define I by JP =
I
,,‘g -P(u) d u.
Then J > 0 and for any r - s 5 L we have from (2.2) 1T(t, s)y 1S (K/J) ly I. This together with B implies the existence of R > 0 such that IT(t,s)y/~RlyIforanyt2s~OandyEY,.
(2.3)
Now let t > s, y E Y, and consider (~--W(v)ylP=
[h4ylPdu s RP
iI
’ / ~(u, s)y IP du by (2.1) and (2.3)
6 (RK)Ply IPby (Nl). Hence it follows that 1T(r, s)yl d KRJyl/(t choose a number L = L(r) > 0 such that
- s) l/P, t > s , y E Y,. So for each 0 < r < 1 we can
y E Y, whenever t - s * L.
lWJ)Yl~~lYl~
(2.4)
Now let t - s 2 L. Then there is an integer n 2 1 such that nL s t - s < (n + l)L. Using the semigroup property (2.1.3) and (2.4) n times and then (2.3), we obtain I T(t, s)yl c r”Rlyt, y E Y,. Now lemma 2.1 yields
I TO, S)Y I G ~e-“ly
I,
y E Y,foranyt
--s SL,
where M = R/r and a = -(log r)/L > 0. Combining this and (2.3) we conclude that
IW,s)yI SMe-“‘lyl,
YE Y,,
t as,
where M = max(M, ReaL). COROLLARY 2.1. Suppose that Y, = Y,,, t 2 0 and T(r, s) = T(t - s) for some one-parameter family T(a) which inherits (2.2) and B. Then two statements below are equivalent:
(Nl’)
I,= I T(~)Y lp dt s KPly Ip,
(N2’) I Ut)y 1s Me-“lyl,
y~Y~forsomep>OandK>O;
yEYoforsomeM?=landa>O.
If we specialize these results to linear operators, 1.1 and 1.3. COROLLARY
2.2.
For any p > 0 theorem
Remark 2.1. Relaxing
measurability linear case.
we have some improvements
in theorems
1.1 and theorem 1.3 remain valid.
the continuity assumption on T(t, s) and g, we may only assume and local boundedness. A typical example of g is Nebr, N > 0, b > 0 as in the
A.
810
ICHIKAWA
Remark 2.2. The condition (Nl) can be replaced by (Nl’) J;” h( / T(t. sjy 1) dr 6 Kh( iyi), K > 0 for some strictly increasing continuous function h which satisfies h(O) = 0, lim h(r) = I-f = and h(st) G k(s)h(t), t c 0, s s 0 for some function k of the same type. We do not require convexity as in [16]. The theorem above can be applied to stochastic systems if p 5 1, but for 0 < p < 1 we need some modification. This is because p-integrable random vectors do not form a Banach space ifp
T(r, s)Y, c Y,, T(t, tjy T(t,
t 3 0,
= y,
2.4)T(u,
(2.5.1)
t as,
s) = T(r,
(2.52)
y EY*,
s)
on Y,,
sSuSt,
(2.5.3)
T(. , s)y for each s 2 0 and y E Y,, is measurable on [s, ~j) x R.
(2.5.4)
THEOREM2.2. Let g be a positive continuous function on [0, ~0) and let p > 0. Suppose T(r, s) satisfy the following: (B’) ElT(r,s)zIP~g(r-s)E/zIP,
r as,
z
EYXp) ‘{Z EY,:EjzlP<
=}.
Then two conditions below are equivalent: (Rl) _/‘ElT(t,s)rlPdr
z E Y&J) for some K > 0;
< KElzlP,
(R2) ;I T(r, s)z lp =z Me-“(‘-‘)E
Iz lp,
z E Y,(p) forsomeM
2 1 anda >O.
The proof of theorem 2.1 goes through with simple modifications. As we see in [ll], we often establish T(r, s) as a map from Y,(q) into Y,(q) with q 3 2. In this case we have: COROLLARY2.3. Suppose T(r, s) : Y,(q) ---, Y,(q) for some q > 2 and let 0
0. Remark 2.3. If T(r, s)y, y E Y forms a Markov process with stationary transition probabilities,
then we may replace z in (B’) by nonrandom y E Y. 3. APPLICATIONS We give three applications of our main results. Here we shall not try to give conditions as general as possible. We shall also omit details of proofs. 3.1. Semilinear evolurion equations Here we consider the evolution equation (1.3): dy/dt
= Ay +f(r>,
Y(O)
=yoEY,
Equivalence of L, stabilirr and exponential stability for a class of nonlinear semigroups in
811
Hilbert space Y under the stated conditions.
