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Obstruction to the bilinear control of the Gross-Pitaevskii equation: an example with an unbounded potential ⁎
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IFAC PapersOnLine 52-16 (2019) 304–309
Obstruction Obstruction to to the the bilinear bilinear control control of of Obstruction to the bilinear control of Gross-Pitaevskii equation: an example Obstruction to the bilinear control of Gross-Pitaevskii equation: an example Gross-Pitaevskii equation: an example an potential Gross-Pitaevskii equation: an example an unbounded unbounded potential an unbounded potential an unbounded potential Thomas Chambrion ∗∗∗ Laurent Thomann ∗∗ ∗∗ ∗∗
the the the with the with with with
Thomas Chambrion ∗ Laurent Thomann ∗∗ Thomas Chambrion Chambrion ∗ Laurent Laurent Thomann Thomann ∗∗ Thomas ∗ ∗∗ ∗ Thomas Chambrion Laurent Thomann e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France ∗ ∗ Universit´ IECL, F-54000 Nancy, France Universit´ e de Lorraine, CNRS, Inria, ∗ ∗ [email protected]). Universit´ee(e-mail: de Lorraine, Lorraine, CNRS, Inria, Inria, IECL, IECL, F-54000 F-54000 Nancy, Nancy, France France Universit´ de CNRS, (e-mail: [email protected]). ∗ ∗∗ Universit´ e(e-mail: dee Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France de Lorraine, CNRS, IECL, F-54000 Nancy, France [email protected]). ∗∗ ∗∗ Universit´ (e-mail: [email protected]). Universit´ e de Lorraine, CNRS, IECL, F-54000 Nancy, France ∗∗ (e-mail: [email protected]). ∗∗ Universit´ (e-mail: [email protected]). e de Lorraine, CNRS, IECL, F-54000 Nancy, France e de Lorraine, CNRS, IECL, F-54000 Nancy, France (e-mail: [email protected]). ∗∗ Universit´ Universit´ e de Lorraine, CNRS, IECL, F-54000 Nancy, France (e-mail: [email protected]). (e-mail: [email protected]). (e-mail: [email protected]). Abstract: In 1982, Ball, Marsden, and Slemrod proved an obstruction to the controllability Abstract: In 1982,with Ball,aMarsden, Slemrod proved obstruction to the an controllability of linear dynamics bounded and bilinear control term.an note presents example of Abstract: In 1982, 1982, Ball, Ball, Marsden, Marsden, and Slemrod proved anThis obstruction to the the controllability controllability Abstract: In and Slemrod proved an obstruction to of linear dynamics with a bounded bilinear control term. This note presents an example of Abstract: In 1982, Ball, Marsden, and Slemrod proved an obstruction to the controllability nonlinear dynamics with respect to the state for which this obstruction still holds while of linear dynamics with a bounded bilinear control term. This note presents an example of of linear dynamics bounded bilinear term. note presents an example of nonlinear dynamics with with arespect to the statecontrol for which thisThis obstruction still holds while the the of linear dynamics with a bounded bilinear control term. This note presents an example of bilinear control term is not continuous. nonlinear dynamics with respect to the state for which this obstruction still holds while the nonlinear dynamics to the state for which this obstruction still holds while the bilinear control termwith is notrespect continuous. nonlinear dynamics with respect to the state for which this obstruction still holds while the bilinear control term is not continuous. bilinear control term is notFederation continuous. © 2019, IFAC (International Automatic Control) Hosting by Elsevier Ltd. All rightsequation reserved. bilinear control term is not continuous. Keywords: Nonlinear control system,ofcontrollability, bilinear control, Gross-Pitaevskii Keywords: Nonlinear control system, controllability, bilinear control, Gross-Pitaevskii equation Keywords: Nonlinear Nonlinear control control system, system, controllability, controllability, bilinear bilinear control, Gross-Pitaevskii equation equation Keywords: control, Gross-Pitaevskii Keywords: NonlinearAND control system, controllability, bilinear control, Gross-Pitaevskii equation 0 1. INTRODUCTION RESULTS have proven that if A generates a C 0 0 semi-group in 1. INTRODUCTION AND RESULTS in X X have proven that if A generates a C semi-group 0 0 semi-group 1. and B is bounded on X, then the attainable set from have proven that if A generates a C X 1. INTRODUCTION INTRODUCTION AND AND RESULTS RESULTS haveBproven that on if AX, generates C 0 semi-group inany X and is bounded then the aattainable set fromin any r 1. INTRODUCTION AND RESULTS have