Exact Controllability of a hermoelastic System with Control in the hermal Component Only George Avalos January 8, 998 Abstract In this work we give a result of exact controllability for a thermoelastic system in which the control term is placed solely in the thermal equation. With such an indirect control input, one is able to control exactly the displacement of the plate, as well as the temperature. his exact controllability occurs in arbitrarily small time. In the case that the moment of inertia parameter for the plate is absent (i.e., = below), then one is provided here with a result of exact controllability for a thermoelastic system which is modelled by the generator of an analytic semigroup. he proof here depends upon a multiplier method so as to attain the associated observability inequality. he particular multiplier invoked is of an operator theoretic nature, and has been used previously by the author in deriving stability results for this pde model. epartment of Mathematics, exas ech University, Lubbock, exas 799{, USA. Research supported in part by the NSF Grant MS-9798.
Introduction. Statement of the Problem Let be a bounded open subset of R with suciently smooth boundary, and >. In this work, we will study the exact controllability problem for the following thermoelastic system, with the control function u [L (;;L ())] : 8 <! tt! tt +!+= on (;); : t +! t = div(u)! = @! @ = on (; ) ; () = on (; ) ;!(t =)=! ;! t (t=)=! ;(t=)= on : ere, the coupling parameter > ; the nonnegative constant is proportional to the thickness of the plate and assumed to be small with M; the constant is also nonnegative. here are other physical constants associated with system, but they have been set here to unity for the sake of simplicity. he operator div denotes the divergence of the vector eld u(x; y) =[u (x; y);u (x; y)]; i.e., div(u) = @u @x + @u @y. As usual, [ ; ] is the unit normal outward to the boundary. he pde model (), without the given interior control, is derived in [], and mathematically describes a Kircho plate subjected to a thermal damping. he displacement of the plate is represented by the function!, and the temperature is denoted by the function. he control term div(u) models a radiative energy ux acting through the volume of the plate. ening the space ; () to be 8 < () = () if > ; () : L () if =, one can show well{posedness of the uncontrolled thermoelastic system (i.e. u = in ()) for initial data [! ;! ; ] () ;() L () (see Proposition. below). In general, for given u [L (;;L ()], the corresponding solution [!;! t ;]isa priori continuous in time into the larger space (A ) (larger with respect to () () ; L ()), this dual space being dened below in (8). We note here at the outset the dichotomy presented by the parameter : With >, the system () is hyperbolic{like; when =, the system is modelled by the generator of
an analytic semigroup, and so corresponds to parabolic{like dynamics (see [8] and [5]). With the basic space of well{posedness established, we are concerned with the following question of exact controllability on a given time interval [;]: For data (terminal) in () ;() L (), is there [! ;! ; ] (initial) and! ;! ; a suitable control u [L (;;L ()] such that the corresponding solution [!;! t ;] to () satises at terminal time ; [!();! t ();()] =! ;! ;? () Controllability properties for this system have been much studied of late, under varying boundary conditions for the displacement, and with dierent choices of controls. he controllability of the system () is initially considered by J. Lagnese in [], with control being implemented in the boundary conditions for! (in this work, free boundary conditions are imposed, instead of the hinged ones in place here). With such a boundary{controlled thermoelastic system, a result of partial exact controllability is obtained for > (the hyperbolic case); that is to say, the displacement! is exactly controlled, provided the coupling parameter is small enough. In a more recent work, valid for > (the hyperbolic case), L. de eresa and E. uazua in [6] derive a result of exact controllability for the displacement! and approximate controllability for the temperature, in the case that interior control is implemented in the Kircho component of (). In each of these works, given that the control term is acting strictly on the plate component of the dynamics, and that >, critical use is made of controllability results for the