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Weak stabilizability of infinite dimensional nonlinear systems
Appl. Math. Lett. Vol. 4, No. 1, pp. 95-98, 1991 Printed in Great Britain. All rights reserved
Infinite ~ABDELLAH
Copyright@
Weak Stabilizability Dimensional Nonlinear BOUNABAT
0893~9659191 $3.00 + 0.00 1991 Pergamon Press plc
of Systems
AND ‘JEAN-PAUL
GAUTHIER
‘Departement de Mathdmatiques FacultC des Sciences de Marrakech 2Laboratoire d’Automatique et de Genie des Pro&d& Universite Claude Bernard (Received
October
1989)
Abstract. In this paper, we deal with some particular nonlinear infinite dimensional stabilization problems. We show that LaSalle invariance principle can be used in the same way in the finite dimensional case in order to get nontrivial stabilization results.
1. INTRODUCTION In this paper, we study a problem of stabilization for a particular class of infinite dimensional nonlinear systems. Our idea in this paper is to generalize to the infinite dimensional case a result by Jurdjevic-Quinn [I]. Basically, this result is an application of LaSalle invariance principle [2]. The pioneer paper [l] nontrivially uses this principle for finite dimensional nonlinear stabilization problems. Other works of one of the authors [3] shows some more nonlinear generalizations of the result of [l]. There is also a paper by Slemrod [4] in the same direction. Slemrod also had the idea to use these techniques for infinite dimensional stabilization problems [5,9]. 0 ur result is slightly different from his results. It is exactly the infinite dimensional version of [l]. The drift of the system is skew-adjoint. In that case the globally stabilizing feedback control law is linear. One should remark that this case (skew-adjoint drift) is not so restrictive: it is almost the WORST case in which one can expect to stabilize. The whole spectrum of the drift is on the imaginary axis. If there is a strictly unstable part in the drift, it is already known in the linear case that it must be finite dimensional. Notice also that in the considered case, no assumption of compactness is needed to get the result. 2. PRELIMINARY Let H be a real Hilbert systems of the form:
RESULTS space
with
AND
STATEMENT
inner
product
4 =A$+~(Bti++b), $(o> = $0, Hl.
bounded f: H
skew-adjoint + H, by:
THEOREM
(e), 11.11. We consider
tl0,
the nonlinear
$EH
u(t) = --lb, +(t>)
Here A is densely defined possibly unbounded, H. We denote D(A) for its Domain.
H2. B is a linear
OF THE
operator
f(v)
skew-adjoint
(A = -A*)
operator
on
on H. Let us define the mapping:
= -h v) (B(cp)+ b) Typeset 95
by A,@-?‘)$
A. BOIJNABAT.J.P. GAUTHIER
96
DEFINITION. Let to, ti, to < ti. A function 9~ WAC((to, tl), H) (the set of weakly absolutely continuous functions from [t0, tl] to H) is a weak solution of (S) if for each w E D(A*) $(rp(t), for almost
4
=
k(t), A*4
+ (f(cp(t)h 4
any t E [~OJl].
It is almost obvious (see Balakrishnan solution of (S) if and only if it satisfies
[6] and Ball [7]) that p(t): [to, ti] -+ H is a weak the variation of constants formula:
t1
P(t) =
A(+-lo)
e
I
PO+
eA(*-“)f(cp(S))dS
for all t E [to,ti].
to
where eA(t-to) is the weak-strong continuous one parameter group of unitary operators whose infinitesimal generator is A (by Stone’s theorem). Weak solutions given by this variation of constant formula are also usually called “mild” solutions. It is very easy to prove that, under assumptions (Hi), (Hz), there is, for any initial value +s E H, a unique weak solution 4(t), t 2 0 (by theorem 1.4 of Pazy [8]) and that ]]$(t)]] is bounded; a) IW(t>ll = IlA(t>ll , since etAis a unitary operator, where A is the solution of the variation of constant equation:
A(t) = A0 -
of(h~(S))
(e-SA(W$(s>>
+ b))dS
J
b) ]]A(t)]]2 is differentiable,
by boundedness
of B and satisfies:
-$ ]]A(t)]]2 = 2(A, A ) = -2](b, It is now clear that
~~l($(t),
etAA(t))12
5 0
b) = 0:
By the above considerations,
llWl12 = llAol12 - /‘((a,$bW2G Vt1 0. 0
Since, as we said A is bounded, decreasing, the integral ~ow((b,+(S)))2dS vergent. Now, (b, 4(S)) is an absolutely continuous function with bounded L2([0,00[). Hence, i9w(b, T/J(S))= 0. NOW we denote by I’(&) {+* E H/3 a sequence t, shows that CLAIM.
I’(&)
I’(&)
must be conderivative in
the weak positive w-limit set of the trajectory $~(t); i.e., I’(&) = + +oo such that weak - ,l$a(q(tn)) = +*}. The above just
c { b}l.
is positively eAt invariant.
PROOF OF THE CLAIM. Let $J* E I’(tio) and 1, be a sequence $f(t, + t) = etAll(t,)
-
J’
such that Ilf(tn) % $J*. But,
e(‘-T)A(b, $(T + t,))(B($(T
+ tn)) + b) dr
0
Hence: I($(tn + t),
J
cp)- (etA$(tn>, PO)1 2 C otItb~IHr+ L))l dr
We& stabilizability of infinite dimensional nonlinear systems F.
or n sufficiently
[since $J(&) s
great,
ti’, ($(&),
](b,$(r
+ t,,))]
< E and ($(tn
+ t),‘~)
,z~
97
(etA+*, cp)Vp E H.
e-A*54l
emAtcp) + (ti*,
To finish, I(+(& +t),cp)
- (eAtll*,cp)I
5 I($J(~~ +t),c~)
(eA”$(L),i41 + I(eA’$(L),d- (eAf+*jdI.
-
n D(Ak) (whenever kCN A is densely defined, ?)s is still dense by theorem 2.7 of Pazy [S]). We can now state our theorem, which is as expected, a nonlinear feedback stabilization theorem for systems (S), under assumptions (Hl), (H2), (H3): NOW
let us make the following
additional
assumption
b E DO =
(Hs):
(H3), moreover assume that the closed linear span of the to H (controllability assumption). Then, (S) is weakly stabilizable, and the linear feedback U(t) = -(b, ti(t)) is a weakly stabilizing control law. THEOREM.
Assume
(Hl),
(H2),
set E = {b, Ab, . . . . Akb, . ..} is equal
3. PROOF
OF THE
Let Z* E I’(&), by (H3) operators. One has:
and
/$-(etAx*,
THEOREM
the fact that
AND
EXAMPLE
etA is a one parameter
b) = (-l)m(etAz*,
Consider
of unitary
Amb) = 0, Vt 2 0.
Hence, for t = 0, (x*,Amb) = 0. By the rank condition, this implies weak w-limit points are 0. It follows that W. ,l&mm($(t)) = 0. EXAMPLE.
group
that
z* = 0. All the
the system:
?I= g + u(t)(B(+) + b), 4@>= $0, Let H = L2(R) d enote the Hilbert
u(t) = -(b, 4(t))
space of all function
f : R + R that are square
We know that, A = 6 is a densely defined skew-adjoint difficult to show that D(A) = H’(R), w h ere H1 (R) denotes f: R -F R and such that 2 is also on L2(R). We define the operator B by: f E L2(R)
+
B(f)
t20 _
= sin(z)
f(-z)
integrable:
operator on L2(R). It is not the Hilbert space of all functions
E L’(R)
B is a linear, bounded operator and skew-adjoint. A straightforward computation shows that B is skew-adjoint in L2(R). Let b(x) = eSxa/‘, it is well known that b(x) together with all its derivatives E H’(R). Moreover its derivatives (Hermite functions) form an orthonormal basis of L2(R) hence all the assumptions of our theorem are satisfied. Then the system (Sl) is weakly stabilizable and a stabilizing feedback is too e-ra/2$(t,
u(t) = -
x) dx, Vt 2 0
J --co
where $(t, z) is the mild-solution
of the nonlinear
system
(Sl).
A. BOUNABAT,
98
J.P. GAUTHIER
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
Jurjevic-Quinn, Controllability and stability, Journal of Diflerenlial Equation8 28 (1978). LaSalle, Stability theory for ordinary differential equations, Jorm. of Difl. Equations 4 (1968). J. P. Gauthier, Structure des systbmes non-lineaires, tdiliona du C.N.R.S. (1984). M. Slemrod, Stabilisation of bilinear control system with applications to nonconservative problems in elasticity, SIAM J. on Control and Opl. 16 (1978). M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Syslems 2, 265-285 (1989). A. V. Balakrishnan, Applied Functional Analysis-Applicalions of Mathemalics, Springer Verlag, New York, (1976). J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Amer Math Society 63, 370-373 (1977). Pazy, Semi-G~oupa of Linear Operalow and Applicalions fo Partial Diflerenlial Equations, Springer Verlag, New York, (1983). J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5, 169-179 (1979).