proven that if A generates a C semi-group in Xa 1.1 Introduction source ψ in X with L controls, r > 1, is contained in and B is bounded on X, then the attainable set from any r 0 r and B is on LX, controls, then the rattainable set from in any source ψ00bounded in X with > 1, is contained a 1.1 Introduction r thensets and B is bounded on X, the of attainable set from in any r countable union of compact X. This represents 1.1 source ψ in X with L controls, r > 1, is contained in 0 1.1 Introduction Introduction source ψ in X with L controls, r > 1, is contained aa 0 union of compact countable sets of X. This represents r 3 1.1 Introduction source ψ in X with L controls, r > 1, is contained in aa deep obstruction to the controllability of bilinear control On the Euclidean space R endowed with its natural norm 0 countable union of of compact sets of of X. X. of This represents union sets This represents a deep obstruction tocompact the controllability bilinear control On the Euclidean space R333 endowed with its natural norm countable countable union of compact setsBanach of X. of This represents a 3 endowed with its natural norm systems in infinite dimensional spaces, since this || ·· |, we study the control problem: deep obstruction to the controllability bilinear control On the Euclidean space R deep obstruction to the controllability of bilinear control endowed with its natural norm On the Euclidean space R systems in infinite dimensional Banach spaces, since this |, we study the control problem: 3 deep obstruction to the controllability of bilinear control endowed with its natural norm On the Euclidean space R 2 3 result implies that the attainable set is meager in Baire systems in infinite dimensional Banach spaces, since this || ·· |, we study the control problem: systems in infinite Banach this |,i∂we study the control problem: + Hψ = − result implies that dimensional the attainable set is spaces, meagersince in Baire tψ Hψ the = u(t)K(x)ψ u(t)K(x)ψ − σ|ψ| σ|ψ|222 ψ, ψ, (t, (t, x) x) ∈ ∈R R× ×R R333 ,, systems inhas infinite Banach this | · |,i∂we study control problem: tψ + sense interior. result implies that dimensional the attainable set is is spaces, meagersince in Baire Baire result implies that the attainable set meager in sense and and has empty empty interior. i∂ttt ψ ψ x) + Hψ Hψ =(x), u(t)K(x)ψ − − σ|ψ| σ|ψ|2 ψ, ψ, (t, (t, x) x) ∈ ∈R R× ×R R3 ,, = ψ ψ(0, i∂ + = u(t)K(x)ψ 0 2 3 result implies that the attainable set is meager in Baire ψ(0, x) = ψ (x), sense and has empty interior. sense and hasresult empty interior. i∂t ψ x) + Hψψ00=(x), u(t)K(x)ψ − σ|ψ| ψ, (t, x) ∈ R × R(1) , The of et ψ(0, ψ(0, x) = = ψ00 (x), 2 senseoriginal and hasresult empty interior. original of Ball Ball et al. al. (1982) (1982) (and (and its its adaptation adaptation (1) The ψ(0, x) = ψ (x), to the Schr¨ o dinger equation Turinici (2000)) been where H = −∆ is the Hamiltonian of the quantum 0 + |x|2 The original result of Ball et al. (1982) (and itshas adaptation (1) The original result of Ball et al. (1982) (and adaptation 2 (1) the Schr¨odinger equation Turinici (2000))its has been exexwhere H = −∆ + |x|2 is the Hamiltonian of the quantum to 1 3 The original result of Ball et al. (1982) (and its adaptation tended to the case of L controls in Boussa¨ ıd et al. (2017). (1) 2 harmonic oscillator on R , u : R −→ R is the control, to the Schr¨ o dinger equation Turinici (2000)) has ex1 where H H= =oscillator −∆ + + |x| |x|onis is the Hamiltonian ofisthe the quantum 3 the Schr¨ odinger equation Turinici (2000)) hasal.been been ex3 , uHamiltonian where −∆ tended to the case of L11 controls in Boussa¨ ıd et (2017). harmonic : R −→ Rof thequantum control, to 2 Rthe 3 3 to the Schr¨ o dinger equation Turinici (2000)) has been ex1 controls More recently, the case where A is non-linear has been where H = −∆ + |x| is the Hamiltonian of the quantum 3 K : R −→ R is a given potential and σ ∈ {0, 1}. tended to the case of L in Boussa¨ ıd et al. (2017). harmonic oscillator on R , u : R −→ R is the control, 3 tended to the case of L controls in Boussa¨ ıd et al. (2017). harmonic on Rpotential , u : R −→ R is the control, More recently, the case where A is non-linear has been K : R33 −→oscillator R is a given and σ ∈ {0, 1}. 1 3 tended to the case of L controls in Boussa¨ ıd et al. (2017). investigated in Chambrion and Thomann (2018b) (for the harmonic oscillator on R , u : R −→ R is the control, 3 −→ More recently, the case where A is non-linear has been K : R R is a given potential and σ ∈ {0, 1}. More recently, the case where A is non-linear has been K :R is a given potential and σ ∈of{0, 1}. investigated in equation) Chambrion andChambrion Thomann (2018b) (forbeen the The Sobolev based on the 3 −→ Rspaces recently, the case where A is non-linear has Klein-Gordon and and Thomann K :R −→ Rspaces is a given and σ ∈of{0, The Sobolev basedpotential on the the domain domain the1}.harmonic harmonic More investigated in Chambrion Chambrion and Thomann Thomann (2018b) (for the the investigated in and (2018b) (for Klein-Gordon equation) and Chambrion and Thomann oscillator are instrumental in the study of dynamics (1). The Sobolev spaces based on the domain of the harmonic The Sobolev based on the harmonic in Chambrion andChambrion Thomann (2018b) (for the (2018a) (for the Gross-Pitaevskii equation (1)). oscillator are spaces instrumental in the the domain study ofof dynamics (1). investigated Klein-Gordon equation) and and Thomann Klein-Gordon equation) and Chambrion and Thomann (2018a) (for the Gross-Pitaevskii equation (1)). The Sobolev based0on the domain the harmonic They for and p 1 oscillator are instrumental study (1). Klein-Gordon oscillator are spaces instrumental in the study ofof dynamics dynamics and Chambrion and Thomann They are are defined, defined, for ss ≥ ≥ 0 in andthe p≥ ≥ 1 by by of (2018a) (for Gross-Pitaevskii equation (1)). (1). (2018a) (for the theequation) Gross-Pitaevskii equation (1)). In (Chambrion and Thomann, 2018a, Theorem 1.6) we oscillator are in study of dynamics (1). s,p s,pinstrumental 3for pthe 3≥ s/2 p 3 They are defined, for s ≥ 0 and p ≥ 1 by They are defined, s ≥ 0 and p 1 by (2018a) (for the Gross-Pitaevskii equation (1)). W = W (R ) = f ∈ L (R ), H f ∈ L (R ) , s,p s,p 3 p 3 s/2 p 3 In (Chambrion and Thomann, 2018a, Theorem 1.6) we s,p s,p 3 p 3 s/2 p 3 1,∞ 3 ≥ W s,pare = defined, W s,p (R3for ) =s f 0∈ and Lp (R ), H f ∈ L (R ) , showed in particular that if K ∈ W (R ), the dynamics They p ≥ 1 by In (Chambrion and Thomann, 2018a, Theorem 1.6) 1,∞ 3 3 s/2 p 3 1,∞ (R 3 ), the dynamics In (Chambrion and that Thomann, 2018a, Theorem 1.6) we we 3 )s= f s∈ L3p (R3 s/2 p 3 showed in particular if K ∈ W s,2 W s,p = =W W s,p (R (RH ), H f ∈ L (R ) , W ) = f ∈ L (R ), H f ∈ L (R ) , 1,∞ 3 = H (R . In (Chambrion and Thomann, 2018a, Theorem 1.6) we 1,∞ (R 3 ), thethe s 3 (1) is non controllable. Under this assumption, map s,p s,p 3 s p) =3W s,2 s/2 p 3 showed in particular that if K ∈ W dynamics s s 3 s,2 showed in particular that if K ∈ W (R ), the dynamics =W . W = W (RH)ss== H f ss∈(R L33 )(R ), s,2 H f ∈ L (R ) , (1) is non controllable. Under this assumption, the map 1,∞ 3 showed in particular W dynamics 1that 3 if K ∈ 1 assumption, 3(R ), thethe H are =H Hdenoted (R )) = = by W s,2 . W s,p and up to (1) The natural norms f (1) is non controllable. Under this assumption, the map H = (R W . is non controllable. Under this map H11 (R33 ) −→ H11 (R33 ) s 3 s,p The natural norms f. W s,p and up to Hs are =(see Hdenoted (R ) = by W s,2 assumption, the map ) −→this H11 (R ) H11 (RUnder equivalence norms e.g. Zhang, 2004, s,p The naturalof norms are denoted by fand W and up up to (1) is non controllable. W The natural by f and to (R33ψ)) − −→ H (R (R33 )) H (R W s,p equivalence ofnorms normsare (seedenoted e.g. (Yajima (Yajima and Zhang, 2004, → Kψ, −→ H H 1 3 1 3 s,p The natural norms are denoted by f and up to Lemma 2.4)), for 1 < p < +∞, we have ψ − → Kψ, W equivalence of for norms (see e.g. (Yajima (Yajima and Zhang, Zhang, 2004, 2004, −→ H (R ) H (R ψ) − equivalence of norms (see e.g. Lemma 2.4)), 1
1 0 {|x|≤1} L (R), then the equation Assume that u ∈ 1 operators in a Banach space X and u : R → R is a realis in ψ). dynamics play a role in r>1 valued control, which involves a control control term uBψ that loc u {|x|≤1} and ψ0 ∈ H (R(1) r loc ). Theorem 1. Let K(x) = ln(|x|)1 r>1 valued control, involves a uBψ that r Φ(R), is bilinear bilinear in (u, (u,which ψ). Such Such dynamics play term a major major role in admits (t)(ψ ). a global flow ψ(t) = L then the equation (1) Assume that u ∈ u 0 L (R), then the equation (1) Assume that u ∈ u (t)(ψ loc r>1 valued control, which involves a control term uBψ that ). admits a global flow ψ(t) = Φ loc r>1 physics and are the subject of a vast literature Khapalov r is bilinear in (u, ψ). Such dynamics play a major role in 0 0 u is bilinear in (u, ψ). Such