uncoupled Kircho plate so as to eventually treat the system () as a perturbation of a Kircho plate. Later still, S. ansen and B. hang in [8] study a one{dimensional version of () under the inuence of a control at one of the boundary conditions for!. With such a single scalar control in place, they are able to obtain a result of exact null controllability; i.e., [!;! t ;] can be driven to zero at time ; this result holds for all. ere, we address the aforementioned question of exact controllability for both the displacement and the temperature, and in both the analytic and nonanalytic cases. Our main result in that direction is as follows: heorem. For all, the system () is exactly controllable in arbitrary time >. hat is to say, for any >, and data [! ;! ; ];! ;! ; in the space () ;() L (), one can nd a control function u [L (;;L ()] such that the corresponding solution [!;! t ;] to () satises [!( );! t ();()] =! ;! ;. he novelties inherent in this theorem are the following: () heorem. states that the displacement of the plate! can be controlled exactly by the indirect means of placing the control input term divu in the thermal component. Note that since the control term is in the heat equation, the proof of
exact controllability will not hinge on perturbation arguments which exploit known controllability results for Kircho plates in the case that >, or Euler{Bernoulli beams in the case that =. he proof of heorem. is necessarily direct. () In the case that =, it has recently been demonstrated in [5] that the thermoelastic system (), under all possible boundary conditions for!, is abstractly modelled by the generator of an analytic semigroup (see () and (6) below). herefore, heorem. constitutes a result of exact controllability for an analytic system in the case that =. (It is well{known that exact controllability results for analytic systems are hard to come by. See [5] and [] for statements of some sucient conditions for the approximate and exact controllability of analytic systems.) In this respect, our work here complements that recently completed by I. Lasiecka and R. riggiani in [6], which gives results of exact null controllability for the thermoelastic model () in the (analytic) case that =, under the inuence of either mechanical or thermal control (as we said earlier, the aforementioned paper [8] also recovers the null controllability of a one{dimensional version of () for = ). he methodology employed in the proving of heorem. is based upon the classical argument of showing the ontoness of the control! terminal state map L (see (5) below for the explicit description of L ). Establishing the surjectivity for L is in turn tantamount to deriving the following (observability) inequality for some C > : kr k C L () k[ ; ; ]k ; () () ; ()L () where is the thermal component of the solution [; t ; ] to the following backwards thermoelastic system, adjoint with respect to (): 8 < tt tt + + = on (;); : t+ t = = @ @ = on (;) ; = on (;) ; ( )= ; t ()= ; ()= on : Amultiplier technique is invoked here to attain the inequality () (see [9] for a treatise of the multiplier method), with the chosen multiplier being of an operator theoretic nature. In fact, the critical multiplier is A, where the operator A denotes the Laplacian with irichlet boundary conditions (see (9) below). his particular multiplier has also seen service in [], [] and [], works which are concerned with ascertaining stability properties of linear and nonlinear variations of ().
Abstract Operator Formulation and Analysis In our proof of controllability (heorem.), the system () and its adjoint (() below) will be considered as abstract evolution equations in a certain ilbert space. o develop these operator models, we must introduce the following denitions and notations. We dene the operator A: L () A! L () to be A= ;with domain (A) = () \ (): (5) A is then positive denite, self{adjoint, and consequently from [7] we have the characterizations (A )= (); (A )= (); (6) (A )= () \ (): In particular, the second characterization in (6) and Green's formula give that for all $, $ (A ), A$; $ h (A = A )i (A $; A $ =($; $ ) ; (7) ) L () and additionally, k$k (A ) = A $ L () L () = k$k L () : (8) We dene A : L () (A )! L () to be A = ; with irichlet boundary conditions, viz. A is also positive denite, self{adjoint, and by [7] (A )= () \ (): (9) (A )= (): () Moreover, this characterization and Green's heorem give that for all, (A ) ha ; i [(A = )] (A ) A ; A L () =(r ; r ) [L ()], () 5
