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On the Dynamics of Boundary Controlled Nonlinear Distributed Parameter Systems‡
Copyright © IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995
ON THE DYNAMICS OF BOUNDARY CONTROLLED NONLINEAR DISTRIBUTED PARAMETER SYSTEMS! C.I. BYRNES·, D.S. GILLIAM·· and V.I. SHUBOV** iThis work was supported in part by a Grant from AFOSR. "Washint on University, St. Louis, Systems Science and Mathemtics ""Texas Tech University , Department of Mathematics
Abstract. This paper is concerned with the effect of boundary feedback control on the dynamics of a class of nonlinear distributed parameter systems governed by convective reaction diffusion equations acting on bounded domains in n-dimensional Euclidean space. Key Words. Distributed parameter systems ; partial differential equations
1. INTRODUCTION
2. FORMULATION OF THE PROBLEM
In this paper we formulate some of the results of our forthcoming work (Byrnes1, 1995) which is devoted to the study of a class of nonlinear distributed parameter systems governed by multidimensional convection reaction diffusion equations. For this class of systems we have introduced a feedback control law implemented through the boundary of the spatial domain . The main objective of this work is to examine the influence of the feedback law on the dynamics of the system. More specifically, we are interested in questions of existence, uniqueness and regularity of solutions and the dependence of the dynamics on the boundary control parameters. This work represents a continuation of our work in (Byrnes6 , 1993 ; Byrnes3 , 1994; Byrnes2 , 1995) devoted to the boundary control of a viscous Burgers ' equation. In fact, the main results of this work are generalizations to a class of nonlinear systems in higher dimensions of our work in (Byrnes2, 1995) .
We now turn to a precise formulation of the problem. Let D C nn (n ~ 1) be a bounded domain with C 2 -boundary r , x = (Xl," · , x n ) E D and t ~ O. We consider systems of the form Wt -
Lw + div F(w)
+ G(w) = h ,
(1)
where the scalar function w(x, t) represents the state of our system F(w) = F(x , t, w), G(w) = G(x , t , w) and h = h(x , t) are allowed to depend on X and t. In (1) we use the following notations . L is a formally self-adjoint uniformly elliptic differential operator Lw
n
a
L ax .
=
i,j=1
(aijWxJ .
(2)
J
With m defined in (12) below , the functions aij E form at each point X E n a positive symmetric matrix , whose minimal eigenvalue VI(X) satisfies: V1(X) ~ Vo > O. The convective term in (1) is given by c2m+1 (n)
One of our long range goals in the control of nonlinear distributed parameter systems consists in the development of systematic strategies for designing feedback laws which can shape or at least influence the response of nonlinear systems. Of considerable practical interest is the question of the effect of boundary control in influencing the steady-state response of nonlinear distributed parameter systems governed by hydrodynamictype equations, such as Navier-Stokes equation , which contain both nonlinear convective terms and diffusive terms (see (Temam , 1988; Constantin, 1985; Ladyzhenskaya, 1975; Ladyzhenskaya, 1991)). The present work is devoted to such a class of equations.
_ n d divF(x , t , w)= L - Fi(x, t , w)= i =1 dX i
n
=L
(Fix ; + FiwWx,) ,
(3)
i=1
where we use the subscript notation for partial derivatives. The precise conditions on the functions Fi , G and h are given below . We are interested in the dynamics generated by (1) on the state space L 2 (D). So, we introduce 837
the initial condition w(x,O) =
we have chosen to deal directly with weak solutions and our proofs are based on classical energy methods . We note that the conditions given below on the nonlinear terms do not guarantee a global in time a priori energy estimate for the solution (see, e.g., (Ladyzhenskaya, 1968)). Introduction of such conditions would exclude from consideration interesting and important classes of equations . For example, the one dimensional Burgers' equation would be excluded . The problem with the feedback law (7) is that , in contrast with the zero dynamics problem , which corresponds to the equation (1) with Dirichlet boundary conditions , i.e. , the system obtained by constraining the output to zero,
(4)
In order to complete the description of our system we now describe the form of the system input , output , feedback control law and hence the desired closed loop system. We assume that a control u can be effected through the boundary f by n
u(x , i) =
L
aij(x)wx,(x, i)7]j(x) ,
(5)
i ,j : l
for x E f , t E [0 , 00) , where 7]j (x) are the components of the unit outward normal vector ij(x) to f at the point x . Our output is the "temperature" on the boundary : y(x, t) = w(x , i) , x E f , i E [0 , 00).
y(x ,i) = w(x , i)
= -k(x)y(x , i) ,
(7)
where the function k E C 2 (f) satisfies k(x) ~ ko > 0 and plays the role of the system gain . The feedback law (7) is nothing more than boundary conditions for the system (1) of the form
Our second main set of results , which we formulate in the next section , concerns the dynamics of our system in the particular case of sufficiently small initial data and forcing term (under less restrictive conditions on the regularity of h, {aij} , Fi and G). We show that in this case our system is locally Lyapunov stable and possesses a local attractor (which is compact and has a finite fractal and Hausdorf dimension) .
n
L
aij(x)w x,7]j(x) + k(x)w(x , t)
= 0,
(9)
the nonlinear convective term persists in the energy balance relation. As it was shown in our work (Byrnes2 , 1995) this problem arises even for a one-dimensional Burgers ' equation with boundary feedback control. In our approach we use the local in time energy estimate to prove the existence of the weak solution on a finite time interval. Then we investigate the regularity properties of this solution and show that it is, in fact , classical for t > O. Next we combine the local in time existence result with a maximum principle to obtain the global solvability.