dynamics play a major role in L (R), then the equation (1) Assume that u ∈ physics and are the subject of a vast literature Khapalov admits u (t)(ψ ). aa global flow r>1 = 1Φ 3 0 global flow ψψ(t) ψ(t) =loc Φ is bilinear inare (u,etthe ψ). Such dynamics play a major role in admits 0 ). attainable set (2010). al. (1982), Marsden, and Slemrod H (R Moreover, for every physics and a literature u (t)(ψ 3 ), the 0 ∈ physics andBall are subject ofBall, a vast vast literature Khapalov 3 (2010). In In Ball etthe al.subject (1982),of Ball, Marsden, andKhapalov Slemrod admits (t)(ψ a global flow ψψ(t) =111Φ ∈ H (R ), the set Moreover, for every 0 ). attainable 0 0 3 physics andBall areet ofBall, a vast literature 3 ), (2010). In al. (1982), Marsden, and Slemrod ∈ attainable set Moreover, ψ u the (2010). In Ball etthe al.subject (1982), Marsden, andKhapalov Slemrod Φ ∈H H11 (R (R ), the attainable set (3) Moreover, for for every every ψ00 T. Chambrion is supported byBall, the grant ”QUACO” ANR-17(t)(ψ ) 3u 0 ) u (t)(ψ T. Chambrion is supported byBall, the grant ”QUACO” ANR-17(2010). In Ball et al. (1982), Marsden, and”BEKAM” Slemrod ∈ H (R ), the attainable set (3) Moreover, for every ψ Φ 0 0 CE40-0007-01. L. Thomann is supported by the grants 0 u r T. Chambrion is supported by the grant ”QUACO” ANR-17u t∈R Φ (t)(ψ ) (3) T. Chambrion supported by the grant ”QUACO” ANR-17u∈Lrloc (R), Φ (t)(ψ0 CE40-0007-01. L. is Thomann is supported by the grants ”BEKAM” 0 ) (3) r (R), t∈R u ANR-15-CE40-0001 and ”ISDEEC” ANR-16-CE40-0013. t∈R u∈L u∈L (R), T. Chambrion is supported by the grant ”QUACO” ANR-17loc CE40-0007-01. L. Thomann is supported by the grants ”BEKAM” r>1 loc Φ (t)(ψ ) (3) r CE40-0007-01. L. Thomann is supported by the grants ”BEKAM” ANR-15-CE40-0001 and ”ISDEEC” ANR-16-CE40-0013. 0 r t∈R t∈R u∈Lr>1(R),
loc u∈Lr>1 loc (R), CE40-0007-01. L. Thomann is supported by the grants ”BEKAM” ANR-15-CE40-0001 and ”ISDEEC” ”ISDEEC” ANR-16-CE40-0013. r ANR-15-CE40-0001 and ANR-16-CE40-0013. r>1 t∈R u∈Lr>1 loc (R), ANR-15-CE40-0001 and ”ISDEEC” ANR-16-CE40-0013. 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) rights reserved. Copyright 446 Hosting by Elsevier Ltd. All r>1 Copyright © 2019 IFAC 446 Peer review under responsibility of International Federation of Automatic Control. Copyright © 446 Copyright © 2019 2019 IFAC IFAC 446 10.1016/j.ifacol.2019.11.796 Copyright © 2019 IFAC 446
2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019
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is a countable union of compact subsets of H1 (R3 ). In this paper, the solutions to (1) are understood in the mild sense t itH u(τ )ei(t−τ )H (Kψ(τ ))dτ ψ(t) = e ψ0 − i 0 t ei(t−τ )H (|ψ|2 ψ)dτ. + iσ 0
305
In the sequel c, C > 0 denote constants the value of which may change from line to line. These constants will always be universal, or uniformly bounded. For x ∈ R3 , we write x = (1 + |x|2 )1/2 . We will sometimes use the notations LpT = Lp ([0, T ]) and LpT X = Lp ([0, T ]; X) for T > 0. 3. PROOF OF THEOREM 1 3.1 Global existence theory for dynamics (1)
1.4 Content of the paper
Notice that using the reversibility of the equation (1), it is enough to consider non-negative times in the proofs.
The rest of this note provides a proof of Theorem 1. The proof crucially relies on classical Strichartz estimates, which we recall in Section 2. The proof itself is split in two parts. The global well-posedness of the problem (1) is established in Section 3.1, using among other some energy estimates. The proof of the obstruction result follows the strategy used in the paper Ball et al. (1982) and is given in Section 3.2.
We first state two technical lemmas which will be useful to control the bilinear term in (1). Lemma 2. Let K(x) = ln(|x|)1{|x|≤1} . Then for all 1 ≤ p < 2 there exists C > 0 such that KψW 1,p (R3 )) ≤ CψH1 (R3 ) .
2. STRICHARTZ ESTIMATES As in Chambrion and Thomann (2018a), the Strichartz estimates play a major role in the argument, let us recall them in the three-dimensional case. A couple (q, r) ∈ [2, +∞]2 is called admissible if 3 2 3 + = , q r 2 and if one defines XT1 := Lq [−T, T ] ; W 1,r (R3 ) , (q,r) admissible
then for all T > 0 there exists CT > 0 so that for all ψ0 ∈ H1 (R3 ) we have (4) eitH ψ0 XT1 ≤ CT ψ0 H1 (R3 ).
Using interpolation theory one can prove that XT1 = L∞ [−T, T ] ; H1 (R3 ) ∩ L2 [−T, T ] ; W 1,6 (R3 ) , so that one can define ψXT1 = ψL∞ ([−T,T ];H1 (R3 )) + ψL2 ([−T,T ];W 1,6 (R3 )) .
