Pierre Lissy. 19 mars 2015

Transcription

1 Explicit lower bounds for the cost of fast controls for some -D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the -D transport-diffusion equation Pierre issy 9 mars 5 bstract In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional aplace operator We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, the Miller s conjecture formulated in [Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J Differential Equations, 4 4, pp -6] is not verified Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection-diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in [J-M Coron and S Guerrero, Singular optimal control: linear -D parabolic-hyperbolic example, symptot nal, 44 5, pp 37-57] concerning this problem is also not verified at least for negative speeds The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method of Fattorini and Russell Introduction Presentation of the problems et us consider the -D aplace operator with domain D := H, and state space H := H, It is well-known that : D H, is a positive definite operator with compact resolvent, the k-th eigenvalue is with eigenvector λ k = k π, kπx e k x := sin Thanks to the continuous functional calculus for positive self-adjoint operators, one can define any positive power of et us consider here some > and let us call / := / lissy@ceremadejussieufr CEREMDE, UMR 7534, Université Paris-Dauphine & CNRS, Place du Maréchal de attre de Tassigny, Paris Cedex 6, France

2 In what follows, we will consider two types of controlled equation on, T,, one of parabolic type, that we write as { y t = / y + bu in, T,, y, = y in,, and one of dispersive type, that we write as { y t = i / y + bu y, = y in, T,, in,, where, for every ϕ D /, ie bϕ = ϕ, b := δ, and u, T, K, K := R for or C for Equation can modelize anomaly fast or slow diffusion see for example [4], whereas can be used to study the energy spectrum of a -D fractional oscillator or for some fractional Bohr atoms see for example [9] For both equations, the most interesting case for physicists is /, ] If N, one can observe, using integrations by parts, that b corresponds to a boundary control on the left side on the / -th derivative of y, so that b can be considered as a natural extension of the boundary control in the case of non-even This kind of controls has already been introduced in [7, Section 33] to give some negative results about the control of fractional diffusion equations with and in [3, Sections 3 and 33] as an application of some results about the cost of fast controls for some classes of abstract parabolic or dispersive equations One can prove, using the result of [8] for diagonal semigroups and scalar control, that b is an admissible control operator see also [3, Section 3 and 33] Moreover, it is well-known that these equations are null-controllable in arbitrary small time see [4] for the parabolic case and for example [3] for the dispersive case Hence, one can easily prove see for example [, Chapter, Section 3] that for every y H, there exists a unique optimal for the, T, K-norm control u opt, T, K bringing y to the equilibrium state, the map y u opt is then linear continuous The norm of this operator is called the optimal null control cost at time T or in a more concise form the cost of the control, denoted C H T,, for equation and C S T,, for equation et us recall that these constants are also the smallest constants C > such that for every y H, there exists some control u driving y to at time T with u,t,k C y H Our first goal is mainly to continue the study done in [3] In this article, the author proved precise upper bounds concerning the cost of the control for some large classes of linear parabolic or dispersive equations including notably and for when the time T goes to, where the underlying elliptic operator was chosen to be self-adjoint or skew-adjoint with eigenvalues roughly as k or ik for some when k + The author also proved some lower bounds that were optimal concerning the power of T involved, but these estimates were not precise enough to understand what was the dependence of the cost of the control with respect to and Here, we will in fact be able to give precise lower-bounds for equations and as soon as > and not only, which will then generalize a little bit the study of lower bounds initiated in [3] Moreover, in the dispersive case, we will see that in the particular case = ie the classical Schrödinger equation controlled on one side of the boundary, we will find again the lower bound that is conjectured to be the optimal one by Miller in [5], but a very surprising result is that in