(6)
Motivated by classical scalar proportional error feedback in linear finite dimensional control theory, we introduce the feedback mechanism: u(x , i)
= 0, for x E f ,
(8)
i, j=l
where x E f , i E [0 , 00) . Thus we consider the initial boundary value problem (1), (4), (8) which we interpret as our "closed loop system" corresponding to the open loop system (1) , (4) and (5) with u == 0 and with the feedback law (7). The forcing term h can be interpreted as a disturbance.
Our third and final set of results from our paper (Byrnes1 , 1995) is a generalization of the notion of high gain limit , extending the concept of root locus from classical automatic controL We show that the solution of the problem (1) , (4 ), (8) converges in some sense to the solution of the zero dynamics (i .e. , the problem (1), (4) with (8) replaced by (9)) as the gain k ---+ 00. In the next section we give a precise formulation of this fact.
Before we turn to the formulation of our main results let us make the following observations. Our first result will be the global in time existence and uniqueness theorem for the problem (1), (4) , (8) with arbitrary L2(0) initial data. We note that equation (1) belongs to a well known class of quasilinear parabolic equations. Unique solvability results for such equations under suitable restrictions on the non linear terms and the data of the problem are also well known (see, e.g. (Ladyzhenskaya, 1968)) . One possible approach to the proof of the existence and uniqueness result would be to use the techniques in (Ladyzhenskaya, 1968) valid for classical solutions and for classes of Holder continuous functions based on the LeraySchauder fixed point theorem . Since we are interested in the dynamics in the state space L2(0) ,
3. STATEMENT OF RESULTS We begin with a statement of the conditions on the nonlinear terms and forcing term in (1) . Let us agree to use the following notations: QT = o x [0 , T], where T > 0; Qto , t, = 0 X [io , id for any il > io > 0 so that QO ,T == QT . Let 1)~ denote the partial derivative with respect to the variable ~ and , in standard multiindex notation , 838
if a
= (al,""
functions u(x , t) for which their classical derivatives VfV'; u E C( Qto ,T) for all p and a such 2p + la I 2. Notice that (14) is a restriction on the behavior of h near the moment t 0 only.
an) is a integer multiindex, then Vexx = Vex, ... VexXn.n Xl
:s
denotes the mixed partial derivative of order lal = al+ ·· ·+a n ·
=
The above conditions on G and Fi are , in fact , slightly stronger than we need . We refer to (Byrnes1 , 1995) for more precise conditions.
At this point we fix T > 0 and consider the problem (1), (4), (8) on the time interval [0, T] . Thus we seek a solution w(x , t) to (1), (4), (8) defined on QT. Since T is arbitrary, all the existence and uniqueness results obtained for t E [0, T] can automatically be extended to the infinite interval [0,00). We assume that the following restrictions are satisfied.
Our first main result consists in the following.
Theorem 1 Assume that all the conditions (10)(15) are satisfied. Then the following statements take place. a) The problem (1), (4), (8) with an arbitrary initial function 'P E L2(D) has a unique weak solution
=
1. G(x , t , 0) 0 (note that this is not really a restriction, but rather an agreement). 2.
(10) 3.
which satisfies Ilw(t) IVfV~VIG(x, t,OI
:s c31~lr +
C4,
--->
0 as t
--->
O.
b) Moreover , w E H 2m ,m(Q t o,T) with m from (12) for any to E (0 , T) . Due to (16) this means that for positive values of time (t > 0) this solution is C 2 ,l-smooth and , therefore , satisfies (1) in a classical sense.
(11)
where r < 2 + 4jn . (11) should be satisfied for any nonnegative integers p and q and multiindex a such that 2p + q + lal 2m, with
:s
m=[n;6]+1.
'PII
c) There is a globally defined dynamical system on the state space L2(D) given in terms of a generalized semigroup
(12)
Here we use the notation [a] for the greatest integer less than or equal a. 4.
This means we have a 2-parameter family of mappings {Stot" 0 :s tl :s t2} such that St3toStotl = St3 t , for tl ::; t2 ::; t3 , and Stt I. These mappings are defined by the formula : Stot, w(td = W(t2) for 0 ::; tl t2, where w is the solution of (1) , (3) with (2) replaced by an arbitrary initial data w(td E L2(D) at the moment t = t l . For time independent h , Fi and G , the generalized semigroup turns into a semigroup {St, t ~ O} by the rule Stot, = St2- t,.
where s < 1+2jn . (13) should be satisfied for any p, q and a such that 2p+q+lal:S 2m+1 , with m defined in (12). In all the conditions (10)-( 13) (x , t , ~) E QT x The constants Ci > 0 (i = 1, ···,6) may depend on T.
=
n. 5.
