A different look at controllability

0.13 MB·

by  Pablo Pedregal

A DIFFERENT LOOK AT CONTROLLABILITY
PABLOPEDREGAL
Abstract. We explore further controllability problems through a standard least square ap-
proach as in [20]. By setting up a suitable error functional E, and putting m(≥ 0) for the
4 infimum, we interpret approximate controllability by asking m=0, while exact controllabil-
1
ity corresponds, in addition, to demanding that m is attained. We also provide a condition,
0
formulated entirely in terms of the error E, which turns out to be equivalent to the unique
2
continuation property, andto approximate controllability. Though werestrictattention here
n
to the 1D, homogeneous heat equation to explain the mainideas, they can be extended in a
a
similar way to many other scenarios some of which have already been explored numerically,
J
duetotheflexibilityoftheprocedureforthenumericalapproximation.
4
1

1. Introduction
]
C
We wouldlike toexplorecontrollabilityproblemsthroughaleastsquareapproximationstrat-
O
egy. As a matter of fact, it has already been considered and proposed in [20].
.
h As in a typical least-square approximation, we will set up a non-negative error functional E,
t
a defined in a suitable space, and let m = infE ≥ 0. Approximate controllability is then defined
m
by demanding m = 0, while exact controllability occurs when, in addition, m is a minimum.
[ The main benefit we have found concerning this viewpoint is that the controllability problem is

1 translated, in an equivalent way, into minimizing such an error functional. As such, from the
v numerical point of view, one can then proceed to produce numerical approximation by utilizing
6
typical descent strategies. Especially in linear situations, such an error functional is convex
3
1 and quadratic (though not necessarily coercive),and this numericalprocedure should workfine.
3 This is indeed so. In addition,it is also true thatm=0 is equivalentto the interesting property
1. E′ =0 implies E =0, so that the only possible critical value for E is zero ([20]).
0 We only treatexplicitly these ideasfor the homogeneousheatequationin (spatial)dimension
4
N = 1, as it will be pretty clear how to extend this philosophy to many other situations.
1
: Specifically, we take Ω=(0,1)⊂R, and T >0, the time horizon.
v
i For initial and final data u (x), and u (x), respectively, we would like to find
X 0 T
the right-point condition f(t), so that the solution of the heat problem
r
a
u −u =0 in (0,T)×(0,1), u(t,0)=0,u(0,x)=u (x),u(t,1)=f(t),
t xx 0
will comply with u(T,x)=u (x).
T
Thisistheboundarycontrollabilitysituation. Thereisalsoaninnercontrollabilitycaseinwhich
we take a fixed subdomain ω ⊂Ω.
E.T.S.IngenierosIndustriales. UniversidaddeCastillaLaMancha. CampusdeCiudadReal(Spain). Research
supportedbyMTM2010-19739oftheMCyT(Spain). e-mail:[email protected] .
1
2 PABLOPEDREGAL
Determinethesourcetermf(t,x)supportedinωinsuchawaythatthesolution
of the problem
u −u =fχ in (0,T)×(0,1), u(t,0)=0,u(0,x)=u (x),u(t,1)=0,
t xx ω 0
will comply with u(T,x)=u (x).
T
We refer to [7], and to [21] for classic results on controllability, and to [9], [13] for a more
recent analysis. [4] also contains important ideas through duality arguments in the context
of the Hilbert Uniqueness Method. See also [5], [11]. The numerical analysis of this kind of
controllability problems has attracted a lot of work. Without pretending to be exhaustive, we
would mention important contributions covering a whole range of methods and approaches in
[2], [4], [8], [12], [14], [15].
2. Approximate controllability
Tokeepthe formalismtoaminimumwithoutcompromisingrigor,letussticktothe situation
describedinthe IntroductionbytakingQ =(0,T)×(0,1),u∈H1(Q )furnishingthedatafor
T T
t=0andt=T,u(0,x)=u (x)andu(T,x)=u (x),respectively,andassumingthatu(t,0)=0
0 T
in the sense of traces. Let
A ={U ∈H1(Q ):U(t,0)=U(0,x)=U(T,x)=0 for x∈(0,1),t∈(0,T)},
0 T
A=u+A .
0
Foru∈A,defineits“corrector”v ∈H1(Q )tobetheuniquesolutionofthevariationalequality
0 T
(2.1) [(u +v)φ+(u +v )φ +v φ ]dxdt=0
t x x x t t
Z
QT
for all φ∈H1(Q ). Note that v is the unique solution of the minimization problem
0 T
1
Minimize in w∈H1(Q ): [(u +w )2+w2+w2]+u w dxdt,
0 T Z (cid:18)2 x x t t (cid:19)
QT
whose equilibrium equation is
(2.2) −(u +v ) −v +v+u =0 in Q .
x x x tt t T
The weak formulation of (2.2) is precisely (2.1). The error functional E : A→R+ is taken to
T
be the size of the corrector
1
E (u)= (v2+v2+v2)dxdt.
T Z 2 x t
QT
It is obvious to realize that if E(u)=0 because u has a vanishing corrector,then u is a solution
of the heat equation (2.2) with v ≡ 0, complying with boundary, initial, and final conditions
providedbyu,i.e. thetraceofuoverx=1istheboundarycontrolsought. LetT:A7→H1(Q )
0 T
be the linear, continuous operator taking u into its correctorv. Our setting is definitely a least-
square approach in the spirit of [3], [10].
Definition 2.1. (1) Wesaythatthedatumu(T,x)isapproximatelycontrollablefromu(0,x)
through the subset {1} of the boundary of (0,1), if for every ǫ>0, there is u ∈A such
ǫ
that E(u )<ǫ.
ǫ
(2) We say that the datum u(T,x) is exactly controllable from u(0,x) through the subset {1}
of the boundary of (0,1), if there is u∈A such that E(u)=0.
A DIFFERENT LOOK AT CONTROLLABILITY 3
(3) The unique continuation property holds, given our framework, if the only function v ∈
imT⊂H1(Q ) with
0 T
(2.3) (U v+U v )dxdt=0
t x x
Z
QT
for all U ∈A is the trivial one v ≡0.
0
Within the framework just introduced through this definition, we can now state our main
result.
Theorem 2.1. Let T >0, and u∈H1(Q ) be given. The following assertions are equivalent:
T
(1) the trace u(T,x) for t = T of u is approximately controllable from u(0,x) through the
end-point {1};
(2) the unique continuation property holds;
(3) E′ (u)=0 implies E (u)=0 for u∈A.
T T
Proof. We will show (2)=⇒(1)=⇒(3)=⇒(2).
Letu∈A, andtakeU ∈A . Considerthe variationu+ηU,preservingboundary,initial,and
0
final data. Let v be the correctorassociatedwith u, and put v+ηV for the correctorassociated
with u+ηU. By linearity, it is elementary to argue that
(2.4) [(U +V)φ+(U +V )φ +V φ ]dxdt=0
t x x x t t
Z
QT
for all φ ∈ H1(Q ), where V, as v itself, belongs to H1(Q ). On the other hand, it is also
0 T 0 T
elementary to compute the derivative of E (u+ηU) with respect to η at η =0. It is given by
T
(vV +v V +v V )dxdt.
x x t t
Z
QT
By using (2.4) for φ=v, we also can write
dE (u+ηU)
hE′ (u),Ui= T =− (U v+U v )dxdt.
T dη (cid:12) Z t x x
(cid:12)η=0 QT
(cid:12)
Recall that T:A=u+A 7→H1(Q ) can b(cid:12)e regarded as a linear, continuous operator taking
0 0 T
u into its corrector v. Then, because of the unique continuation property, hE′(u),Ui=0 for all
U ∈ A if and only if Tu = 0. Therefore, it is a standard fact in Functional Analysis that over
0
the quotient space A /kerT both quantities kTuk and
0
sup hE′ (u),Ui= sup (U v+U v )dxdt
T Z t x x
U∈A0,kUk=1 U∈A0,kUk=1 QT
should be equivalent norms. Hence, for some positive constant C >0,
kE′ (u)k≡ sup hE′ (u),Ui≥CkTuk=CE (u)1/2.
T T T
U∈A0,kUk=1
If, starting out at arbitrary u0 ∈ A, we follow the flow of −E′ , we would eventually reach a
T
certain u∈A so that kE′ (u0+u)k<ǫ. This, together with the previous inequality, yields the
T
approximate controllability result.
Assume now that u˜ ∈ A in a critical point of E . Under the approximate controlability
T
property, we would like to conclude that u is indeed a solution of the controlabilityproblem. To
this end, notice that:
4 PABLOPEDREGAL
• the infimum of E over A vanishes: this is the approximate controlability property;
T
• E is a non-negative, convex functional.
T
As a consequence of the convexity,if u˜ is a criticalpoint ofE , it has to be a minimizer as well.
T
But then the infimum becomes a minimum, and it has to vanish, i.e. E (u˜)=0.
T
Finally, let v˜ ∈ imT be such that (2.3) holds for all U ∈ A . Let u˜ ∈ A be such that
0
Tu˜=v˜. By the computations performed above,(2.3) implies E′ (u˜)=0, and so, by hypothesis,
T
kv˜k2 =E (u˜)=0, that is v˜≡0. (cid:3)
T
Theequivalencestatedinthistheoremimpliesthattheaboveconceptofapproximatecontrol-
lability might be a bit more flexible than the classic one, at least for data sets which are traces
for t=0 and t=T of H1(Q )-functions.
T
Proposition 2.2. For every positive time T >0, the unique continuation property in Definition
2.1 holds.
Proof. Simply notice that H1(Q )⊂A , and so we can take U =v in (2.3) to obtain
0 T 0
(v v+v2)dxdt=0.
Z t x
QT
Because v(0,x) = v(T,x) = 0 for all x ∈ (0,1), we conclude that v ≡ 0 in Q . This together
x T
with the vanishing boundary conditions v(t,0)=v(t,1)=0 implies v ≡0. (cid:3)
3. Extension
The setting described in the previous section admits some straightforward variations. The
choice of the space for the correctors v ∈ H1(Q ) can be changed. For example, one can take
0 T
v ∈H1(Q )forabroadersituation,andinthiscasethecorrectorwillenjoythenaturalboundary
T
condition all around Q : u = 0 for x = 0 and x = 1, while u = 0 for t = 0 and t = T. But
T x t
intermediate alternatives are also possible: v = 0 for t = 0, and t =T, and so u =0 for x= 0
x
and x=1, or v ∈A , as well. Another possibility is to define the corrector v for a.e. time slice
0
as a minimization problem only in space. This can again be easily set up in more or less the
same terms (see [20]).
Rather than considering these various possibilities which are straightforward variations, we
would like to explore the most general framework that this approachmay allow for u instead of
for v. Our ambient space will now be
A ={U ∈L2(0,T;H˜1(0,1)):U ∈L2(0,T;H˜−1(0,1)),
0 0 t
U(0,x)=U(T,x)=0 for x∈(0,1),t∈(0,T)}.
We are taking here
H˜1(0,1))={U ∈H1(0,1):U(0)=0},
0
while H˜−1(0,1) is its dual. Notice that every U ∈ A belongs to the space C([0,T];L2(0,1))
0
so that traces of U are defined for every time t ∈ [0,T] ([6]). If u ∈ L2(0,T;H˜1(0,1)) (and so
0
u(t,0)=0 for a.e. t ∈(0,T)), with u ∈L2(0,T;H˜−1(0,1)), furnishes initial and final data, we
t
will put as before A=u+A . This time initial and final data, u(0,x), u(T,x) merely belong to
0
L2(0,1).
A DIFFERENT LOOK AT CONTROLLABILITY 5
For u ∈ A, define its “corrector” v ∈ H1(Q ) to be the unique solution of the variational
0 T
problem
[(u +v )φ +(v −u)φ +vφ]dxdt =0
x x x t t
Z
QT
for all φ∈H1(Q ). Note that v is the unique solution of the minimization problem
0 T
1
Minimize in w∈H1(Q ): [(u +w )2+(w −u)2+w2] dxdt.
0 T Z (cid:18)2 x x t (cid:19)
QT
The error functional E :A→R+ is taken to be the size of the corrector
1
E(u)= (v2+v2+v2)dxdt.
Z 2 x t
QT
As above, we investigate the derivative of the error functional. To this end, put u+ηU for
U ∈A , and v+ηV, its corresponding corrector,with v the corrector for u. Then
0
[(U +V )φ +(V −U)φ +Vφ]dxdt=0
x x x t t
Z
QT
for all φ∈H1(Q ). In the same way,
0 T
hE′(u),Ui= (v V +v V +vV)dxdt= (Uv −U v )dxdt,
x x t t t x x
Z Z
QT QT
by taking φ=v in the last identity.
The unique continuation property, and the equivalence with approximate controllability are
established in the same way as before.
4. Exact controllability
Within this framework,exactcontrollabilitycanbe deducedasa consequenceofthe factthat
therangeofthemapTisclosed,inadditiontotheuniquecontinuationproperty. Moreprecisely,
recall the definition of the map T:u+A 7→H1(Q ), taking every feasible u∈u+A into its
0 0 T 0
corrector v, in the analytical framework of Section 2. The error functional corresponds to the
least-square problem
1
Minimize in u∈u+A : kTuk2.
0
2
Exact controllability can then be achieved as a consequence of two facts:
(1) the infimum m is in fact a minimum;
(2) m does vanish.
The unique continuationpropertyis relatedto the secondissue,but the firstis equivalenttothe
factthatthe rangeofTis closed. Exceptforgeneralremarksinvolvingthe adjointoperatorT∗,
the exactcontrollabilityissueinvolvessubtle anddelicateideasaboutCarlemaninequalitiesand
observability. This elegant theory is very well established (see some of the references indicated
in the Introduction).
6 PABLOPEDREGAL
5. The inner controllability case
Let ω ⊂(0,1) be an interval. Put Q =(0,T)×(0,1), q =(0,T)×ω. Let A be takennow
T T 0
as the space
A ={U ∈L2(0,T;H2(0,1)∩H1(0,1)):U ∈L2(Q ),U =0 on ∂Q },
0 0 t T T
and u ∈ L2(0,T;H2(0,1)∩H1(0,1)) with u ∈ L2(Q ), carrying the boundary (around ∂Ω),
0 t T
initial, and final data. For feasible functions u ∈ A ≡ u+A , let v be its associated corrector,
0
the unique solution of the problem
(5.1) v +v =χ (t,x)(u −u ) in Q , v =0 on ∂Q ,
tt xx QT\qT t xx T T
and the error functional E :A7→R+
T
1
E (u)= (v2+v2)dxdt.
T 2Z t x
QT
We also put T:A7→H1(Q ) for the linear, continuous mapping taking u into its corrector v.
0 T
Assume that u∈A is such that E(u)=0. Then v ≡0, and
χ (u −u )≡0 in Q .
QT\qT t xx T
Hence if we take f =(u −u )∈L2(Q ), then
t xx T
u −u =χ f in Q ,
t xx qT T
and the restriction f becomes the sought control.
Definition 5.1. (1) Wesaythatthedatumu(T,x)isapproximatelycontrollablefromu(0,x)
through the subset ω of Ω, if for every ǫ>0, there is u ∈A such that E(u )<ǫ.
ǫ ǫ
(2) We say that the datum u(T,x) is exactly controllable from u(0,x) through the subset ω
of Ω, if there is u∈A such that E(u)=0.
(3) The unique continuation holds if the only v ∈imT⊂H1(Q ) with v ≡0 in q and
0 T T
(5.2) (−Uv +U v )dxdt=0
t x x
Z
QT\qT
for all U ∈A is the trivial one v ≡0.
0
(4) We say that E is an error functional if E′ (u)=0 implies E (u)=0.
T T T
Just as in the boundary situation, the integral occurring in the unique continuation property
is precisely the integral that appears when computing the Gateaux derivative
dE (u+ǫU)
T =hE′ (u),Ui.
dǫ (cid:12) T
(cid:12)ǫ=0
(cid:12)
Indeed, because of linearity, (cid:12)
(5.3) V +V =χ (U −U ) in Q , V =0 on ∂Q ,
tt xx QT\qT t xx T T
if V is the variation produced in v by U ∈A in u. Then
0
hE′ (u),Ui= (v V +v V )dxdt.
T Z t t x x
QT
By using (5.3), we obtain
hE′ (u),Ui=− (U −U )vdxdt.
T Z t xx
QT\qT
A DIFFERENT LOOK AT CONTROLLABILITY 7
Let us focus on the second term
U vdxdt.
xx
Z
QT\qT
A first integration by parts yields
− U v dxdt+ U vdS.
x x x
Z Z
QT\qT ∂(QT\qT)
But since v =0 around ∂Q , we find that the boundary integral equals
T
U vdS =− U vdxdt.
x xx
Z Z
∂qT qT
WecanalwaystakeU tobearbitraryinq ,andindependentoftime,sothatv ≡0inq . Hence,
T T
altogether,
(5.4) hE′ (u),Ui=− (U v+U v )dxdt.
T Z t x x
QT\qT
This isthe basiccomputationforanequivalenceasinthe boundarysituation. The prooffollows
exactly along the same lines as with the boundary counterpart.
Theorem 5.1. Let T > 0, and ω ⊂ Ω be given. Let also u ∈ A furnish initial and final data.
The following are equivalent:
(1) the trace u(T,x) for t=T of u is approximately controllable from u(0,x) through ω;
(2) the corresponding unique continuation holds;
(3) E is an error functional in the sense of Definition 5.1.
T
In this setting, it is also immediate to check that the unique continuation condition holds,
so that we have approximate controllability as well. Just notice that the corrector v, being the
solution in (5.1), is a feasible direction U because H2(Q )∩H1(Q )⊂A . By taking U =v in
T 0 T 0
(5.4), we conclude immediately that v ≡0.
6. Final comments
The formalism introduced here, and described in detail for the linear, homogeneous heat
equationin(spatial)dimensionN =1canbe formallyextended, inaratherdirectway,tomany
other frameworks because of its flexibility. The specific treatment of the unique continuation
propertymayhoweverchangefromsituationto situation. Forinstance,it is well-knownthatfor
the wave equation the unique continuation property requires a certain size of the horizon T due
to the finite speed of propagation. Some of those situations include, but are not limited to:
• higher dimension N >1;
• inhomogeneous heat equation;
• wave equation;
• systems of differential equations;
• situations for degenerate equations;
• non-linear problems.
Especiallyinlinearcases,thisviewpointnaturallyleadstoaniterativeapproximationscheme
based on a standard descent method. It has already been tested in various scenarios and, at
least numerically, it performs very well (see [1], [16], [17], [18], [19]).
8 PABLOPEDREGAL
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ETSI Industriales, Universidadde Castilla-La Mancha,13071CiudadReal,Spain
E-mail address: [email protected]