IDCpartement de Mat&matiques FacultC des Sciences de Marrakech Bld S& Marrakeck (MAROC) 2Laboratoire d’Automatique et de G&ie des Pro&d& UniversitC Claude Bernard Lyon 1 - Bat 721 - 43 Bld du 11 Novembre 1918 - 69622 Villeurbanne cedex France
Infinite ~ABDELLAH
Copyright@
Weak Stabilizability Dimensional Nonlinear BOUNABAT
0893~9659191 $3.00 + 0.00 1991 Pergamon Press plc
of Systems
AND ‘JEAN-PAUL
GAUTHIER
‘Departement de Mathdmatiques FacultC des Sciences de Marrakech 2Laboratoire d’Automatique et de Genie des Pro&d& Universite Claude Bernard (Received
October
1989)
Abstract. In this paper, we deal with some particular nonlinear infinite dimensional stabilization problems. We show that LaSalle invariance principle can be used in the same way in the finite dimensional case in order to get nontrivial stabilization results.
1. INTRODUCTION In this paper, we study a problem of stabilization for a particular class of infinite dimensional nonlinear systems. Our idea in this paper is to generalize to the infinite dimensional case a result by Jurdjevic-Quinn [I]. Basically, this result is an application of LaSalle invariance principle [2]. The pioneer paper [l] nontrivially uses this principle for finite dimensional nonlinear stabilization problems. Other works of one of the authors [3] shows some more nonlinear generalizations of the result of [l]. There is also a paper by Slemrod [4] in the same direction. Slemrod also had the idea to use these techniques for infinite dimensional stabilization problems [5,9]. 0 ur result is slightly different from his results. It is exactly the infinite dimensional version of [l]. The drift of the system is skew-adjoint. In that case the globally stabilizing feedback control law is linear. One should remark that this case (skew-adjoint drift) is not so restrictive: it is almost the WORST case in which one can expect to stabilize. The whole spectrum of the drift is on the imaginary axis. If there is a strictly unstable part in the drift, it is already known in the linear case that it must be finite dimensional. Notice also that in the considered case, no assumption of compactness is needed to get the result. 2. PRELIMINARY Let H be a real Hilbert systems of the form:
RESULTS space
with
AND
STATEMENT
inner
product
4 =A$+~(Bti++b), $(o> = $0, Hl.
bounded f: H
skew-adjoint + H, by:
THEOREM
(e), 11.11. We consider
tl0,
the nonlinear
$EH
u(t) = --lb, +(t>)
Here A is densely defined possibly unbounded, H. We denote D(A) for its Domain.
H2. B is a linear
OF THE
operator
f(v)
skew-adjoint
(A = -A*)
operator
on
on H. Let us define the mapping:
= -h v) (B(cp)+ b) Typeset 95
by A,@-?‘)$
A. BOIJNABAT.J.P. GAUTHIER
96
DEFINITION. Let to, ti, to < ti. A function 9~ WAC((to, tl), H) (the set of weakly absolutely continuous functions from [t0, tl] to H) is a weak solution of (S) if for each w E D(A*) $(rp(t), for almost
4
=
k(t), A*4
+ (f(cp(t)h 4
any t E [~OJl].
It is almost obvious (see Balakrishnan solution of (S) if and only if it satisfies
[6] and Ball [7]) that p(t): [to, ti] -+ H is a weak the variation of constants formula:
t1
P(t) =
A(+-lo)
e
I
PO+
eA(*-“)f(cp(S))dS
for all t E [to,ti].
to
where eA(t-to) is the weak-strong continuous one parameter group of unitary operators whose infinitesimal generator is A (by Stone’s theorem). Weak solutions given by this variation of constant formula are also usually called “mild” solutions. It is very easy to prove that, under assumptions (Hi), (Hz), there is, for any initial value +s E H, a unique weak solution 4(t), t 2 0 (by theorem 1.4 of Pazy [8]) and that ]]$(t)]] is bounded; a) IW(t>ll = IlA(t>ll , since etAis a unitary operator, where A is the solution of the variation of constant equation:
A(t) = A0 -
of(h~(S))
(e-SA(W$(s>>
+ b))dS
J
b) ]]A(t)]]2 is differentiable,
by boundedness
of B and satisfies:
-$ ]]A(t)]]2 = 2(A, A ) = -2](b, It is now clear that
~~l($(t),
etAA(t))12
5 0
b) = 0:
By the above considerations,
llWl12 = llAol12 - /‘((a,$bW2G Vt1 0. 0
Since, as we said A is bounded, decreasing, the integral ~ow((b,+(S)))2dS vergent. Now, (b, 4(S)) is an absolutely continuous function with bounded L2([0,00[). Hence, i9w(b, T/J(S))= 0. NOW we denote by I’(&) {+* E H/3 a sequence t, shows that CLAIM.