(14)
hE H 2m ,m(Q to ,T )
d) This semigroup is continuous in the following sense: each St,t, is a continuous mapping, for each 'P E L2(D) the function St,t2('P) is a continuous function of tl and t2 with respect to L 2 (D)-norm.
(15)
with m defined in (12), for any to E (0, T). By H 2m ,m(Q t o,T) we denote the Hilbert space of all functions u(x, t) which have weak derivatives in L2( Qto,T) for all p and a such that 2p+ lal :s m.
e) Each mapping St,t2 is compact for t2
> tl'
f) If h = 0 and Fi and G are independent of time, then there exists a positive continuous monotone increasing function a(O, ~ ~ 0 such that a(O) = 0 and
We remark that the following embedding theorem takes place (see, e.g., (Ladyzhenskaya, 1968»
H 2m ,m(Q to,T ) C C 2,1(Q to ,T ) ,
:s
( 16)
IISt'Pll
where C 2 ,l( Qto ,T) denotes the Banach space of all
:s a(II'PII)
for any t E [0 , (0) ,
839
which means that the closed loop system is globally Lyapunov stable .
This attractor is locally compact (i .e., its intersection with any ball is compact) , but it may be unbounded .
We make several brief remarks concerning the proof of this theorem. The main statements of the theorem are a) and b). The statements c)-f) are obtained in the process of proving a) and b). To obtain our result we first prove in (Byrnesl, 1995) the existence of the weak solution on a small time interval [0, T(llcplI, h)], where T(llcpll , h) --> as Ilcpll ---- 00 or IIhIILoo([O ,T] ,H-'(n)) ---- 00 . We have to do this since, as it has already been mentioned in the Introduction , our assumptions do not guarantee a global in time energy estimate. Next we investigate the regularity of the weak solution and show that it belongs to the space H 2m ,m (Q1G ,T) with m from (12) and < to < T ::; T(llcpll, h). This allows us to conclude that this solution is , in fact , classical for t > 0. This step is technically the most difficult part of the entire proof of the theorem. Finally, we combine the local in time solvability result with a global in time a priori estimate obtained from a classical maximum principle for parabolic equations (Ladyzhenskaya, 1968) and extend the local result to an arbitrary time interval [0 , T]. We notice that the condition (10) is introduced exactly to ensure that the maximum principle is valid . We note that the local dynamics problem for one dimensional (n = 1) convection reaction diffusion equations is examined in detail in (Okasha, 1995).
\Ve now turn to the se con main result of our work , which is closely related to Theorem 1. This result deals with the unique global in time solvability of (1) , (4), (8) in the case of sufficiently small initial data and disturbance h. While this case is does not apply to general initial data in L2(Q) , the restrictions on aij , Fi, G and h are considerably relaxed and , furthermore , our result gives important information about the dynamics of our system in a small neighborhood of the origin , when the disturbance is also small.
°
°
Theorem 2 Assume that aij E Hl(Q) , (12) is satisfied only for G (i .e., p = q = 0 0' = 0) and (13) is satisfied only for Fi and V';V:Fi with q+ 10'1 = 1. Assume , also, that G(x , t , 0) = 0, Fi(x, t , 0) = 0 and that h satisfies (14) . In this case there exists numbers p > and ~ > 0, such that if
°
Ilcpll ::;
(19)
p
and ess sup Ilh(t)IIH-l ( n ) ::;~, tE[O ,T ) then
(20)
°
a) for any T > there exists a unique weak solution w E LOO ([0 , T], L2(Q)) n L2([O , T], Hl(Q)) of the problem (1), (4) , (8) ;
We now turn to the the question of the existence of an attractor for our dynamical system (see (Constantin , 1985; Ladyzhenskaya, 1975; Ladyzhenskaya, 1991 ; Temam , 1988; Shubov , 1992)). Assume that h, Fi and G are time independent . In this case our system generates a continuous semigroup {Sl,t ~ o} on the state space L2(Q) such that all the operators SI with t > are compact. This is , however , not sufficient to conclude that the system has a global absorbing ball , i.e. , a ball in the state space into which every trajectory of the system eventually enters and remains. The only case in which we can establish the existence a global attractor is for a zero disturbance h as discussed in part f) of Theorem 1. Namely, fix any R> and let BR = {lP E L2(Q) : IllPll ::; R} be the ball in L2(Q) of radius R centered at the origin. Here and below by 11·11 we mean the norm in L2(Q). Consider our dynamical system {St , t > O} restricted to the state space MR Ut~OSt(BR) (it is obvious that MR is invariant by its definition) It follows from (18) that this new system has an absorbing ball Ba(R), where a is exactly the function from (18) . Since all operators {St , t ~ o} are compact we can conclude that this system has a compact attractor AR . Notice that AR, C AR2 for Rl ::; R2 and , therefore , the entire system defined on L2 (Q) has the global attractor A = UR~ o AR .
b) this solution is continuous with respect to t in the L2(Q)-norm , i.e.,
Ilw(t
°
+ At) -
w(t)11
~ 0,
as t
-->
0
for any t E [0, T) ; c) this solution has the estimate
Ilw(t)11 ::; p, t E [0, (0) ;
(21 )
d) if h = 0 the w satisfies
°
Ilw(t)11 ::; pe--yt ,
(22)
for a number I > 0;
=
e) if, in addition , h E LOC([0 , 00) , L2(Q)) and satisfies ess sup
11 h(t)11
::; ~l,
(23)
tE[O,oo )
with a sufficiently small an estimate 1
~l
Ilw(t)IIHl (n) ::; t 1 / 2 M(t, 840
>
0, then w has
Ilcpll), t > 0
(24)
on the gain k( x). Denote the corresponding solution of the zero dynamics problem by w(x, t) . Our final main result consists in the following .