T
(5) where q and r are the H¨ older conjugate of q and r. In particular, they are related by 2 3 7 (6) + = . q r 2 We refer to Poiret (2012) for a proof. Let us point out that (5) implies that
t i(t−τ )H
F (τ )dτ
≤ CT F1 L1 ([−T,T ],H1 (R3 )) +F2 L2 ([−T,T ],W 1,6/5 (R3 )) ,(7) for any F1 , F2 such that F1 + F2 = F , which will prove useful. e
0
1 XT
with 1/q1 = 1/p − 1/2 and 1/q2 = 1/p − 1/6. Now we use the expression of K to get KLq1 < ∞ (since q1 < ∞). Then we write, by (2) KW 1,q2 ≤ CxKLq2 + C∇KLq2 , and since |∇K| ≤ C|x|−1 , we check that KW 1,q2 < ∞ because q2 < 3.
The case p = 2 does not hold true in the previous statement, but we have the following bound after averaging in time. Lemma 3. Let K(x) = ln(|x|)1{|x|≤1} and T > 0. Then for all 2 ≤ q < ∞, there exists CT > 0 such that for all ψ ∈ XT1 KψLq ([−T,T ];H1 (R3 )) ≤ CT ψXT1 . Proof. Firstly by (2) we have Kψ 1 ≤ c∇Kψ 2 + cK∇ψ H
We will also need the inhomogeneous version of the Strichartz estimates: for all T > 0, there exists CT > 0 so that for all admissible couple (q, r) and function F ∈ Lq ([−T, T ]; W 1,r (R3 )), t ei(t−τ )H F (τ )dτ X 1 ≤ CT F Lq ([−T,T ],W 1,r (R3 )) , 0
Proof. Let 1 ≤ p < 2, then by (Chambrion and Thomann, 2018a, Lemma A.1), Kψ 1,p ≤ CKLq1 ψH1 + CKW 1,q2 ψL6 , (8) W
447
L
L2
+ cKxψ L2 . (9)
• Let us study the first term in (9). Since |∇K| ≤ C|x|−1 , we can use the Hardy inequality −1 |x| ψ 2 3 ≤ C ψ 1 3 (10) L (R )
H (R )
(we refer to (Tao, 2006, Lemma A.2) for the general statement and proof of this inequality), and therefore the contribution of the first term reads ∇Kψ Lq L2 ≤ 1 . CT 1/q ψL∞ H1 ≤ CT ψXT T
T
• To bound the contribution of the two last terms in (9), we will use that K ∈ Lp (R3 ) for any 1 ≤ p < ∞. Given 2 ≤ q < ∞, we choose r > 2 such that the couple (q, r) is (Strichartz) admissible and write, using H¨older K∇ψ 2 ≤ KLp ∇ψLr ≤ cKLp ψW 1,r , L with 1/p + 1/r = 1/2. Thus K∇ψ q 2 ≤ cKLp ψLq W 1,r ≤ cKLp ψX 1 . L L T T T
Similarly, Kxψ q 2 ≤ cKLp ψLq W 1,r ≤ cKLp ψX 1 , L L T T T
which achieves the proof.
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We now state a global existence result for (1) adapted to our control problem. Proposition 4. Let r > 1 and u ∈ Lrloc (R). Set K(x) = ln(|x|)1{|x|≤1} . Let ψ0 ∈ H1 (R3 ), then the equation (1) admits a unique global solution ψ ∈ C(R; H1 (R3 )) ∩ L2loc (R; W 1,6 (R3 )) which moreover satisfies the bound ψL∞ ([−T,T ];H1 (R3 )) + ψL2 ([−T,T ];W 1,6 (R3 )) ≤ C T, ψ0 H1 (R3 ) , uLr (−T,T ) .(11)
Proof. The proof is in the spirit of the proof of (Chambrion and Thomann, 2018a, Proposition 1.5), but here we use moreover the Hardy inequality (10) to control the bilinear term in (1). Energy bound: Assume for a moment that the solution exists on a time interval [0, T ]. For 0 ≤ t ≤ T , we define σ E(t) = ψHψ + |ψ|2 + |ψ|4 dx 2 3 R σ = |∇ψ|2 + |x|2 |ψ|2 + |ψ|2 + |ψ|4 dx. 2 R3
Then, using that ∂t ψ = −i(Hψ + σ|ψ|2 ψ) + iu(t)K(x)ψ, we get E (t) = 2 ∂t ψ ψ + Hψ + σ|ψ|2 ψ dx R3 = −2u(t) KψHψdx 3 R = 2u(t) ψ∇K · ∇ψdx.
Using that |∇K| ≤ C|x| we get
R3 −1
, by the Hardy inequality (10)
As a consequence, 2 Φ(ψ)XT1 ≤ cψ0 H1 + cψL∞ H1 ψL2 L∞ T T
+CT uLrT ψL∞ H1 . T By the Gagliardo-Nirenberg and Sobolev inequalities on R3 , 1/2
1/2
1/2
1/2
ψL∞ ≤ CψL6 ψW 1,6 ≤ CψH1 ψW 1,6 , 1/2
1/2
thus ψL2T L∞ ≤ cT 1/4 ψL∞ H1 ψL2 W 1,6 , and for ψ ∈ T T BT,R we get Φ(ψ)XT1 ≤ cψ0 H1 + cT 1/2 R3 + CT RuLrT .
We now choose R = 2cψ0 H1 . Then for T > 0 small enough, Φ maps BT,R into itself. With similar estimates we can show that Φ is a contraction in BT,R , namely Φ(ψ1 ) − Φ(ψ2 )XT1 ≤ cT 1/2 R2 + CT uLrT ψ1 − ψ2 XT1 . As a conclusion there exists a unique fixed point to Φ, which is a local solution to (1). The local time of existence only depends on u and on the H1 -norm. Therefore one can use the energy bound to show the global existence. Proof of the bound (11): It remains to prove that ψL2 ([−T,T ];W 1,6 (R3 )) ≤ C T, ψ0 H1 (R3 ) , uLr (−T,T ) . (12) The argument follows the main lines of (Chambrion and Thomann, 2018a, Bound (1.18)), so we only give the key steps. Similarly to Chambrion and Thomann (2018a), for any δ > 0 small enough we can prove the bound ψL2 ([τ,τ +δ];W 1,6 ) ≤ cψL∞ H1 T 2 +cδ 1/2 ψ L∞ H1 ψL2 ([τ,τ +δ];W 1,6 ) T
+CT ψL∞ H1 uLr (−T,T ) . T
E (t) ≤ 2|u(t)| ψ∇K L2 ∇ψ L2 ≤ C|u(t)| |x|−1 ψ L2 ∇ψ L2 2 ≤ C|u(t)| ψ H1
≤ C|u(t)|E(t). Thus, using that σ ≥ 0, we deduce the first part of the bound (11). Local existence and global existence: We consider the map t ei(t−τ )H (|ψ|2 ψ)dτ Φ(ψ)(t) = eitH ψ0 + iσ 0 t u(τ )ei(t−τ )H (Kψ)dτ, −i 0
and we will show that it is a contraction in the space BT,R := ψXT1 ≤ R ,
with R > 0 and T > 0 to be fixed.
Then we choose δ = δ(T ) > 0 such that 2 1 cδ 1/2 ψ L∞ H1 = , T 2 thus the previous estimate gives ψL2 ([τ,τ +δ];W 1,6 ) ≤ 2ψL∞ H1 c + CT uLr (−T,T ) , T
and by summing up in δ, we get (12). 3.2 Meagerness of the attainable set
Let > 0 and let u, un ∈ L1+ ([0, T ]; R) such that un u weakly in L1+ ([0, T ]; R). This implies a bound un L1+ ≤ C(T ) for some C(T ) > 0, uniformly in n ≥ 1. T We have t ψ(t) = eitH ψ0 − i u(τ )ei(t−τ )H (Kψ(τ ))dτ 0 t ei(t−τ )H (|ψ|2 ψ)dτ, +iσ 0
and
By the Strichartz inequalities (4) and (5) Φ(ψ)XT1 ≤ cψ0 H1 + c |ψ|2 ψ L1 H1 + cuKψL1T H1 . T
Then, for all r > 1, there exists q < ∞ sur that 1/r + 1/q = 1, and we have by Lemma 3 uKψL1T H1 ≤ uLrT KψLqT H1 ≤ CT uLrT ψXT1 . 448
ψn (t) = e
itH
ψ0 − i
+iσ
t
un (τ )ei(t−τ )H (Kψn (τ ))dτ 0
t 0
ei(t−τ )H (|ψn |2 ψn )dτ.
2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019
Thomas Chambrion et al. / IFAC PapersOnLine 52-16 (2019) 304–309
We set zn = ψ − ψn , then zn satisfies
zn = L(ψ, ψn ) + N (ψ, ψn ),
with
L(ψ, ψn ) = −i
t
(13)
u(τ ) − un (τ ) ei(t−τ )H (Kψ)dτ 0 t un (τ )ei(t−τ )H K(ψ − ψn ) dτ −i
By the Minkowski inequality, the unitarity of eiτ H and the H¨older inequality t un (τ ) − u(τ ) ei(t−τ )H (Kψ(τ ))dτ H1 (R3 ) tn t un (τ ) − u(τ ) Kψ(τ ) 1 3 dτ ≤ H (R ) tn
≤ un − uL1+ KψLq
N (ψ, ψn ) = iσ
t
e 0
i(t−τ )H
+iσ
0
T
ei(t−τ )H (ψ − ψn )ψn2 dτ.
Let us prove that zn tends to 0 in L∞ ([0, T ]; H1 (R3 )). Lemma 5. Denote by t un (τ ) − u(τ ) ei(t−τ )H (Kψ(τ ))dτ ∞ . n := Then n tends to 0, when n tends to +∞.
Proof. We proceed by contradiction. Assume that there exists > 0, a subsequence of un (still denoted by un ) and a sequence tn −→ t ∈ [0, T ] such that tn ≥ . un (τ ) − u(τ ) ei(tn −τ )H (Kψ(τ ))dτ H1 (R3 )
(14) Up to a subsequence, we can assume that for all n ≥ 1, tn ≤ t or tn ≥ t. We only consider the first case, since the second is similar. By the Minkowski inequality and the unitarity of eiτ H
=
T 0
+∞
+∞
(2k + 1)|αk (τ )|2 and
k=0
(2k + 1)|αk (τ )|
k=0
2 q /2
dτ
1/q
.
(19)
k=0
Denote by ρ = supn≥0 un − uL1+ . We claim that there T exists M > 0 large enough such that the function g(τ, x) = M k=0 αk (τ )hk (x) satisfies v−gLq ([0,T ];H1 (R3 )) ≤ /(4ρ). Actually, T 1/q +∞ q /2 v − gLqT H1 = (2k + 1)|αk (τ )|2 dτ
tn un (τ ) − u(τ ) ei(tn −τ )H− ei(t−τ )H (Kψ(τ ))dτ 1 3 H (R ) 0 itn H itH (Kψ(τ )) q ≤ un − uL1+ e −e , 1 3
0
k=M +1
tends to zero when M −→ +∞, by the Lebesgue theorem and (19), hence the claim.
Lτ ∈[0,T ] H (R )
where 1 < q < ∞ is such that 1/(1 + ) + 1/q = 1. Now, by Lemma 3, we have (15)
Now we apply (Chambrion and Thomann, 2018a, Lemma 3.2) (with d = 3 and s = 1) together with the previous lines, and we get that t n un (τ )−u(τ ) ei(tn −τ )H −ei(t−τ )H (Kψ(τ ))dτ 1 3 0
so that we have v(τ, ·)2H1 =
This implies in particular that +∞ q /2 (2k + 1)|αk (τ )|2 ∈ LqT .
Then by H¨ older
KψLqT H1 (R3 ) ≤ CT, ψXT1 < ∞.
k=0
v
H (R )
T
t Let us now prove that 0 un (τ )−u(τ ) ei(t−τ )H (Kψ(τ ))dτ tends to 0 in H1 (R3 ), to reach a contradiction with (14). We set v(τ ) = ei(t−τ )H (Kψ(τ )). Then by the unitarity of H, we have v(τ )H1 = Kψ(τ )H1 , thus by (15), v ∈ Lq ([0, T ]; H1 (R3 )). We expand v on the Hermite functions (hk )k≥0 (which are the eigenfunctions of H) which form a Hilbertian basis of L2 (R3 ) +∞ αk (τ )hk (x), v(τ, x) =
LqT H1
tn un (τ )− u(τ ) ei(tn −τ )H− ei(t−τ )H (Kψ(τ ))dτ 1 3 H (R ) 0 tn ≤ un (τ )− u(τ ) ei(tn −τ )H − ei(t−τ )H (Kψ(τ )) dτ H1 (R3 ) 0 tn it H itH un (τ ) − u(τ ) (Kψ(τ )) 1 3 dτ. = e n −e 0
H (R )
0
LT H1 (R3 )
0
(17)
T
Using Lemma 3 and the fact that un − uL1+ ≤ C, we T deduce that the term (17) tends to 0. We combine this with (16) to deduce tn un (τ ) − u(τ ) ei(tn −τ )H (Kψ(τ ))dτ − 0 t un (τ ) − u(τ ) ei(t−τ )H (Kψ(τ ))dτ −→ 0.(18) 1 3
0
H1 (R3 )
≤ |t − tn |1/q un − uL1+ KψL2q H1 (R3 ) .
(ψ − ψn )(ψ + ψn )ψ dτ
t
τ ∈[tn ,t]
T
0
and
307
H (R )
(16)
tends to zero when n tends to +∞.
449
We have t t M un (τ )−u(τ ) αk (τ )dτ. un (τ )−u(τ ) g(τ )dτ = hk 0
k=0
0
Then, by (19), for all k ≥ 0, αk ∈ LqT , which implies
M
t 0
2 un (τ ) − u(τ ) g(τ )dτ H1 (R3 ) =
t 2 s (2k + 1) un (τ ) − u(τ ) αk (τ )dτ −→ 0,
k=0
0
2019 IFAC NOLCOS 308 Vienna, Austria, Sept. 4-6, 2019
Thomas Chambrion et al. / IFAC PapersOnLine 52-16 (2019) 304–309
by the weak convergence of (un ). Finally, for n large enough, t un (τ ) − u(τ ) v(τ )dτ H1 (R3 ) 0 t ≤ un − u L1+ + un (τ ) − u(τ ) g(τ )dτ T 4ρ H1 (R3 ) 0 ≤ , 2 which together with (14) and (18) gives the contradiction. Thanks to (Chambrion and Thomann, 2018a, Lemma A.3) we get N (ψ, ψn )(t) H1 (R3 ) t ≤ (ψ − ψn )(ψ + ψn )ψ H1 (R3 ) dτ 0 t (ψ − ψn )ψn2 H1 (R3 ) dτ + 0 t zn H1 (R3 ) ψ 2W 1,6 + ψn 2W 1,6 dτ. ≤c
(20)
1/2
1/2
H zn (t) = −iH 0 t i(t−τ )H 1/2 Kzn dτ + H 1/2 N (ψ, ψn )(t). un (τ )e H −i 0
where p < 2 is given by
=
−
2 1+ ),
(21)
see (6).
≤ C3 (T )n , and this latter quantity tends to 0 when n tends to +∞. Lemma 6. (Gr¨onwall inequality). Let f, A, B : [0, T ] −→ R+ be non-negative functions with f (0) = 0, and assume that A is an increasing function. Assume that t f (t) ≤ A(t) + B(τ )f (τ )dτ, ∀ 0 ≤ t ≤ T. (23)
t
0
B(τ )dτ .
(24)
Then by the classical Gr¨onwall inequality (A is constant in s) we get s f (s) ≤ A(t) exp B(τ )dτ ,
for all 0 ≤ s ≤ t, which implies (24) for s = t.
4. CONCLUSION
un H 1/2 Kzn L1+ Lp (R3 ) t t 1/(1+) ≤C |un (τ )|1+ zn (τ ) 1+ . (22) H1 (R3 ) dτ 0
As a conclusion, from (20), (21) and (22) we infer
This note provides an example of a Ball-Marsden-Slemrod like obstruction to the controllability of a nonlinear partial differential equation with a bilinear control term. The novelty of the result lies in the unboundedness of the biinear control term. The possible relations of this obstruction result and the concepts introduced in Boussa¨ıd et al. (2013) will be the subject of further investigations in future works.
zn (t) H1 (R3 ) ≤ t 1/(1+) n + C |un (τ )|1+ zn (τ ) 1+ + H1 (R3 ) dτ 0 t zn (τ ) H1 (R3 ) ψ(τ ) 2W 1,6 + ψn (τ ) 2W 1,6 dτ. +c
REFERENCES Ball, J.M., Marsden, J.E., and Slemrod, M. (1982). Controllability for distributed bilinear systems. SIAM J. Control Optim., 20(4), 575–597. doi:10.1137/0320042. URL https://doi.org/10.1137/0320042. Boussa¨ıd, N., Caponigro, M., and Chambrion, T. (2013). Weakly coupled systems in quantum control. IEEE Trans. Automat. Control, 58(9), 2205–2216. doi:10.1109/TAC.2013.2255948. URL https://doi.org/10.1109/TAC.2013.2255948.
0
We now apply Lemma 6 with t 1/(1+) A(t) = n + C |un (τ )|1+ zn (τ ) 1+ dτ 1 3 H (R ) 0
thus
0
|un (τ )|1+ dτ
0
By Lemma 2 H 1/2 Kzn Lp (R3 ) = Kzn W 1,p (R3 ) ≤ C zn H1 (R3 ) , which in turn implies
and
T
0
t
1 7 3(2
Proof. In the case A constant, the result is classical (see e.g. (Tao, 2006, Theorem 1.10)). Now assume that A is an increasing function and let 0 ≤ s ≤ t ≤ T , then by (23) s B(τ )f (τ )dτ, ∀ 0 ≤ s ≤ t. f (s) ≤ A(t) +
Thus from the Strichartz inequality (7) we deduce
1 p
≤ C2 (T )n exp C2 (T )
f (t) ≤ A(t) exp
u(τ ) − un (τ ) ei(t−τ )H (Kψ)dτ
+ N (ψ, ψn )(t) H1 (R3 ) ,
0
zn (t) H1 (R3 ) ≤ n + un H 1/2 Kzn L1+ Lp (R3 )
0
Finally, by the classical Gr¨onwall inequality we deduce
Then for all 0 ≤ t ≤ T ,
By (13) we have
t
0
where in the last line, we used (11). We raise the previous inequality to the power (1 + ), and get t 1+ dτ . zn (t) H1 (R3 ) ≤ C2 (T ) n+ |un (τ )|1+ zn (τ ) 1+ 1 3 H (R ) zn 1+ H1 (R3 ) L∞ T
0
zn (t) H1 (R3 ) t 1/(1+) ≤ n + C |un (τ )|1+ zn (τ ) 1+ dτ 1 3 H (R ) 0 t c ψ(τ ) 2W 1,6 + ψn (τ ) 2W 1,6 dτ e 0 t 1/(1+) , ≤ C1 (T ) n + |un (τ )|1+ zn (τ ) 1+ H1 (R3 ) dτ
B(τ ) = c ψ(τ ) 2W 1,6 + ψn (τ ) 2W 1,6 ,
450
2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019
Thomas Chambrion et al. / IFAC PapersOnLine 52-16 (2019) 304–309
Boussa¨ıd, N., Caponigro, M., and Chambrion, T. (2017). On the ball–marsden–slemrod obstruction in bilinear control systems. Preprint hal-01537743. Chambrion, T. and Thomann, L. (2018a). On the bilinear control of the gross-pitaevskii equation. Preprint arXiv:1810.09792. Chambrion, T. and Thomann, L. (2018b). A topological obstruction to the controllability of nonlinear wave equations with bilinear control term. Preprint arXiv:1809.07107, to appear in SICON. Khapalov, A.Y. (2010). Controllability of partial differential equations governed by multiplicative controls, volume 1995 of Lecture Notes in Mathematics. SpringerVerlag, Berlin. doi:10.1007/978-3-642-12413-6. URL https://doi.org/10.1007/978-3-642-12413-6. Poiret, A. (2012). Solutions globales pour l’´equation de schr¨ odinger cubique en dimension 3. Preprint arXiv:1207.1578. Tao, T. (2006). Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI. doi:10.1090/cbms/106. URL https://doi.org/10.1090/cbms/106. Local and global analysis. Turinici, G. (2000). On the controllability of bilinear quantum systems. In Mathematical models and methods for ab initio quantum chemistry, volume 74 of Lecture Notes in Chem., 75–92. Springer, Berlin. Yajima, K. and Zhang, G. (2004). Local smoothing property and Strichartz inequality for Schr¨ odinger equations with potentials superquadratic at infinity. J. Differential Equations, 202(1), 81–110.
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