and k k = A (A ) L () = kr k [L ()] : () For, we dene the operator P by and here consider two cases: P I + A, () (i) In the case that the parameter >;we dene a space ; () equivalent to () with its inner product being dened as ( $; $ ) ; () ($; $ ) L () + (r$; r$ ) L () 8 $; $ (); () and with its dual denoted as ;(). he characterization (), () and two extensions by continuity will then yield that P L ;(); ;(), with (5) hp! ;! i =(! ;! ; () ) ;() : (6) ; () Furthermore, the obvious ;(){ellipticity ofp and Lax{Milgram give that P is boundedly invertible, i.e. P L ;(); ;() : (7) In addition, the operator P : L () (P )! L (), being positive denite and self{adjoint, has its square root P as a well{dened operator with (P )= ;() (after using the interpolation theorem in [], p. ); it then follows from () and (6) that for $ and $ ;(); P $ L () = k$k + L () kr$k = k$k ; (8) [L ()] ; () P $; P $ L () (ii) In the case that =, then P = I, and we simply set =($; $ ) ; () : (9) ;() = ;() = L (). () 6
We denote the ilbert space to be (A ) ;() L (); () with the inner product @!! e! 5 ; e! e = A! ; A e! 5A L () + P! ;P e! L () + ; e L () With the above denitions, we then set A : (A )! to be I A @ P A P A A A A I with (A )= We dene a (control) operator B L by having for u [L ()], n [! ;! ; ] (A )(A )(A ) and such that A! ;() : Bu = [L ()], (A ) L () divu : () () h (A ) i 5 : () Finally, we dene the map L :[L (;;L ()]! byhaving for all u [L (;;L ())], L u e A ( t) Bu(t): (5) If we take the initial data [! ;! ; ]tobein, and control u [L (;;L ()], then the coupled system () becomes formally the operator theoretic model!! d! t 5 = A! t 5 + Bu!()! t () () 5 = 7!! 5 : (6)
Similar to what was done in [] and in [] for the thermoelastic system () with higher order boundary conditions in place, one can show the existence of an associated semigroup e A t. In particular, we have t Proposition. (well{posedness) Again with the parameter, A, as dened in (), generates a C semigroup of contractions e A t on the energy space t : as With these dynamics in hand, the solution [!;! t ;] to () may be written explicitly!(t)! t (t) (t) 5 = e A t!! 5 + t e A (t s) Bu(s)ds; (7) and as we show below that BL [L (;;L ())] ; (A ) (see Proposition. and Remark. below), then a priori this input to state map gives that [!;! t ;] C [;]; (A ) : Given the representation (7) for the solution [!;! t ;], proving the asserted exact controllability at given time >(heorem.) is then equivalent to the functional analytical principle of showing the surjectivity of the operator L, where L is dened in (5) (see [9] and []). Note that L L [L (;;L ())] ; A control operator L is well{dened as an element of, asbis (see Remark. below); therefore, the as a mapping into the state space makes sense a priori only as an unbounded operator with some given domain of denition. owever, in what follows below, we show that the map L can be extended to all of [L (;;L ())]. In particular, we have: Lemma. he operator L L [L (;;L ())] ;. he proof of this result follows from a chain of propositions. Proposition. he ilbert space adjoint A of A is given by A = @ I A P A A, A A I P with (A )=(A ): (8) One can work to show that in fact [!;! t ;] C([;]; ) for. 8
Proof: ene the operator :! as = @ P I A P A A A I with ( )=(A ): hen for [! ;! ; ] and he! ; e! ; e i (A )wehave A, (9) @A =! 5; e! e! e! A! ;A e! 5A L () P + P P A ;P e! L () (A + I) ; e P A! ;P e! L () A! ; e L () L () =! ; Ae! [(A )] (A ) A! ; A e! + ( ;A e! ) L! () ;A e L () ; (A + I) e L () L () = = A! ; A e! + P! ; P L () P + A e + ; (A + I) e L () @!! 5 ; e! e! e 5 : A P! ;P L () P Ae! L () +( ;A e! ) L () 9
hus, ( ) (A ) and A ( = : () ) as On the other hand, one can explicitly compute the inverse of A L( ;(A )) A = @ A A (A + I) A A P A A (A + I) I (A + I) A (A + I) A ; () in turn, its ilbert space adjoint A L( ;(A )) can be computed as A = @ A A (A + I) A A P A A (A + I) I (A + I) A (A + I) A : hus for [! ;! ; ],wehave that @ A A (A + I) A A P A A (A + I) I (A + I) A (A + I) A!! 5 = and so In addition, A A (A + I) A! + A P! A A (A + I)! (A + I) A! (A + I) (A ) ( A ) ( A ) (A ): 5 ; A (A + I) A! + P! A (A + I) ; (): From these two containments, the denition of ( ) in (9), and (), we then deduce that the adjoint A is as given in (8). Remark. Since div L [L ()] ; (), we have from () and (8) that BL [L ()] ; (A ). Using the form of A given in Proposition., and its associated semigroup e A t,we quickly have the following: t