Here M(t,O is a continuous monotone increasing function in t, ( ~ 0; f) the weak solution w depends continuously on the initial data 'P and the forcing term h in the following sense. If Wl and W2 are the solutions of (1) with different initial conditions 'PI and 'P2 and different forcing terms hI and h2 , respectively, then
Ilwl(t) - w2(t)11 2 2
C (t)
+
1t
::;
Theorem 3 Assume that 9 and h in both problems are the same and that all conditions of Theorem 1 are satisfied. Then
max Ilwk(t) - w(t)11 --- 0
tE[O,T)
if min Ik(x)l--
(25)
xEr
(1191 - 92112 + Ilhl (T) -
00 ,
(26)
where w(t) denotes the function w as a function of x E n with t fixed.
h2(7)11~-1(D) dT)
Moreover, for any T
where C(t) is a continuous function of t E [0, (0).
> to > 0
max Ilwk(t) - w(t)IIHq(D ) --- 0
tE [t o,T)
as min Ik(x)I--xEr
Explicit expressions for p, a- and I in terms of the known parameters of the problem (independent of 9 and h) can be given.
where q Hq(n)
= [%]
+ 1.
00 ,
(27)
Due to the embedding
c C(n), (27) implies that for t > 0
Corollary 1 Under the conditions in Theorem 2
maxlwk(x , t) - w(x, t)I--- 0
we have the following results.
xED
1. the ball Bp = {1P E L2(n) : 1111>11 ::; p} is invariant with respect to the dynamics of our system. This means , in particular, that the system is locally Lyapunov stable.
as min Ik(x)I--xEr
(28)
If only the conditions of Theorem 2 with (23) replaced by (20) are satisfied then (26) still takes place and instead of (27) we have
2. In this case, when F; , G and h are time independent, we have a continuous semigroup {St, t ~ O} with the state space Bp. It follows from (24) and the compactness of the embedding Hl(n) C L2(n) that all the mappings {St, t > O} are corn pact. Based on the well known results of attractor theory (see, e.g. , (Constantin, 1985 ; Ladyzhenskaya, 1975; Ladyzhenskaya , 1991 ; Shubov , 1992)) we can conclude that our system has a compact local attractor in Bp. The Hausdorff and fractal dimensions of this attractor are finite.
sup IIw k (t) - W(t)IIH1(D ) ---.. 0 tE[to ,T]
as min Ik(x)l-xEr
00.
(29)
4. REFERENCES C.I. Byrnes, D.S. Gilliam, V.l. Shubov, (1995) Boundary control for convection reaction diffusion equations, Preprint.
C.l. Byrnes, D.S. Gilliam, V.I. Shubov , (1995)
3. The estimate (22) implies that the equilibriumsolution w Oof(l), (4), (8) with h 0 is exponentially stable.
=
00.
Local and global dynamics for a boundary controlled viscous Burgers' equation, Preprint.
=
C.I. Bvrnes, D.S. Gilliam, V.I. Shubov, (1994)
Bou~dary Control, Feedback Stabilization and the Existence of Attractors for a Viscous Burgers Equation, Preprint
Finally we turn to our third and final main result which describes a nonlinear enhancement of classical root locus. To formulate this result we mention that a theorem similar to Theorem 1 takes place for the zero dynamics problem (1) , (4), (9), i.e., for the case of Dirichlet boundary conditions. The proof of such a theorem is actually much easier than the proof of Theorem 1, since in the case of Dirichlet conditions we have a global in time a priori energy estimate. We refer to our forthcoming work (Byrnes1 , 1995) for details. Let us denote the solution of the controlled problem (1), (4), (8) by wk(x, t) to emphasize that it depends
C.l. Bvrnes, D.S. Gilliam, V .l. Shubov, (1994) Global Lyapunov stabilization of a nonlinear distributed parameter system, Proceedings of the 33rd International IEEE Conference on Decision and Control, Orlando, Florida.
C.l. Byrnes, D.S. Gilliam, V.l. Shubov, (1994) High gain limits of trajectories and attractors for a boundary controlled viscous Burgers' equation , to appear in Journal of Mathematical Systems, Estimation and Control.