PROPOSITION
a nonnegative
3.1. Let f: Y- Y be Lipschitz continuous function u(y) on Y with properties
with f(0) = 0. Suppose there exists
(i) u(y) s c/y/P, y E Y for some c > 0 and p > 0, (ii) u(y) is F rCch et d’ff I erentiable except possibly at y = 0 and (u,(y), Ay +f(y)) yEa forsome d>O.
c -dlyjP,
Then ly(t; yO) 1< Me-” 1~01,yo E Y for some M 2 1 and a > 0, where y(t; yO) is the mild solution of (1.3). The condition B is assured by proposition 1.1 for T(t)yo = y(r; ~0). Thus we can apply corollary 2.1. But to show (Nl’ ) we cannot use the Liapunov argument directly to (1.3) because we are dealing with mild solutions. So we approximate them by differentiable functions which satisfy dy/dr
= Ay + h,(t),
y(0) = Y, E a(A),
(3.1)
where h,(t) + y(t; yo) uniformly on compact intervals andy, ---, yo. We next apply the Liapunov argument to (3.1) and u(y). Then passing to the limit n--, 30 we obtain (Nl’) [9]. We may equally use dy/dt
= AY + R~f(y),
Y(O) =RAYO
(3.2)
where R* = ,IR(,I, A) and R(L, A) is the resolvent of A [9]. COROLLARY
3.1. Suppose there exists a nonnegative
2WPy,Ay +f(y))
G
-dlyl*,
operator
0 < P E Z(Y)
y E Ed(A) forsomed
such that
>O.
Then Iy(6~0)
I6
yoEYforsomeMZ1
Me-%0I,
anda >O.
Remark 3.1. The function u(y) = (Py, Y) plays the role of a Liapunov (Py, y)@ is not equivalent to jyl. Remark 3.2. In view of corollary
2RePy,Ay)
function even though
3.1 we may replace (D3) by s -4~
12,
yEa(A)forsomed>O.
Remark 3.3. If we take Y = R” and A = 0, then (1.3) is a nonlinear
differential
equation.
3.2. Stochastic differential equations [6, 151
Consider a stochastic differential dy =f(r) where f:R”-*R”, w(t) is a standard
equation
dt + G(Y) dw(0,
Y(O) =~a ER”,
(3.3)
G(.):R”+R”“” are Lipschitz continuous with f(0) = 0, G(0) = 0 and Wiener process in R”. There is a unique solution y(t; yo) and for each
p >
0, .%a Yo)Ip < NebijyolP for some N = N(p) 2 1 and 6 = b(p) > 0. In this case one can
show: PROPOSITIONS 3.2. Suppose there exists a nonnegative the origin) function u(y) on R” such that U(Y) + IYIIUY(Y)I + IYI21 UYV(Y)I c c IY I py
WY)
twice differentiable
y ER”forsomec
(except possibly at
>O andp >O,
y E R n for some d > 0,
s -d IY Ip,
where ?E is the differential MU
generator = (1)
(3.4.1) (3.4.2)
of (3.3):
tr . G(Y)G
r(y)uyY(y) + (u,(Y),~(Y) ),
uyy is the Hessian of u(y) and tr. denotes the trace of a matrix. Then .W~YO)
lp 6 Me-(2fIy01P,
yoER”forsomeM>l
anda >O.
(3.5)
We shall show how to apply theorem 2.2. For each 9;,-measurable random variable z with IzI < m w.p.1. we define T(t, s)z = y(t; s, z) where y(t; s, z) is the unique solution of dy =f(y)
dr + G(y) dw(r),y(s)
= z.
(3.4)
Let Y, be the space of n-dimensional random vectors which are 9;,-measurable and finite w.p.1. Then it is known that T(t, s)Y, C Y, and T(t, s)z is continuous on [s, =) w.p.1. If ElzjP < ~0, then Ito’s lemma applied to u(y) and (3.4) yields
iI
nEIT(r,s)zI”dts(c/d)ElzlP.
(3.7)
Thus by theorem 2.2 we obtain El T(t, s)z 1~c Me-“‘El z IP for some M 5 1 and a > 0. If, in particular, s = 0 and z = yo E R”, then this yields (3.5). If we assume p 2 1, then we may apply theorem 2.1 directly to obtain (3.5). In this case we take Y, the space of p-integrable 9,-measurable random vectors. Then T(r, s) : Y, -+ Yt and if we denote by I / the norm in 2 = L, (Q, 8, ,u; R”), then all conditions in theorem 2.1 are satisfied. On the other hand (3.4) assures (3.7) and hence (3.5) as well. 3.3 Semilinear stochastic evolution equations As a final application we consider (1.8) with the stated conditions. It is usually convenient to establish a solution in C([O, T]; L,(R, 9, CL,Y)), q 3 2 [lo, 111. So let Y, = Y,(q) be the space of 9”,-measurable q-integrable random variables in Y for some q 5 2. Define r(t, s)z = y(t; s, z) where y(t; s, z) is the unique mild solution of dy = [AY +f(r)]
dr + G(Y) dw(t),y(s)
=z EY,.
(3.8)
Then y(t; s, z) forms a Markov process and T(t, s) satisfies (2.5) [ 111. Thus sufficient conditions for L, stability assure exponential stability as well. PROPOSITION3.3. Let f: Y+ Y and G: Y-, y(H, Y) be Lipschitz continuous with f(0) = 0 and G(0) = 0. Suppose there exists a nonnegative twice FrCchet differentiable (except possibly
813
Equivalence of L, stability and exponential stability for a class of nonlinear semigroups
at the origin) function o(y) on Y such that
O(Y) + lYlby(Y)l Zu(y)
+ lY12b,,(Y)l
= b,(y),Ay
y E Y forsomec
-w,
+f(r))
+ (Qtr.