I’(&)
I’(&)
must be conderivative in
the weak positive w-limit set of the trajectory $~(t); i.e., I’(&) = + +oo such that weak - ,l$a(q(tn)) = +*}. The above just
c { b}l.
is positively eAt invariant.
PROOF OF THE CLAIM. Let $J* E I’(tio) and 1, be a sequence $f(t, + t) = etAll(t,)
-
J’
such that Ilf(tn) % $J*. But,
e(‘-T)A(b, $(T + t,))(B($(T
+ tn)) + b) dr
0
Hence: I($(tn + t),
J
cp)- (etA$(tn>, PO)1 2 C otItb~IHr+ L))l dr
We& stabilizability of infinite dimensional nonlinear systems F.
or n sufficiently
[since $J(&) s
great,
ti’, ($(&),
](b,$(r
+ t,,))]
< E and ($(tn
+ t),‘~)
,z~
97
(etA+*, cp)Vp E H.
e-A*54l
emAtcp) + (ti*,
To finish, I(+(& +t),cp)
- (eAtll*,cp)I
5 I($J(~~ +t),c~)
(eA”$(L),i41 + I(eA’$(L),d- (eAf+*jdI.
-
n D(Ak) (whenever kCN A is densely defined, ?)s is still dense by theorem 2.7 of Pazy [S]). We can now state our theorem, which is as expected, a nonlinear feedback stabilization theorem for systems (S), under assumptions (Hl), (H2), (H3): NOW
let us make the following
additional
assumption
b E DO =
(Hs):
(H3), moreover assume that the closed linear span of the to H (controllability assumption). Then, (S) is weakly stabilizable, and the linear feedback U(t) = -(b, ti(t)) is a weakly stabilizing control law. THEOREM.
Assume
(Hl),
(H2),
set E = {b, Ab, . . . . Akb, . ..} is equal
3. PROOF
OF THE
Let Z* E I’(&), by (H3) operators. One has:
and
/$-(etAx*,
THEOREM
the fact that
AND
EXAMPLE
etA is a one parameter
b) = (-l)m(etAz*,
Consider
of unitary
Amb) = 0, Vt 2 0.
Hence, for t = 0, (x*,Amb) = 0. By the rank condition, this implies weak w-limit points are 0. It follows that W. ,l&mm($(t)) = 0. EXAMPLE.
group
that
z* = 0. All the
the system:
?I= g + u(t)(B(+) + b), 4@>= $0, Let H = L2(R) d enote the Hilbert
u(t) = -(b, 4(t))
space of all function
f : R + R that are square
We know that, A = 6 is a densely defined skew-adjoint difficult to show that D(A) = H’(R), w h ere H1 (R) denotes f: R -F R and such that 2 is also on L2(R). We define the operator B by: f E L2(R)
+
B(f)
t20 _
= sin(z)
f(-z)
integrable:
operator on L2(R). It is not the Hilbert space of all functions
E L’(R)
B is a linear, bounded operator and skew-adjoint. A straightforward computation shows that B is skew-adjoint in L2(R). Let b(x) = eSxa/‘, it is well known that b(x) together with all its derivatives E H’(R). Moreover its derivatives (Hermite functions) form an orthonormal basis of L2(R) hence all the assumptions of our theorem are satisfied. Then the system (Sl) is weakly stabilizable and a stabilizing feedback is too e-ra/2$(t,
u(t) = -
x) dx, Vt 2 0
J --co
where $(t, z) is the mild-solution
of the nonlinear
system
(Sl).
A. BOUNABAT,
98
J.P. GAUTHIER
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
Jurjevic-Quinn, Controllability and stability, Journal of Diflerenlial Equation8 28 (1978). LaSalle, Stability theory for ordinary differential equations, Jorm. of Difl. Equations 4 (1968). J. P. Gauthier, Structure des systbmes non-lineaires, tdiliona du C.N.R.S. (1984). M. Slemrod, Stabilisation of bilinear control system with applications to nonconservative problems in elasticity, SIAM J. on Control and Opl. 16 (1978). M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Syslems 2, 265-285 (1989). A. V. Balakrishnan, Applied Functional Analysis-Applicalions of Mathemalics, Springer Verlag, New York, (1976). J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Amer Math Society 63, 370-373 (1977). Pazy, Semi-G~oupa of Linear Operalow and Applicalions fo Partial Diflerenlial Equations, Springer Verlag, New York, (1983). J. M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim. 5, 169-179 (1979).
IDCpartement de Mat&matiques FacultC des Sciences de Marrakech Bld S& Marrakeck (MAROC) 2Laboratoire d’Automatique et de G&ie des Pro&d& UniversitC Claude Bernard Lyon 1 - Bat 721 - 43 Bld du 11 Novembre 1918 - 69622 Villeurbanne cedex France