3 the case of the heat equation controlled on one side of the boundary ie with =, our lower bound will be twice bigger than the one expected according to the conjecture done by Miller in [6], and commonly accepted up to now see notably [3], [3] or [9] We will then formulate a new conjecture for this problem Remark Here, for the sake of simplicity and because we think that it is enough for our purpose, we chosed to treat only the case of equations and However, the results given below might be adapted to the following more general cases or y t + y = bu y t + iy = bu, where is a positive selfadjoint operator on some Hilbert Space H with eigenvalues λ n the corresponding eigenvector being denoted e n, with the assumption that λ n n is a regular increasing sequence of positive numbers verifying moreover that there exist some > and some R > such that λ n = Rn + O n, n and b is a scalar control input, ie b D and u, T, K, where K := R or C, and the sequence < b, e k > D,D k N is bounded from above and below see [3] Understanding the behavior of fast controls is of interest in itself but it may also be applied at least in some cases to study the uniform controllability of transport-diffusion equations in the vanishing viscosity limit as explained in [] and [] because of the strong connection existing between these problems and highlighted in these references In fact, the technique of the proof we will give here to estimate the cost of fast controls for equations and can also be used to obtain a new result for the transport-diffusion problem that we introduce now et us consider some constant M > and some viscosity coefficient ε > We are interested in the following family of transport-diffusion equations y t εy xx My x = in, T,, y, = vt in, T, 3 y, = in, T, y, = y in,, with initial condition y H, and control v, remark that the speed of the convection term is negative If ε is taken equal to and if the initial condition y is taken in,, we obtain a transport equation at constant speed: y t My x = in, T,, y, = in, T, 4 y, = y in,, which is known to be null-controllable if and only if T /M, the optimal control in -norm is in this case the null function since we do not act on the equation s before, one can define for equation 3 some cost of the control C T D T,, M, ε, and in the sequel we will precisely study its dependence with respect to ε at fixed T,, M Such a family of equations will be said uniformly controllable at time T if and only if C T D T,, M, ε as ε and non-uniformly controllable otherwise s we will see later, the typical behavior of this kind of equations is that the cost of the control explodes for small enough T and decreases exponentially for large enough T when ε tends to Our goal here will be to give a new lower bound for the minimal time needed to ensure uniform controllability 3

4 State of the art We will restrict here mainly to recall results in the -D case the situation is far more complicated in the multidimensional case, see for example [5] and [6] The first results concerning the cost of fast boundary controls have been obtained in the case of heat and Schrödinger equations Concerning the one-dimensional heat equation on, T, with boundary control on one side, the time-dependence of the cost of the boundary control is exp β + /T for some constant β > see [7] for the lower bound and [8] for the upper bound, where the notation β + means that we simultaneously have that the cost of the control is expβ/t and expk/t for every K > β as close as β as we want the implicit constant in front of the exponential may explode when we get closer to β because it seems to be a fraction of some power of T The constant β verifies /4 β 3 /4 The best upper bound was obtained in [9] and the lower bound in [6] These estimates on β were the best that were known up to now For the Schrödinger equation on, T, with boundary control on one side, one also has that the dependence in time of the cost of the boundary control is under the form exp β + /T for some constant β > The constant β verifies /4 β 3 / The upper bound is obtained in [9] and the lower bound in [5] These estimates on β are the best that are known up to now In both cases, it was conjectured that the lower bound is optimal, ie that one can choose β = β = /4 We will call from now on these conjectures on β and β the Miller s conjectures et us mention that, in the case of the heat equation, there exists another conjecture concerning sharp integral observability estimates and that is stronger than the previous one, see [3] and [], which concerns the observability of the heat equation More precisely, it was proved in [3] that there exists some constant C int T, such that T e t ϕt, x dxdt C int T, x ϕt, dt, 5 where ϕ is a solution on the forward free heat equation ϕ t ϕ xx = in, T,, ϕ, = in, T, ϕ, = in, T, ϕ, = ϕ in, 6 with ϕ, However, since 5 was obtained thanks to a reasoning by contradiction, the authors were unable to estimate precisely the constant C int T, natural conjecture cf [3, Section, Section 3, Section 5] and also [] would be that the constant C int T, does not blow up in a too violent way, in the following sense: For every δ > and >, one can choose C int T, such that C int T, = O e δ T, T because this would notably give, after some easy computations, the Miller s conjecture see [3] and [] and also the Coron-Guerrero conjecture for positive speeds cf [], with initial 4