Ely(t;
YO) Ip s Me-a’lyolP,
We need approximations
*(Y)u~.~Y)
s -d
IY 14
(3.9.2)
d > 0.
Y E%(A),
Then
Gb)WG
(3.9.1)
>O andp >O,
y. E Y for some M 3 1 and a > 0.
of (3.8) based on R(A, A) [S-lo] to obtain from (3.9)
~Ely(r;r,z)lPdr~KElzlp~KIElz~~]p’q. i5 Then corollary 2.3 yields EJy(t; s, z)jp c Me-“[EIzjq]J’/q, z E Y,(q) for some M z 1 and a > 0. Setting s = 0, z = y. we obtain the conclusion. We may also apply theorem 2.1 if p 3 2. If we assume f = 0 and G E 2 (Y, y (H, Y)), then we obtain a generalization of theorem 1.2. COROLLARY 3.2. Let y(t; yo) be the mild solution of the linear equation (1.5). Then (Sl’) and (S2’) in Section 1 are equivalent for any p > 0. If there exists a function u(y) satisfying (3.9) with f = 0, then
Ely(t; yO) 1~c Me-“‘lyolP, yo E Y for some M 2 1 and a > 0. COROLLARY
3.3.
If
f = 0 and G = 0, then we obtain theorem 1.1 for a Hilbert space Y and
p>o. Remark 3.4. Datko’s original version of theorem
his result is used in the latter. So proposition
1.1 does not follow from theorem 1.2 since 3.3 is a more natural extension.
3.4. An example
We give an illustrative example. Consider the heat equation a/Jty(x,
t> = a’/a+J(*?
y(0, t) = y(I, t) = 0, where b E Lz(O, 1) and c/y]. For this example H'(0, 1). Then
t) +
@)f
(Y( -7 t)>,
f is a real Lipschitz continuous function on Lz(O, 1) with we take Y = Lz(O, 1) and A = d2/dx2 @Y,Y)
s -duly\‘,
=
(u,(Y),
function on Y. We define
AY + bf(Y) ),
Y EQ(A).
Then ~dlY12
= ~(Y,AY s -2dlyl’+
If(y))
+ bf(Y))
2161
s -2(n2 - c) ly 12.
If(
IYI
S
with i%(A) = Hb(0, 1) II
for anyy E ED(A).
We assume 161 = 1. Let u(y) be a Frechet differentiable zdU(Y)
(3.10)
Y(X, 0) =yo(x),
?td
by
A. ICHIKAWA
81-l
Thus the Liapunov function 1y /’ gives the region of asymptotic stability :c < .x’. r\iow consider P E 9?( Y) given by
where e, = fi sin rr,-r~ and for each g, h E Y, g 0 h E Z(Y) is defined by (g 0 h)y = g(g, h) E Y. Then P is a self-adjoint positive nuclear operator and is in fact. the solution of (1.2). Obviously P is not strictly positive. But we have xd(Py,y)=
Z(Py,Ay +bf(y)) s -(l
- 2c]Pb()]yl’.
Thus by corollary 3.1 the system (3.10) is exponentially stable if c < l/(2/ Pb I). If ! Pb / < l/212’, then this region is larger than {c < ,n?). Take, for example, b = e,, m > 1. Then 1Pbl = 1/2m*d and (3.10) is exponentially stable if c < rn’?. Consider now the stochastic version of (3.10) dy(x, t) = a’/dx’y(x, t) dr + b(x)f(y(
., t)) dw(t),
(3.11)
r(O,t)=y(l,t) =O,y(x,O) =yo(x),
where w(t) is a real standard function on Y. Define ZJ by
Wiener process. Let u(y) be a twice Frechet differentiable
%u(Y> = (U,(Y), AY > +
(l/2) (4yP,
b ) If(y) 1:
Then
Hence if cz < 2,$, the system (3.11) is exponentially hand
stable in the mean square. On the other
s -(l - (Pb, b)c*)ly/t
Thus by proposition 3.3 we obtain the region of exponential stability: c? < l/( Pb, b). This is m > 1, then (Pb, b) = 1/2m’,n?. The system largerthan{2<2n?}if(Pb,b)<1/22.Ifb=e,, (3.11) is exponentially stable in the mean square sense if 2 < 2m’;i. 4,FINALREMARKS
Since theorems 2.1 and 2.2 involve two parameter
semigroups, it is possible to replace (1.3), semigroups of bounded
(1.8) and (3.3) by time varying systems. The theory of two-parameter
linear operators is available [14]. We assumed all nonlinearities to be Lipschitz in Section 3 just to assure the conditions (B) or (B’). But theorems in Section 2 do not require this.
Equivalence of L, stability and exponential stability for a class of nonlinear semigroups
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