Corollary.5 For terminal data [ ; ; ], the function [; t ; ] C([;]; ), dened by (t) t (t) (t) 5e A ( t) 5; () is a weak solution of the following system (adjoint with respect to ()): 8 < : tt tt + + = t+ t = on (;); = @ @ = on (;) ; () = on (;) ; ( )= ; t ()= ; ()= on : Remark.6 Note that for data [ ; ; ](A ), the system () may be written abstractly as (see (8) and ()) 8 >< P tt = A + A in ; (); t = A t +(A +I) in L (); () >: [( );(); ()] = [ ; ; ]: Concerning this adjoint system, we have the following additional regularity and energy relation: Proposition.7 he component of the solution [; t ; ] to the backward system () satises L ; ; (A ). Indeed, we have the following relation valid for all data [ ; ; ] : = i hk[(); t (); ()]k k[ ; ; ]k A (t) L () k (t)k. (5) L ()
Proof: aking [ ; ; ] (A ), we have, via Corollary.5, d (t) (t) (t) t (t) 5 = @A t (t) 5 ; t (t) 5A (t) (t) (t) = A t (t); A (t) A (t); A t (t) L () L () + (A (t); t (t)) L () (A t (t); (t)) L () + A (t);a (t) + k (t)k : L L () () Integrating both sides of this equation from to and using the terminal condition of () gives the relation (5), at least for smooth [ ; ; ]. he fact that L ; ; (A ), with continuous dependence on the data, now comes from the contraction of the semigroup e A t. A density argument then concludes the t proof. Proposition.8 he operator B e A ( ) L ;[L (;;L ())], and for every [ ; ; ] B e A ( ) 5 = r, (6) where is the thermal component of the solution [; t ; ] to the adjoint system (). Proof: Using the denition of B in () and Remark., we compute its adjoint B L (A ); [L (;;L ())] : For every u [L (;;L ())] and [ ; ; ] (A ); * Bu; + 5 = @A Bu(t); A [(A )] (A ) 5A = @ A A (A + I) divu (A + I) divu (after using () and (8)) 5 ; P A P A A (A + I) 5A
= A (A + I) divu; [(A )] (A ) (A + I) divu; A L () + hdivu; i = hdivu; i [(A )] (A ) = (u; r ) = [L @u; B ()] [(A )] (A ) 5 : (7) A his form of the adjoint B, the fact that e A () L (A );C([;]; (A ), Proposition.7, and a density argument give the result. Proof of Lemma. :From a computationalong the lines of that undertaken for (A (7), it can be shown that the adjoint L L ) ;[L (;;L ())] has the classical form L 5 () =B e A ( ) 5 (8) (see [])). Proposition.8 and the use of duality then provide the asserted Lemma.. Proof of heorem. As mentioned above, showing the exact controllability for given > isequivalent to showing the ontoness of the operator L L [L (;;L ())] ; (see (5) and Lemma.). In turn, by the classical functional analysis, the surjectivity ofl is equivalent to showing that there exists a constant C > such that the following injectivity condition holds for all [ ; ; ] : L 5 [L (; ;L ())] C 5 : (9) It is this inequality which we will proceed to verify. Note that in \pde form", the inequality (9) becomes (see (8), Proposition.8 and ()) A C 5, () L ()
where is the (thermal) component of the solution [; t ; ] to the backwards system (). Recall, that using the semigroup generated by A,[; t ; ] has the explicit form given in (). Accordingly, one can show, in a fashion similar to that in employed in [], [] and [], that for data [ ; ; ], the corresponding solu-, enjoys the following tion [; t ; ] to (), besides residing in C regularity: [;]; C([;]; (A)); A A t C([;]; ()); C([;]; ()): () In view of this regularity for solutions corresponding to smooth initial data, a density argument will therefore allow the assumption throughout that [; t ; ] has the regularity needed to justify the computations performed below. Also, we will have frequent need throughout of the following Green's heorem which is derived in [] for functions $ and $ \smooth enough": @$ ( $)$ d=a($; $ )+ @ +( )@B $ $ d @ [$ +( )B $] @$ @ d ; () where the bilinear form a(; ) is dened by a ($; $ ) $xx $ + xx $ yy$ + yy $ xx$ + yy $ yy$ xx + ( )$xy $ xy ere, (; )ispoisson's ratio, and the boundary operators B i are given by d: () B $ @ $ @x@y @ $ @ $ @y @x ; () B $ ( ) @ $ @x@y + @ $ @ $ : @y @x Step (Proof of a requisite trace result). We rst derive a trace regularity result for the adjoint system () which does not follow from the standard Sobolev trace theory, and which is critical in obtaining the estimate (). his result is analogous to that proved in [] and []. We note that related trace regularity results for Euler{ Bernoulli plates were proved in [7], and for Kircho plates in [].