5 conditions and not H initial conditions, but it has only a neglecting impact on the cost of the control From now on, we will call this conjecture the Ervedoza-Zuazua conjecture These results were later generalized to other self-adjoint or skew-adjoint elliptic operators by the author in [3] More precisely, it was proved that if we consider some abstract linear control system with boundary control and where the elliptic operator associated to the system is skewadjoint or self-adjoint with eigenvalues having a behaviour roughly as Rk or irk when, then the cost of the control is bounded from above by exp K/RT / where K is some explicit constant depending on, and is bounded from below by exp C/T /, where C is some non-explicit constant independant of T but depending on R and However, in this case, because of the lack of explicit lower bound and some lack of optimization in the computations of the upper bound, it was impossible to deduce some reasonable conjecture concerning the exact behaviour of the cost of the control Concerning the transport-diffusion equation, let us recall the known results in the case of negative speed, which is interesting us here Since one can prove see [, ppendix ] that the solution of 3 with initial condition y, converges in some sense to the one of 4 when ε, one might reasonably expect that C T D T,, M, ε + for T < /M and C T D T,, M, ε for T > /M the fact that we consider initial conditions in H here is not a problem and only comes from the fact that we want to consider an admissible control operator, it has only a neglecting impact on the cost of the control However, it is proved in [] that one has C T D T,, M, ε Ce K ε for some constants C, K independent of ε if T < /M for M > This surprising result led the authors to make the following conjecture concerning positive results for the uniform controllability of the family of equations 3 for large enough times: C T D T,, M, ε as ε + as soon as T > /M From now on, we will call this conjecture the Coron-Guerrero conjecture In [], it is proved the exponential decay of the cost of the control when ε + for sufficiently large time, the estimate on this time was improved in [5] and then [], the later article making the link between this problem and the cost of fast controls for the heat equation This study was also extended to varying in time and space and regular enough speed M and arbitrary space dimension in [6] 3 Main results and comments In this section, we are going to give the main results of this paper and some additional comments The first result of this article is the following, which concerns equation : Theorem For every T >, > and >, one has C H T,, C π T π T π T + sin π exp sin π Notably, applying 7 for =, we have 8 π 4 T 4 C H T,, C π T 6π T + 4 exp T π T, 5 T π T 7

6 which is twice bigger than the usual conjecture s a consequence, the usual Miller s conjecture made in [6], and the stronger Ervedoza-Zuazua conjecture made in [3] and studied in details in [], are not verified Our second result concerns equation : Theorem For every T >, > and >, one has C S T,, C π T π T π T + Notably, applying 8 for =, we have sin π exp 8 / π 4 T 4 C S T,, C π T 6π T + 4 /4 exp 4T sin π, T and we find again the Miller s conjecture made in [5] for the Schrödinger equation The last result is the following, and concerns equation 3: Theorem 3 For every M >, T >, > and ε >, one has M 3 + ε 3 / C T D T,, M, ε ε 3 3 πε T Notably, C T D T,, M, ε explodes as soon as + M 8πε exp M ε M T 4ε π εt 8 9 M T 4ε < M ε, ie T <, M which is very surprising s a consequence, the Coron-Guerrero conjecture given in [] is also not verified for negative speeds et us give additional remarks: Remark The same computations for positive speed of propagation for 3 which would correspond to M < here in equation 3 do not improve the existing result given in [] ie T > / M as a lower bound for the time needed for the uniform controllability If we compare the results given in Theorems and to the one given in [3] concerning upper bounds, we see that they do not really have the same shapes In fact, the quantity sinπ/ which was in the parabolic case in [3] appears here in the dispersive case and conversely for the quantity sinπ/ The author was not able to understand deeply the reason of this lack of unity However, one possible explanation is that the moment method, as it is usually applied that is to say study a Weierstrass product issued from the eigenvalues, and then compensate it with some appropriate multiplier in order to apply the Paley-Wiener Theorem is maybe not totally adapted from the viewpoint of the cost of the control 6

