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August  2022, 11(4): 1087-1148. doi: 10.3934/eect.2021037

Stabilization of higher order Schrödinger equations on a finite interval: Part II

1. 

Department of Mathematics, Bilkent University, Bilkent, Ankara, 06800 Turkey

2. 

Department of Mathematics, İzmir Institute of Technology, Urla, İzmir, 35430 Turkey

*Corresponding author: Türker Özsarı

Received  February 2021 Published  August 2022 Early access  July 2021

Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [3] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the $ L^2 $-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.

Citation: Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part II. Evolution Equations and Control Theory, 2022, 11 (4) : 1087-1148. doi: 10.3934/eect.2021037
References:
[1]

G. P. Agrawal, Nonlinear Fiber Optics, 3$^{rd}$ edition, Academic Press, San Diego, 2001. doi: 10.1007/3-540-46629-0_9.

[2]

A. Batal and T. Özsarı, Output feedback stabilization of the linearized Korteweg-de Vries equation with right endpoint controllers, Automatica, 109 (2019), 108531. doi: 10.1016/j.automatica.2019.108531.

[3]

A. Batal, T. Özsarı and K. C. Yılmaz, Stabilization of higher order schrödinger equations on a finite interval: Part I, Evolution Equations & Control Theory, available online. doi: 10.3934/eect.2020095.

[4]

E. Bisognin, V. Bisognin and O. P. Vera Villagrán, Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping, Electron. J. Differential Equations, (2007), 18 pp.

[5]

J. L. BonaS. M. Sun and B. -Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[6]

X. Carvajal and F. Linares, A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations, 16 (2003), 1111-1130. 

[7]

X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), 10 pp.

[8]

X. Carvajal, Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.  doi: 10.1007/s00041-005-5028-3.

[9]

M. M. CavalcantiW. J. CorreaM. A. Sepulveda and R. V. Asem, Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping, Stud. Univ. Babes-Bolyai Math., 64 (2019), 161-172.  doi: 10.24193/subbmath.2019.2.03.

[10]

J. C. Ceballos V., R. Pavez F. and O. P. Vera Villagrán, Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients, Electron. J. Differential Equations, (2005), No. 122, 31 pp.

[11]

E. Cerpa and J. -M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.

[12]

M. Chen, Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), 15 pp. doi: 10.1007/s12044-018-0410-7.

[13]

J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120.  doi: 10.1016/j.matpur.2014.03.004.

[14]

P. N. da Silva and C. F. Vasconcellos, On the stabilization and controllability for a third order linear equation, Port. Math., 68 (2011), 279-296.  doi: 10.4171/PM/1892.

[15]

O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395.  doi: 10.1016/j.sysconle.2010.05.001.

[16]

C. E. Kenig and G. Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3 (1997), 417-433.  doi: 10.1007/BF02649104.

[17]

Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.  doi: 10.1007/BF01008354.

[18]

Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524.  doi: 10.1109/JQE.1987.1073392.

[19]

M. Krstic, B.-Z. Guo and A. Smyshlyaev, Boundary controllers and observers for Schrödinger equation, in 2007 46th IEEE Conference on Decision and Control, (2007), 4149–4154.

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607.

[21]

C. Laurey, The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.  doi: 10.1016/S0362-546X(96)00081-8.

[22]

W. Liu, Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim., 42 (2003), 1033-1043.  doi: 10.1137/S0363012902402414.

[23]

S. Marx and E. Cerpa, Output feedback stabilization of the Korteweg–de Vries equation, Automatica J. IFAC, 87 (2018), 210-217.  doi: 10.1016/j.automatica.2017.07.057.

[24]

T. Özsarı and E. Arabacı, Boosting the decay of solutions of the linearised Korteweg–de Vries–Burgers equation to a predetermined rate from the boundary, Internat. J. Control, 92 (2019), 1753-1763.  doi: 10.1080/00207179.2017.1408923.

[25]

T. Özsarı and A. Batal, Pseudo-backstepping and its application to the control of Korteweg–de Vries equation from the right endpoint on a finite domain, SIAM J. Control Optim., 57 (2019), 1255-1283.  doi: 10.1137/18M1211933.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

A. Smyshlyaev and M. Krstic, Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. Automat. Control, 49 (2004), 2185-2202.  doi: 10.1109/TAC.2004.838495.

[28]

A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs, Systems Control Lett., 54 (2005), 613-625.  doi: 10.1016/j.sysconle.2004.11.001.

[29]

A. Smyshlyaev and M. Krstic, Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary, Systems Control Lett., 58 (2009), 617-623.  doi: 10.1016/j.sysconle.2009.04.005.

[30]

G. Staffilani, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations, 10 (1997), 777-796. 

[31]

H. Takaoka, Well-posedness for the higher order nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 10 (2000), 149-171. 

[32]

S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion, in IEEE 2013 American Control Conference, Washington, DC, (2013), 3302–3307.

[33]

S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries with anti-diffusion by boundary feedback with non-collocated observation, in IEEE 2015 American Control Conference, 2015. doi: 10.1109/ACC.2015.7171020.

[34]

Z. XuL. LiZ. Li and G. Zhou, Soliton interaction under the influence of higher-order effects, Optics Communications, 210 (2002), 375-384.  doi: 10.1016/S0030-4018(02)01803-5.

show all references

References:
[1]

G. P. Agrawal, Nonlinear Fiber Optics, 3$^{rd}$ edition, Academic Press, San Diego, 2001. doi: 10.1007/3-540-46629-0_9.

[2]

A. Batal and T. Özsarı, Output feedback stabilization of the linearized Korteweg-de Vries equation with right endpoint controllers, Automatica, 109 (2019), 108531. doi: 10.1016/j.automatica.2019.108531.

[3]

A. Batal, T. Özsarı and K. C. Yılmaz, Stabilization of higher order schrödinger equations on a finite interval: Part I, Evolution Equations & Control Theory, available online. doi: 10.3934/eect.2020095.

[4]

E. Bisognin, V. Bisognin and O. P. Vera Villagrán, Stabilization of solutions to higher-order nonlinear Schrödinger equation with localized damping, Electron. J. Differential Equations, (2007), 18 pp.

[5]

J. L. BonaS. M. Sun and B. -Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[6]

X. Carvajal and F. Linares, A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations, 16 (2003), 1111-1130. 

[7]

X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Differential Equations, (2004), 10 pp.

[8]

X. Carvajal, Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl., 12 (2006), 53-70.  doi: 10.1007/s00041-005-5028-3.

[9]

M. M. CavalcantiW. J. CorreaM. A. Sepulveda and R. V. Asem, Finite difference scheme for a high order nonlinear Schrödinger equation with localized damping, Stud. Univ. Babes-Bolyai Math., 64 (2019), 161-172.  doi: 10.24193/subbmath.2019.2.03.

[10]

J. C. Ceballos V., R. Pavez F. and O. P. Vera Villagrán, Exact boundary controllability for higher order nonlinear Schrödinger equations with constant coefficients, Electron. J. Differential Equations, (2005), No. 122, 31 pp.

[11]

E. Cerpa and J. -M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.

[12]

M. Chen, Stabilization of the higher order nonlinear Schrödinger equation with constant coefficients, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), 15 pp. doi: 10.1007/s12044-018-0410-7.

[13]

J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl. (9), 102 (2014), 1080-1120.  doi: 10.1016/j.matpur.2014.03.004.

[14]

P. N. da Silva and C. F. Vasconcellos, On the stabilization and controllability for a third order linear equation, Port. Math., 68 (2011), 279-296.  doi: 10.4171/PM/1892.

[15]

O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Systems Control Lett., 59 (2010), 390-395.  doi: 10.1016/j.sysconle.2010.05.001.

[16]

C. E. Kenig and G. Staffilani, Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3 (1997), 417-433.  doi: 10.1007/BF02649104.

[17]

Y. Kodama, Optical solitons in a monomode fiber, J. Stat. Phys., 39 (1985), 597-614.  doi: 10.1007/BF01008354.

[18]

Y. Kodama and A. Hasegawa, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524.  doi: 10.1109/JQE.1987.1073392.

[19]

M. Krstic, B.-Z. Guo and A. