June  2018, 8(2): 383-395. doi: 10.3934/mcrf.2018015

Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback

1. 

Department of Mathematics, Tianjin University of Commerce, Tianjin 300134, China

2. 

School of Mathematics, Tianjin University, Tianjin 300354, China

* Corresponding author: Dongyi Liu

Received  February 2016 Revised  March 2017 Published  March 2018

Fund Project: This research is supported by the Natural Science Foundation of China grant NSFC-61573252.

Design of controller subject to a constraint for a Schrödinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and exponential stability of the system are discussed with the LaSalle invariance principle and Riesz basis method, respectively. In the end, a numerical simulation illustrates the feasibility of the suggested feedback control law.

Citation: Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control and Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015
References:
[1]

M. Aassila, Exact controllability of the Schrödinger equation, Applied Mathematics and Computation, 144 (2003), 89-106.  doi: 10.1016/S0096-3003(02)00394-6.

[2]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[3]

I. AksikasJ.J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor, Systems& Control Letters, 56 (2007), 122-132.  doi: 10.1016/j.sysconle.2006.08.012.

[4]

B. d'Andréa-Novel and J.M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[5]

V. Barbu, Nonlinear Differential Equations of Monotone types in Banach Spaces, Springer New York Dordrecht Heidelberg London, 2010.

[6]

P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Advances in Mathematical Sciences and Applications, 12 (2002), 817-827. 

[7]

P. Bégout, Maximum decay rate for the nonlinear Schrödinger equation, Nonlinear Differential Equations and Applications NoDEA, 11 (2004), 451-467.  doi: 10.1007/s00030-004-2003-7.

[8]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Univerisity Press, New York, 1998.

[9]

C. Chen and D.S. Elliott, Measurements of optical phase variations using interfering multiphoton ionization processes, Physical Review Letters, 65 (1990), 1737-1740.  doi: 10.1103/PhysRevLett.65.1737.

[10]

R. CipolattiE. Machtyngier and E. San Pedro Siqueira, Nonlinear boundary feedback stabilization for Schrödinger equations, Differential and Integral Equations, 9 (1996), 137-148. 

[11]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM Journal on Mathematical Analysis, 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.

[12]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, Journal of Functional Analysis, 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[13]

P. GrossD. Neuhauser and H. Rabitz, Teaching lasers to control molecules in the presence of laboratory field uncertainty and measurement imprecision, The Journal of Chemical Physics, 98 (1993), 4557-4566.  doi: 10.1063/1.465017.

[14]

B. Guo and J. Liu, Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schröinger equation subject to boundary control matched disturbance, International Journal of Robust and Nonlinear Control, 24 (2014), 2194-2212.  doi: 10.1002/rnc.2977.

[15]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions, Lecture Notes in Mathematics, Vol. 841, Springer-Verlag, New York, 1981.

[16]

W. HeX. He and S.S. Ge, Vibration Control of Flexible Marine Riser Systems with Input Saturation, IEEE/ASME Transactions on Mechatronics, 21 (2016), 254-265.  doi: 10.1109/TMECH.2015.2431118.

[17]

T. Kato, Nonlinear semigroups and evolution equations, Journal of the Mathematical Society of Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940508.

[18]

Y. Kōmura, Nonlinear semi-groups in Hilbert space, Journal of the Mathematical Society of Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940493.

[19]

R. KosloffS. A. RiceP. GaspardS. Tersigni and D. J. Tannor, Wavepacket dancing: Achieving chemical selectivity by shaping light pulses, Chemical Physics, 139 (1989), 201-220.  doi: 10.1016/0301-0104(89)90012-8.

[20]

M. Guo and B. Kristic, Boundary controllers and observers for the linearized schrödinger equation, SIAM Journal on Control and Optimization, 49 (2011), 1479-1497.  doi: 10.1137/070704290.

[21]

I. Lasiecka and T. Seidman, Strong stability of elastic control systems with dissipative saturating feedback, System and Control Letters, 48 (2003), 243-252.  doi: 10.1016/S0167-6911(02)00269-4.

[22]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential and Integral Equations, 5 (1992), 521-535. 

[23]

I. Lasiecka and R. Triggiani, Well-posedness and sharp uniform decay rates at the $ L_2(\Omega) $-level of the Schrödinger equation with nonlinear boundary dissipation, Journal of Evolution Equations, 6 (2006), 485-537.  doi: 10.1007/s00028-006-0267-6.

[24]

I. LasieckaR. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part Ⅰ: $ H_1(\Omega) $-estimates, Journal of Inverse Ill-posed Problems, 12 (2004), 43-123. 

[25]

I. LasieckaR. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part Ⅱ: $ L_2(\Omega) $-estimates, Journal of Inverse and Ill-posed Problems, 12 (2004), 183-231. 

[26]

Z. LiuJ. Liu and W. He, Partial differential equation boundary control of a flexible manipulator with input saturation, International Journal of Systems Science, 48 (2017), 53-62.  doi: 10.1080/00207721.2016.1152416.

[27]

D. LiuL. ZhangZ. Han and G. Xu, Stabilization of the timoshenko beam system with restricted boundary feedback controls, Acta Applicandae Mathematicae, 141 (2016), 149-164.  doi: 10.1007/s10440-015-0008-3.

[28]

E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256. 

[29]

E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM Journal Control and Optimization, 32 (1994), 24-34.  doi: 10.1137/S0363012991223145.

[30]

S. Nicaise and S. Rebiai, Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback, Portugaliae Mathematica, 68 (2011), 19-39. 

[31]

N. H. Pavel, Nonlinear Evolution Operators and Semigroups, Springer-Verlag, Berlin, Heidelberg, 1987.

[32]

S. ShiA. Woody and H. Rabitz, Optimal control of selective vibrational excitation in harmonic linear chain molecules, The Journal of Chemical Physics, 88 (1988), 6870-6883.  doi: 10.1063/1.454384.

[33]

R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, 1997.

[34]

M. Slemrod, Feedback Stabilization of a Linear Control System in Hilbert Space with an a priori Bounded Control, Mathematics of Control, Signals and Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[35]

G. Xu and B. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM Journal on Control and Optimization, 42 (2003), 966-984.  doi: 10.1137/S0363012901400081.

[36]

G. Xu and D. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application, IMA Journal of Applied Mathematics, 67 (2002), 357-370.  doi: 10.1093/imamat/67.4.357.

[37]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.

show all references

References:
[1]

M. Aassila, Exact controllability of the Schrödinger equation, Applied Mathematics and Computation, 144 (2003), 89-106.  doi: 10.1016/S0096-3003(02)00394-6.

[2]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[3]

I. AksikasJ.J. Winkin and D. Dochain, Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor, Systems& Control Letters, 56 (2007), 122-132.  doi: 10.1016/j.sysconle.2006.08.012.

[4]

B. d'Andréa-Novel and J.M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[5]

V. Barbu, Nonlinear Differential Equations of Monotone types in Banach Spaces, Springer New York Dordrecht Heidelberg London, 2010.

[6]

P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Advances in Mathematical Sciences and Applications, 12 (2002), 817-827. 

[7]

P. Bégout, Maximum decay rate for the nonlinear Schrödinger equation, Nonlinear Differential Equations and Applications NoDEA, 11 (2004), 451-467.  doi: 10.1007/s00030-004-2003-7.

[8]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Univerisity Press, New York, 1998.

[9]

C. Chen and D.S. Elliott, Measurements of optical phase variations using interfering multiphoton ionization processes, Physical Review Letters, 65 (1990), 1737-1740.  doi: 10.1103/PhysRevLett.65.1737.

[10]

R. CipolattiE. Machtyngier and E. San Pedro Siqueira, Nonlinear boundary feedback stabilization for Schrödinger equations, Differential and Integral Equations, 9 (1996), 137-148. 

[11]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM Journal on Mathematical Analysis, 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.

[12]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, Journal of Functional Analysis, 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[13]

P. GrossD. Neuhauser and H. Rabitz, Teaching lasers to control molecules in the presence of laboratory field uncertainty and measurement imprecision, The Journal of Chemical Physics, 98 (1993), 4557-4566.  doi: 10.1063/1.465017.

[14]

B. Guo and J. Liu, Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schröinger equation subject to boundary control matched disturbance, International Journal of Robust and Nonlinear Control, 24 (2014), 2194-2212.  doi: 10.1002/rnc.2977.

[15]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions, Lecture Notes in Mathematics, Vol. 841, Springer-Verlag, New York, 1981.

[16]

W. HeX. He and S.S. Ge, Vibration Control of Flexible Marine Riser Systems with Input Saturation, IEEE/ASME Transactions on Mechatronics, 21 (2016), 254-265.  doi: 10.1109/TMECH.2015.2431118.

[17]

T. Kato, Nonlinear semigroups and evolution equations, Journal of the Mathematical Society of Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940508.

[18]

Y. Kōmura, Nonlinear semi-groups in Hilbert space, Journal of the Mathematical Society of Japan, 19 (1967), 493-507.  doi: 10.2969/jmsj/01940493.

[19]

R. KosloffS. A. RiceP. GaspardS. Tersigni and D. J. Tannor, Wavepacket dancing: Achieving chemical selectivity by shaping light pulses, Chemical Physics, 139 (1989), 201-220.  doi: 10.1016/0301-0104(89)90012-8.

[20]

M. Guo and B. Kristic, Boundary controllers and observers for the linearized schrödinger equation, SIAM Journal on Control and Optimization, 49 (2011), 1479-1497.  doi: 10.1137/070704290.

[21]

I. Lasiecka and T. Seidman, Strong stability of elastic control systems with dissipative saturating feedback, System and Control Letters, 48 (2003), 243-252.  doi: 10.1016/S0167-6911(02)00269-4.

[22]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control, Differential and Integral Equations, 5 (1992), 521-535. 

[23]

I. Lasiecka and R. Triggiani, Well-posedness and sharp uniform decay rates at the $ L_2(\Omega) $-level of the Schrödinger equation with nonlinear boundary dissipation, Journal of Evolution Equations, 6 (2006), 485-537.  doi: 10.1007/s00028-006-0267-6.

[24]

I. LasieckaR. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part Ⅰ: $ H_1(\Omega) $-estimates, Journal of Inverse Ill-posed Problems, 12 (2004), 43-123. 

[25]

I. LasieckaR. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part Ⅱ: $ L_2(\Omega) $-estimates, Journal of Inverse and Ill-posed Problems, 12 (2004), 183-231. 

[26]

Z. LiuJ. Liu and W. He, Partial differential equation boundary control of a flexible manipulator with input saturation, International Journal of Systems Science, 48 (2017), 53-62.  doi: 10.1080/00207721.2016.1152416.

[27]

D. LiuL. ZhangZ. Han and G. Xu, Stabilization of the timoshenko beam system with restricted boundary feedback controls, Acta Applicandae Mathematicae, 141 (2016), 149-164.  doi: 10.1007/s10440-015-0008-3.

[28]

E. Machtyngier and E. Zuazua, Stabilization of the Schrödinger equation, Portugaliae Mathematica, 51 (1994), 243-256. 

[29]

E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM Journal Control and Optimization, 32 (1994), 24-34.  doi: 10.1137/S0363012991223145.

[30]

S. Nicaise and S. Rebiai, Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback, Portugaliae Mathematica, 68 (2011), 19-39. 

[31]

N. H. Pavel, Nonlinear Evolution Operators and Semigroups, Springer-Verlag, Berlin, Heidelberg, 1987.

[32]

S. ShiA. Woody and H. Rabitz, Optimal control of selective vibrational excitation in harmonic linear chain molecules, The Journal of Chemical Physics, 88 (1988), 6870-6883.  doi: 10.1063/1.454384.

[33]

R. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, 1997.

[34]

M. Slemrod, Feedback Stabilization of a Linear Control System in Hilbert Space with an a priori Bounded Control, Mathematics of Control, Signals and Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[35]

G. Xu and B. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM Journal on Control and Optimization, 42 (2003), 966-984.  doi: 10.1137/S0363012901400081.

[36]

G. Xu and D. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application, IMA Journal of Applied Mathematics, 67 (2002), 357-370.  doi: 10.1093/imamat/67.4.357.

[37]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer-Verlag, New York, 1990.

Figure 1.  Real part of $w(x, t)$
Figure 2.  Imaginary part of $w(x, t)$
Figure 3.  Real and imaginary parts of $w(1, t)$
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