7 3 It is very surprising that the dispersive case gives a lower bound that is twice less that the one in the parabolic case In fact, if we think a little bit about the computations done in many articles concerning the dispersive case [5], [9] or [3] for example, we always obtain an upper bound for the dispersive case which is twice the one for the parabolic case because of the study of the Weierstrass product that is used in the moment method, where the asymptotic upper bound at infinity is different in the two cases Hence, it seems more logical that the cost of the control for the dispersive case is the same or twice as for the parabolic case, and not half 4 By using the results given in [], we see that if we assume that we were able to prove that C S T,, e /T, then we would obtain new upper bound for the transport-diffusion problem T > / M and T> +/ M resp for positive and negative speeds 5 Since the Ervedoza-Zuazua conjecture is not verified, one can think on how to replace it natural substituation would be the following one: for every solution ϕ of 6, we have T e t ϕt, x dxdt C int T, x ϕt, dt, where C int T, is growing subexponentially in /T Unfortunately, this inequality is not verified et us prove it by contradiction If this inequality where true, then, using the computations of [, Page ], we would obtain the uniform controllability of 3 as soon as T > + 3/M 73/M, which cannot be true because of Theorem 3 and the fact that 8 The results and preceding remarks lead us to the following open questions: Open Questions re the lower bounds given in Theorems, and 3 optimal? re the lower bounds in the case of the heat and Schrödinger equations ie = optimal? The author believes that this might be true at least for the heat equation or more generally for equation, but is more sceptical concerning equation and has no idea for equation 3 Moreover, according to the previous remark, the author thinks that it might not be possible to find some integral observability estimate similar to 5 with subexponential in /T constant in the right-hand side Proofs of Theorems, and 3 The proofs are based on the following idea: we are going to consider the optimal control associated to the first eigenfunction, and then we will study the Fourier transform of this control, which is an entire function of exponential type and with some prescribed zeros In some sense, this idea comes from the moment method of [4], but we use it in a reverse way compared to what is done usually: we do not construct the control thanks to the Paley-Wiener Theorem this will only give upper bounds but we assume that the control exists and we see what we can deduce if we remark that it verifies the moment problem fter some rescaling and translations, we are then led to study an entire function of exponential type with some prescribed zeros, and we use a representation formula for functions of exponential type in order to make a link between the value and the functions and the repartition of its zeros on the upper half-plane et us mention that this idea has already been used in [] to derive lower bounds for the problem of the uniform controllability of the transport-diffusion equation The main differences here are that we were able to find a better result in the case of negative speed, and we also that were able to extend 7

8 significantly the scope of the method to other cases than a singular limit, ie to the case of study of the cost of fast controls for and, where the eigenvalues have a very different behaviour from the ones of equation 3, which is interesting in itself and highlights one more time the connection between the uniform controllability and the cost of fast controls Remark 3 n alternative proof of Theorems, and 3 would have been to consider the control associated to some eigenfunction e N for some N large enough depending on, T and and to do the same computations In fact, this will not give better results than the proof presented here, and we can say that in some sense, the rescaling and translation arguments that are appearing during the proof of the theorems is quite equivalent to looking at high frequencies Proof of Theorem In all what follows, C will always be a numerical constant independant of the parameters We define y H, as follows: πx y x := sin ccording to [, Page 6], with ε = and M, there exists some numerical constant C such that y H, C3 We consider u the optimal control associated to this initial condition, which verifies by definition and thanks to estimate u, C H T,, y H, CC H T,, 3/ Proceeding as in [, Page 6-7], we obtain because of the fact that yt, = and the definition by transposition of the solutions of kπ T k π π kπ ut exp t dt = sin sin dx 3 et us define the complex function v by vz := Using 3 and 4, we deduce that v i π T/ T/ u t + T exp istdt 4 = π exp π T, 5 and for every k N with k > we have v i k π = 6 We deduce, using 4 and, that T Imz T vz exp ut dt C H T,, T Imz T exp y H, CC H T,, T Imz T exp 3/ 7 8