C.l. Byrnes, D.S. Gilliamand V.l. Shubov , (1993) 841
Boundary control, for a viscous burgers ' equation, Identification and Control in Systems
Governed by Partial Differential Equations , H.T . Banks, R.H. Fabino and K. Ito (Eds.) , SIAM 1993 , 171-185. P. Constantin, C. Foias , R. Temam , (1985) Attractors representing turbulent flows, Memoirs of A.M.S . 53, No. 314. O.A., Ladyzhenskaya, (1991) Attractors for Semigroups and Evolution Equations , Cambridge Univ . Press. O.A. Ladyzhenskaya, (1975) The dynamical system that is generated by the N avier-Stokes equations, Journal of Soviet Mathematics , 3, 458-
479 . O.A . Ladyzhenskaya, V.A. Solonnikov , N.N. Ural'ceva, (1968) Linear and Quasilinear Equations of Parabolic Type , Translations of AMS, VoJ. 23. N.A. Okasha, Existence , regularity and long-time behavior of solutions for a class of boundary controlled convective reaction diffusion equations, Ph.D . Dissertation , Texas Tech Univer-
sity, 1995 . Shubov, V. (1992) Long time behavior of infinite dimensional dissipative dynamical systems, Journal of Math . Systems, Estimation
and Control, 2, 381-427. R. Temam, (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68 . Springer-Verlag.
842
ON THE DYNAMICS OF BOUNDARY CONTROLLED NONLINEAR DISTRIBUTED PARAMETER SYSTEMS! C.I. BYRNES·, D.S. GILLIAM·· and V.I. SHUBOV** iThis work was supported in part by a Grant from AFOSR. "Washint on University, St. Louis, Systems Science and Mathemtics ""Texas Tech University , Department of Mathematics
Abstract. This paper is concerned with the effect of boundary feedback control on the dynamics of a class of nonlinear distributed parameter systems governed by convective reaction diffusion equations acting on bounded domains in n-dimensional Euclidean space. Key Words. Distributed parameter systems ; partial differential equations
1. INTRODUCTION
2. FORMULATION OF THE PROBLEM
In this paper we formulate some of the results of our forthcoming work (Byrnes1, 1995) which is devoted to the study of a class of nonlinear distributed parameter systems governed by multidimensional convection reaction diffusion equations. For this class of systems we have introduced a feedback control law implemented through the boundary of the spatial domain . The main objective of this work is to examine the influence of the feedback law on the dynamics of the system. More specifically, we are interested in questions of existence, uniqueness and regularity of solutions and the dependence of the dynamics on the boundary control parameters. This work represents a continuation of our work in (Byrnes6 , 1993 ; Byrnes3 , 1994; Byrnes2 , 1995) devoted to the boundary control of a viscous Burgers ' equation. In fact, the main results of this work are generalizations to a class of nonlinear systems in higher dimensions of our work in (Byrnes2, 1995) .
We now turn to a precise formulation of the problem. Let D C nn (n ~ 1) be a bounded domain with C 2 -boundary r , x = (Xl," · , x n ) E D and t ~ O. We consider systems of the form Wt -
Lw + div F(w)
+ G(w) = h ,
(1)
where the scalar function w(x, t) represents the state of our system F(w) = F(x , t, w), G(w) = G(x , t , w) and h = h(x , t) are allowed to depend on X and t. In (1) we use the following notations . L is a formally self-adjoint uniformly elliptic differential operator Lw
n
a
L ax .
=
i,j=1
(aijWxJ .
(2)
J
With m defined in (12) below , the functions aij E form at each point X E n a positive symmetric matrix , whose minimal eigenvalue VI(X) satisfies: V1(X) ~ Vo > O. The convective term in (1) is given by c2m+1 (n)
One of our long range goals in the control of nonlinear distributed parameter systems consists in the development of systematic strategies for designing feedback laws which can shape or at least influence the response of nonlinear systems. Of considerable practical interest is the question of the effect of boundary control in influencing the steady-state response of nonlinear distributed parameter systems governed by hydrodynamictype equations, such as Navier-Stokes equation , which contain both nonlinear convective terms and diffusive terms (see (Temam , 1988; Constantin, 1985; Ladyzhenskaya, 1975; Ladyzhenskaya, 1991)). The present work is devoted to such a class of equations.
_ n d divF(x , t , w)= L - Fi(x, t , w)= i =1 dX i
n
=L
(Fix ; + FiwWx,) ,
(3)
i=1
where we use the subscript notation for partial derivatives. The precise conditions on the functions Fi , G and h are given below . We are interested in the dynamics generated by (1) on the state space L 2 (D). So, we introduce 837
the initial condition w(x,O) =
we have chosen to deal directly with weak solutions and our proofs are based on classical energy methods . We note that the conditions given below on the nonlinear terms do not guarantee a global in time a priori energy estimate for the solution (see, e.g., (Ladyzhenskaya, 1968)). Introduction of such conditions would exclude from consideration interesting and important classes of equations . For example, the one dimensional Burgers' equation would be excluded . The problem with the feedback law (7) is that , in contrast with the zero dynamics problem , which corresponds to the equation (1) with Dirichlet boundary conditions , i.e. , the system obtained by constraining the output to zero,
(4)
In order to complete the description of our system we now describe the form of the system input , output , feedback control law and hence the desired closed loop system. We assume that a control u can be effected through the boundary f by n
u(x , i) =
L
aij(x)wx,(x, i)7]j(x) ,
(5)
i ,j : l
for x E f , t E [0 , 00) , where 7]j (x) are the components of the unit outward normal vector ij(x) to f at the point x . Our output is the "temperature" on the boundary : y(x, t) = w(x , i) , x E f , i E [0 , 00).
y(x ,i) = w(x , i)
= -k(x)y(x , i) ,
(7)
where the function k E C 2 (f) satisfies k(x) ~ ko > 0 and plays the role of the system gain . The feedback law (7) is nothing more than boundary conditions for the system (1) of the form
Our second main set of results , which we formulate in the next section , concerns the dynamics of our system in the particular case of sufficiently small initial data and forcing term (under less restrictive conditions on the regularity of h, {aij} , Fi and G). We show that in this case our system is locally Lyapunov stable and possesses a local attractor (which is compact and has a finite fractal and Hausdorf dimension) .
n
L
aij(x)w x,7]j(x) + k(x)w(x , t)
= 0,
(9)
the nonlinear convective term persists in the energy balance relation. As it was shown in our work (Byrnes2 , 1995) this problem arises even for a one-dimensional Burgers ' equation with boundary feedback control. In our approach we use the local in time energy estimate to prove the existence of the weak solution on a finite time interval. Then we investigate the regularity properties of this solution and show that it is, in fact , classical for t > O. Next we combine the local in time existence result with a maximum principle to obtain the global solvability.
(6)
Motivated by classical scalar proportional error feedback in linear finite dimensional control theory, we introduce the feedback mechanism: u(x , i)
= 0, for x E f ,
(8)
i, j=l
where x E f , i E [0 , 00) . Thus we consider the initial boundary value problem (1), (4), (8) which we interpret as our "closed loop system" corresponding to the open loop system (1) , (4) and (5) with u == 0 and with the feedback law (7). The forcing term h can be interpreted as a disturbance.
Our third and final set of results from our paper (Byrnes1 , 1995) is a generalization of the notion of high gain limit , extending the concept of root locus from classical automatic controL We show that the solution of the problem (1) , (4 ), (8) converges in some sense to the solution of the zero dynamics (i .e. , the problem (1), (4) with (8) replaced by (9)) as the gain k ---+ 00. In the next section we give a precise formulation of this fact.
Before we turn to the formulation of our main results let us make the following observations. Our first result will be the global in time existence and uniqueness theorem for the problem (1), (4) , (8) with arbitrary L2(0) initial data. We note that equation (1) belongs to a well known class of quasilinear parabolic equations. Unique solvability results for such equations under suitable restrictions on the non linear terms and the data of the problem are also well known (see, e.g. (Ladyzhenskaya, 1968)) . One possible approach to the proof of the existence and uniqueness result would be to use the techniques in (Ladyzhenskaya, 1968) valid for classical solutions and for classes of Holder continuous functions based on the LeraySchauder fixed point theorem . Since we are interested in the dynamics in the state space L2(0) ,
3. STATEMENT OF RESULTS We begin with a statement of the conditions on the nonlinear terms and forcing term in (1) . Let us agree to use the following notations: QT = o x [0 , T], where T > 0; Qto , t, = 0 X [io , id for any il > io > 0 so that QO ,T == QT . Let 1)~ denote the partial derivative with respect to the variable ~ and , in standard multiindex notation , 838
if a
= (al,""
functions u(x , t) for which their classical derivatives VfV'; u E C( Qto ,T) for all p and a such 2p + la I 2. Notice that (14) is a restriction on the behavior of h near the moment t 0 only.
an) is a integer multiindex, then Vexx = Vex, ... VexXn.n Xl
:s
denotes the mixed partial derivative of order lal = al+ ·· ·+a n ·
=
The above conditions on G and Fi are , in fact , slightly stronger than we need . We refer to (Byrnes1 , 1995) for more precise conditions.
At this point we fix T > 0 and consider the problem (1), (4), (8) on the time interval [0, T] . Thus we seek a solution w(x , t) to (1), (4), (8) defined on QT. Since T is arbitrary, all the existence and uniqueness results obtained for t E [0, T] can automatically be extended to the infinite interval [0,00). We assume that the following restrictions are satisfied.
Our first main result consists in the following.
Theorem 1 Assume that all the conditions (10)(15) are satisfied. Then the following statements take place. a) The problem (1), (4), (8) with an arbitrary initial function 'P E L2(D) has a unique weak solution
=
1. G(x , t , 0) 0 (note that this is not really a restriction, but rather an agreement). 2.
(10) 3.
which satisfies Ilw(t) IVfV~VIG(x, t,OI
:s c31~lr +
C4,
--->
0 as t
--->
O.
b) Moreover , w E H 2m ,m(Q t o,T) with m from (12) for any to E (0 , T) . Due to (16) this means that for positive values of time (t > 0) this solution is C 2 ,l-smooth and , therefore , satisfies (1) in a classical sense.
(11)
where r < 2 + 4jn . (11) should be satisfied for any nonnegative integers p and q and multiindex a such that 2p + q + lal 2m, with
:s
m=[n;6]+1.
'PII
c) There is a globally defined dynamical system on the state space L2(D) given in terms of a generalized semigroup
(12)
Here we use the notation [a] for the greatest integer less than or equal a. 4.
This means we have a 2-parameter family of mappings {Stot" 0 :s tl :s t2} such that St3toStotl = St3 t , for tl ::; t2 ::; t3 , and Stt I. These mappings are defined by the formula : Stot, w(td = W(t2) for 0 ::; tl t2, where w is the solution of (1) , (3) with (2) replaced by an arbitrary initial data w(td E L2(D) at the moment t = t l . For time independent h , Fi and G , the generalized semigroup turns into a semigroup {St, t ~ O} by the rule Stot, = St2- t,.
where s < 1+2jn . (13) should be satisfied for any p, q and a such that 2p+q+lal:S 2m+1 , with m defined in (12). In all the conditions (10)-( 13) (x , t , ~) E QT x The constants Ci > 0 (i = 1, ···,6) may depend on T.
=
n. 5.
(14)
hE H 2m ,m(Q to ,T )
d) This semigroup is continuous in the following sense: each St,t, is a continuous mapping, for each 'P E L2(D) the function St,t2('P) is a continuous function of tl and t2 with respect to L 2 (D)-norm.
(15)
with m defined in (12), for any to E (0, T). By H 2m ,m(Q t o,T) we denote the Hilbert space of all functions u(x, t) which have weak derivatives in L2( Qto,T) for all p and a such that 2p+ lal :s m.
e) Each mapping St,t2 is compact for t2
> tl'
f) If h = 0 and Fi and G are independent of time, then there exists a positive continuous monotone increasing function a(O, ~ ~ 0 such that a(O) = 0 and
We remark that the following embedding theorem takes place (see, e.g., (Ladyzhenskaya, 1968»
H 2m ,m(Q to,T ) C C 2,1(Q to ,T ) ,
:s
( 16)
IISt'Pll
where C 2 ,l( Qto ,T) denotes the Banach space of all
:s a(II'PII)
for any t E [0 , (0) ,
839
which means that the closed loop system is globally Lyapunov stable .
This attractor is locally compact (i .e., its intersection with any ball is compact) , but it may be unbounded .
We make several brief remarks concerning the proof of this theorem. The main statements of the theorem are a) and b). The statements c)-f) are obtained in the process of proving a) and b). To obtain our result we first prove in (Byrnesl, 1995) the existence of the weak solution on a small time interval [0, T(llcplI, h)], where T(llcpll , h) --> as Ilcpll ---- 00 or IIhIILoo([O ,T] ,H-'(n)) ---- 00 . We have to do this since, as it has already been mentioned in the Introduction , our assumptions do not guarantee a global in time energy estimate. Next we investigate the regularity of the weak solution and show that it belongs to the space H 2m ,m (Q1G ,T) with m from (12) and < to < T ::; T(llcpll, h). This allows us to conclude that this solution is , in fact , classical for t > 0. This step is technically the most difficult part of the entire proof of the theorem. Finally, we combine the local in time solvability result with a global in time a priori estimate obtained from a classical maximum principle for parabolic equations (Ladyzhenskaya, 1968) and extend the local result to an arbitrary time interval [0 , T]. We notice that the condition (10) is introduced exactly to ensure that the maximum principle is valid . We note that the local dynamics problem for one dimensional (n = 1) convection reaction diffusion equations is examined in detail in (Okasha, 1995).
\Ve now turn to the se con main result of our work , which is closely related to Theorem 1. This result deals with the unique global in time solvability of (1) , (4), (8) in the case of sufficiently small initial data and disturbance h. While this case is does not apply to general initial data in L2(Q) , the restrictions on aij , Fi, G and h are considerably relaxed and , furthermore , our result gives important information about the dynamics of our system in a small neighborhood of the origin , when the disturbance is also small.
°
°
Theorem 2 Assume that aij E Hl(Q) , (12) is satisfied only for G (i .e., p = q = 0 0' = 0) and (13) is satisfied only for Fi and V';V:Fi with q+ 10'1 = 1. Assume , also, that G(x , t , 0) = 0, Fi(x, t , 0) = 0 and that h satisfies (14) . In this case there exists numbers p > and ~ > 0, such that if
°
Ilcpll ::;
(19)
p
and ess sup Ilh(t)IIH-l ( n ) ::;~, tE[O ,T ) then
(20)
°
a) for any T > there exists a unique weak solution w E LOO ([0 , T], L2(Q)) n L2([O , T], Hl(Q)) of the problem (1), (4) , (8) ;
We now turn to the the question of the existence of an attractor for our dynamical system (see (Constantin , 1985; Ladyzhenskaya, 1975; Ladyzhenskaya, 1991 ; Temam , 1988; Shubov , 1992)). Assume that h, Fi and G are time independent . In this case our system generates a continuous semigroup {Sl,t ~ o} on the state space L2(Q) such that all the operators SI with t > are compact. This is , however , not sufficient to conclude that the system has a global absorbing ball , i.e. , a ball in the state space into which every trajectory of the system eventually enters and remains. The only case in which we can establish the existence a global attractor is for a zero disturbance h as discussed in part f) of Theorem 1. Namely, fix any R> and let BR = {lP E L2(Q) : IllPll ::; R} be the ball in L2(Q) of radius R centered at the origin. Here and below by 11·11 we mean the norm in L2(Q). Consider our dynamical system {St , t > O} restricted to the state space MR Ut~OSt(BR) (it is obvious that MR is invariant by its definition) It follows from (18) that this new system has an absorbing ball Ba(R), where a is exactly the function from (18) . Since all operators {St , t ~ o} are compact we can conclude that this system has a compact attractor AR . Notice that AR, C AR2 for Rl ::; R2 and , therefore , the entire system defined on L2 (Q) has the global attractor A = UR~ o AR .
b) this solution is continuous with respect to t in the L2(Q)-norm , i.e.,
Ilw(t
°
+ At) -
w(t)11
~ 0,
as t
-->
0
for any t E [0, T) ; c) this solution has the estimate
Ilw(t)11 ::; p, t E [0, (0) ;
(21 )
d) if h = 0 the w satisfies
°
Ilw(t)11 ::; pe--yt ,
(22)
for a number I > 0;
=
e) if, in addition , h E LOC([0 , 00) , L2(Q)) and satisfies ess sup
11 h(t)11
::; ~l,
(23)
tE[O,oo )
with a sufficiently small an estimate 1
~l
Ilw(t)IIHl (n) ::; t 1 / 2 M(t, 840
>
0, then w has
Ilcpll), t > 0
(24)
on the gain k( x). Denote the corresponding solution of the zero dynamics problem by w(x, t) . Our final main result consists in the following .
Here M(t,O is a continuous monotone increasing function in t, ( ~ 0; f) the weak solution w depends continuously on the initial data 'P and the forcing term h in the following sense. If Wl and W2 are the solutions of (1) with different initial conditions 'PI and 'P2 and different forcing terms hI and h2 , respectively, then
Ilwl(t) - w2(t)11 2 2
C (t)
+
1t
::;
Theorem 3 Assume that 9 and h in both problems are the same and that all conditions of Theorem 1 are satisfied. Then
max Ilwk(t) - w(t)11 --- 0
tE[O,T)
if min Ik(x)l--
(25)
xEr
(1191 - 92112 + Ilhl (T) -
00 ,
(26)
where w(t) denotes the function w as a function of x E n with t fixed.
h2(7)11~-1(D) dT)
Moreover, for any T
where C(t) is a continuous function of t E [0, (0).
> to > 0
max Ilwk(t) - w(t)IIHq(D ) --- 0
tE [t o,T)
as min Ik(x)I--xEr
Explicit expressions for p, a- and I in terms of the known parameters of the problem (independent of 9 and h) can be given.
where q Hq(n)
= [%]
+ 1.
00 ,
(27)
Due to the embedding
c C(n), (27) implies that for t > 0
Corollary 1 Under the conditions in Theorem 2
maxlwk(x , t) - w(x, t)I--- 0
we have the following results.
xED
1. the ball Bp = {1P E L2(n) : 1111>11 ::; p} is invariant with respect to the dynamics of our system. This means , in particular, that the system is locally Lyapunov stable.
as min Ik(x)I--xEr
(28)
If only the conditions of Theorem 2 with (23) replaced by (20) are satisfied then (26) still takes place and instead of (27) we have
2. In this case, when F; , G and h are time independent, we have a continuous semigroup {St, t ~ O} with the state space Bp. It follows from (24) and the compactness of the embedding Hl(n) C L2(n) that all the mappings {St, t > O} are corn pact. Based on the well known results of attractor theory (see, e.g. , (Constantin, 1985 ; Ladyzhenskaya, 1975; Ladyzhenskaya , 1991 ; Shubov , 1992)) we can conclude that our system has a compact local attractor in Bp. The Hausdorff and fractal dimensions of this attractor are finite.
sup IIw k (t) - W(t)IIH1(D ) ---.. 0 tE[to ,T]
as min Ik(x)l-xEr
00.
(29)
4. REFERENCES C.I. Byrnes, D.S. Gilliam, V.l. Shubov, (1995) Boundary control for convection reaction diffusion equations, Preprint.
C.l. Byrnes, D.S. Gilliam, V.I. Shubov , (1995)
3. The estimate (22) implies that the equilibriumsolution w Oof(l), (4), (8) with h 0 is exponentially stable.
=
00.
Local and global dynamics for a boundary controlled viscous Burgers' equation, Preprint.
=
C.I. Bvrnes, D.S. Gilliam, V.I. Shubov, (1994)
Bou~dary Control, Feedback Stabilization and the Existence of Attractors for a Viscous Burgers Equation, Preprint
Finally we turn to our third and final main result which describes a nonlinear enhancement of classical root locus. To formulate this result we mention that a theorem similar to Theorem 1 takes place for the zero dynamics problem (1) , (4), (9), i.e., for the case of Dirichlet boundary conditions. The proof of such a theorem is actually much easier than the proof of Theorem 1, since in the case of Dirichlet conditions we have a global in time a priori energy estimate. We refer to our forthcoming work (Byrnes1 , 1995) for details. Let us denote the solution of the controlled problem (1), (4), (8) by wk(x, t) to emphasize that it depends
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