Lemma. he component of the solution [; t ; ] of () satises j L (;;L ( )), with the accompanying estimate kk L ( ) C A L () where C does not depend on the parameter : + P t L () + A L () + k[ ; ; ]k ; (5) Proof: We start by multiplying the rst equation of () by the quantity h r, where h(x; y) [h (x; y);h (x; y)] is a C () vector eld such that hj =[ ; ] on, and subsequently integrate from to so as to obtain the equation We now estimate the left hand side. tt tt + + ;hr =: (6) L () (i) o start o, ( tt ;hr) =( L () t;hr) L () ( t ;hr t ) L () = ( t ;hr) L () + = ( t ;hr) L () t [h x + h y ] d + div t h d t [h x + h y ] d; (7) after making use of the divergence theorem and the fact that t = on. (ii) Next, ( tt ;hr) L () = (r t; r (h r)) [L ()] (r t ; r (h r t )) [L ()] (after using in part the fact that h r= @ @ = on ) 5
= (r t ; r (h r)) [L ()] + tx h x tx h y + ty h y d + ty h x div jr t j h d d [ tx ty h x + tx ty h y ] d = (r t ; r(h r)) [L ()] + tx h y + ty h x tx h x ty h y d [ tx ty h x + tx ty h y ] d; (8) after again using the divergence theorem and the fact that R div jr tj h d= R jrt j d =(as t (t) ()). (iii) o handle the fourth order term in the expression (6), we use the Green's heorem () and the fact that h r= on to obtain ; h r = a (; h r) L () ( +( )B ) @ d : (9) @ We note at this point that we can rewrite the rst term on the right hand side of (9) as a (; h r) = R where O A L () h r xx + yy + xx yy + ( ) xy d + O A L () ; (5) denotes a series of terms which can be majorized 6
by R A L (). We consequently have by the divergence theorem that a (; h r) = +O A h r xx + yy + xx yy + ( ) xy d L () = div h xx + yy + xx yy + ( + O A L () ) xy = = xx + yy + xx yy + ( ) xy d + O A L () () + O A L () ; (5) where in the last step above, we have used the fact (as reasoned in [], Ch. ) that j = @ @ = implies that + xx yy + xx yy + ( ) xy =() on. o handle the second term on the right hand side of (9), we note that B = on, which implies that =+( )B = @ on : (5) @ he insertion of (5) into (9), followed by the consideration of (5) then yields that ; h r L () = kk L ( + O A 7 L () ) : (5)
(iv) o handle the last term on the left hand side of equation (6), we use the classical Green's theorem and the fact that h r = on to obtain ( ;hr) = L (r ; r (h r)) () [L : (5) ()] o nish the proof, we rewrite (6) by collecting the relations given above in (7), (8), (5) and (5) to obtain the relation kk L = (r ; r (h r)) ( ) [L + O ()] A L () + O P t +( t ;hr) L () + (r t ; r (h r)) [L : ()] L () A majorization of this quantity, which in part uses the contraction of the semigroup e A t, gives the desired inequality (5). t Step (Proof of the Inequality ()) We multiply the rst equation in () by A to obtain and integrate in time and space tt tt + + ;A =; (55) L () and proceed to estimate this quantity. (A.) ealing with R tt tt ;A : Using an integration by parts L () and the second dierential equation of () yields tt tt ;A L () = = t ;A h + t ;A L () h + r t ; ra [L ()] i t ;A t + r L () t; ra t [L ()] h i k t k + L kr () tk L () t ; I + A L () + r L () t; r I + A + r t ; ra [L ()] 8 i [L ()] : (56)
In regards to the last two terms of this relation, we haveby Green's heorem and the fact that t () for >, that for any t [;] Moreover for t ; r t (t);ra (t) = ( [L ()] t (t); (t)) L () C k t k C([; ];L ()) + 8 k (t)k L () : (57) t (t);a (t) L () 8 k t(t)k + L C () A (t) L () 8 k t(t)k L () + C k (t)k [(A )] 8 k t(t)k L () + C k k C([; ]; ()) ; (58) after using (). Combining (57) and (58), using the contraction of the semigroup e A t t and (56), we then have the estimate C tt tt ;A L () k t k L () A L () i hk t k L() + kr tk [L()] + kr t k [L ()] A L () + k ; ; k +C k t k C([; ];L ()) + k k C([; ]; ()) P t L () + k ; ; k +C + k t k C([; ];L ()) + k k C([; ]; ()) A L () : (59) where the constant C does not depend on, M: (A. ) ealing with R!; A : Another application of Green's theorem in () and the fact that A = give ; A L () = a ; A ; @A @ L ( ) : (6) 9
Estimating the right hand side of (6) yields, after the use of trace theory, elliptic regularity and the mean inequality, C ; A L () A A L () L () + kk L C + C ( ) A L () (where the inverted C is the same constant present in (5)) C + A A L () L () A +C A L () L () + P t L () + k[ ; ; ]k (by Lemma.) A +C L () A + L () P t L () + k[ ; ; ]k ; (6) after the use of the mean inequality. (A.) ealing with R ;A L () : Easily we have ;A L () = A ;A L () = k k : L () (6) (A.) Combining (55), (59), (6) and (6) thus results in the following: For > small enough there exists a constant C> (independent of ) such that the solution
[!;! t ;] of () satises ( ) C A A hk t k L() + kr tk [L()] i L () L () + k t k C([; ];L ()) + k k C([; ]; ()) + k[ ; ; ]k, (6) where the noncrucial dependence of C upon has not been noted. (A.5) he Conclusion of the Proof of heorem. : o majorize the norm of the component, wemultiply the rst equation of () by, integrate from to and integrate by parts to thereby obtain the relation P t ;P = L () A L () + P t L () A ;A Concerning the rst term in this relation, we have for 8 t [;] P t (t);p (t) L () P t (t) L () L () P + C (t) k[ ; ; ]k P +C (t) : (6) L () L () : (65) Combining (6) and (65), we eventually arrive at the following estimate for > small enough: ( ) A P t + P L () L () + C A C([; ];L ()) L () + k[ ; ; ]k ; (66) where the noncrucial dependence of C upon has not been noted.
hus, if is small enough, we then have, upon combining (6) and (66), the existence of a constant C such that A + k t k + kr L L () tk + k L () k L () () + ( )( ) k[ ; ; ]k +C + P C([; ];L ()) A L () + k t k C([; ];L ()) + k k C([; ]; ()) : (67) From here, we apply the relation (5) and its inherent property that k[(t); t (t); (t)]k k[(); t (); ()]k 8 t [;] (recall that [; t ; ] solves the backward problem ()) to obtain k[ ; ; ]k A (t) k (t)k L () = k[(); t (); ()]k L () k[(t); t (t); (t)]k + ( )( ) k[ ; ; ]k +C A + P L () + k t k + k C([; ];L C([; ];L ()) k C([; ]; : (68) ()) ()) aking > small enough in (68), we then have the following preliminary inequality valid for all and >: k[ ; ; ]k C A + P L () + k t k + k C([; ];L C([; ];L ()) k : (69) C([; ]; ()) ()) he inequality () is nally obtained after invoking the following proposition which can be derived through a (by now) classical compactness/uniqueness argument (see e.g., [] and []. Proposition. he inequality (69) implies the existence ofaconstant C that the corresponding solution [; t ; ] of () satises P + k t k + k C([; ];L ()) k C([; ]; ()) C([; ];L ()) C A L () such : (7)
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