9 et us consider some numerical parameter β > to be chosen later We introduce z iβ fz := v 8 Inequality 7 becomes fz CC H T,, T exp One has, for k N and k >, and thanks to 6, T Imz β T / 3/ 9 fb k =, where b k verifies ie We also have, thanks to 5, where b k i β = ik π T, b k := i T k π β + fb = π exp π T, b := i β + T π 3 Using the usual representation of the functions of exponential type given for example in [, Theorem p56], we have, for every z such that Imz >, fz = z al + σx + x fτ z a l π R τ z dτ, where σ is the type of f, which verifies thanks to 9 that σ T 4 We apply this equality at point b, then we use 3 remark that b is a pure imaginary number and 4 to obtain fb b a l + s a l β T where the a k are all the roots of f of positive imaginary part et us study the right-hand side of this equality First term of the right-hand side: We study I := l= + T π + b fτ π R τ dτ, 5 + b b a l b a l 9

10 One remark that we have, due to the fact that b ir and that Ima l >, for every l, b a l b a l < Hence, we deduce that b b l I = s b l x T π /β We use the change of variables Hence we obtain We call + x T π /β I β T π k T π / β + k + T π / dx τ := πt x β πt β Using an integration by parts, we obtain τ + τ dτ = + One can write It is well-known that + τ dτ = 6 τ + τ dτ 7 := πt 8 β + τ dτ = Γ + τ dτ Γ + dτ 9 + τ dτ 3 + τ Using the Euler reflection formula for the Γ function and the relation Γz + = zγz, we deduce that + τ dτ = π sin π 3 Concerning the second term of 3, we have hence dτ, + τ dτ 3 + τ Putting together 6, 8, 9, 3 and 3, we deduce that b a l β + s a l π T β sin π T + 33

11 Concerning the third time of the right-hand-side, an easy changing of variables gives dτ b τ + b = π R Hence, using the fact that τ is real and 9, we deduce that b fτ π R τ + b dτ β + CC T H T,, T 3/ 34 Using, 5, 33 and 34, we deduce that π T π β + β π T sin π T + β T hence there exists a numerical constant C such that C H T,, C / π T π T π T + β + + π T + CC HT,, T 3/, exp β sin π T β T Now, we optimize β by trying to maximize what is inside the exponential We find β = and we deduce C H T,, C π T π T π T + Proof of Theorem sin π sin π, exp π T sinπ/ T 35 π T The computations are very similar to the one of the previous part, hence we are going to skip some details We define y H, as in We consider u the optimal control associated to this initial condition, which verifies by definition and thanks to estimate u, C S T,, y H, CC S T,, 3/ 36 Proceeding as before, we obtain T ik π ut exp kπ et us define the complex function v by vz := Using 37 and 38, we deduce that v π T/ T/ t dt = π kπ sin sin dx 37 ut + T exp istdt 38 = π exp iπ T 39

12 and for every k N with k > we have v k π = 4 We also have, using 38 and 36, that T Imz T vz exp ut dt CC S T,, T Imz T exp 3/ 4 et us consider some numerical parameter β > to be chosen later We introduce z + iβ fz := v 4 Inequality 4 becomes fz CC S T,, T exp One has, for k N and k >, and thanks to 4, where b k verifies T Imz β T 3/ 43 fb k =, 44 We also have, thanks to 5, b k := T k π + i β 45 fb = π exp iπ T, 46 where b := T k π + i β 47 Using the same representation theorem as in the proof of Theorem, we have for every z such that Imz >, where fz = z al + σx + x fτ z a l π R τ z dτ, σ T We apply this equality at point b and use 47 and 48 to obtain 48 fb b a l + s a l β T + fτ + τ dτ, 49 + b where the a k are all the roots of f of positive imaginary part et us study the right-hand side of this equality

13 First term of the right-hand side: We study b a l I := b a l s before, one obtains that b b l I = s b l + l= x T π /β k T π / β + k + T π / x T π /β 5 We use the same change of variables so that we obtain τ := I β πt πt β / πt β x, τ + τ dτ 5 Using an integration by parts, we obtain τ dτ = + τ + where was defined in 8 We have Using 3, we deduce that dτ = + τ Concerning the second term of 53, we still have dτ + τ dτ, 5 + τ dτ 53 + τ + τ dτ = π sin π 54 Putting together 8, 5, 5, 54 and 55, we deduce that b a l β + s a l π T / dτ 55 + τ β sin π T + 56 Concerning the third time of the right-hand-side, we obtain exactly as before and according to 47 b fτ β π R τ dτ + CC + b T S T,, T 3/ 57 3

14 Using 46, 49, 56 and 57, we deduce that β + π π T / β / hence there exists a numerical constant C such that C S T,, C sin π T / π T π exp T π T + 4β + β T β sin π T Now, we optimize β by trying to maximize what is inside the exponential We find and we deduce C S T,, C π T 3 Proof of Theorem 3 β = / π T π T +, sin π sin π exp + + CC S T,, T 3/, β T sin π T The computations are very similar to the ones done in [, Pages 6-9], so we are going to skip some points First of all, we choose the initial condition as πx Mx y x := sin exp ε Using [, Pages 6-7], one has y H, C ε3 3 M 3 + ε 3 We consider u the optimal control associated to this initial condition, which verifies by definition u, C T D T,, M, ε y ε 3 3 H, CC T D T,, M, ε M 3 + ε 3 59 Following [, Page 7], we see that if we consider we have vz := T/ T/ v i π = πε exp π εt 58 u t + T exp istdt, 6 M T 8ε and for every k N with k > we have v i k π = 6 6 4

15 We deduce, using 6 and 59, that T Imz T vz exp ut dt C T D T,, M, ε T Imz T exp y H, ε 3 3 / C M 3 + ε 3 C T DT,, M, ε T Imz T exp 63 et us introduce s fs := v 64 4ε Then inequality 63 becomes ε 3 3 / fz C M 3 + ε 3 C T DT,, M, ε T exp T Imz 65 8ε One has, for k N and k > and thanks to 6 where b k verifies We also have, thanks to 6, fb = πε exp π εt where fb k =, 66 b k := i M + 4k ε π 67 M T 8ε, 68 b := i M + 4ε π 69 Using the same representation theorem, one has, for every z such that Imz >, fz = that we apply at point b : fb = z al + σx + x fτ z a l π R τ z dτ, b a l + T M + επ b a l 8ε + b π R et us study separatly the terms of the right-hand side fτ τ dτ 7 + b First term of the right-hand side: we can proceed as we did before, and we obtain b a l k ε π ε π x b a l M / + k +ε π M / + ε π x dx We use the change of variables τ := πε M x 5

16 Hence we obtain b a l M b a l πε Using an integration by parts, we easily obtain πε M τ + τ dτ b a l M + b a l 8πε M + 7 ε Third term: using the fact that τ is real, we have Imτ = and then by 65 and straightforward computations b fτ ε 3 3 / T π τ dτ + b M 3 + ε 3 CC T DT,, M, ε 7 R Conclusion: by using 68, 7, 7 and 7, we deduce that πε π εt + M 8πε Hence, we obtain M T 8ε M ++ ε ε 3 3 M 3 + ε 3 M 3 + ε 3 / C T D T,, M, ε C ε 3 3 References / CC T DT,, M, ε T + T M T exp πε + M 8πε M ε M T 4ε 8ε + επ T π εt [] Coron, J-M, Control and nonlinearity, Volume 36 of Mathematical Surveys and Monographs merican Mathematical Society, Providence 7 [] Coron, J-M and Guerrero, S, Singular optimal control: a linear -D parabolic-hyperbolic example symptot nal, 443-4:37-57, 5 [3] Ervedoza, S and Zuazua, E, Sharp observability estimates for the heat equations, rch Ration Mech nal Volume, no 3, [4] Fattorini, H O, and Russell, D, Exact controllability theorems for linear parabolic equations in one space dimension,rch Ration Mech nal, Volume 43 97, Issue 4, pp 7-9 [5] Glass, O, complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, J Funct nal [6] Guerrero, S and ebeau, G, Singular optimal control for a transport-diffusion equation, Comm Partial Differential Equations [7] Güichal, E, lower bound of the norm of the control operator for the heat equation, J Math nal ppl, :5957, 985 [8] Ho, and Russell, D, dmissible input elements for systems in Hilbert space and a Carleson measure criterion, SIM J Control Optim, 4:64-64, 983 [9] Guo, X and Xu, M, Some physical applications of fractional Schrödinger equation J Math Phys, 478:84, 9, 6 6

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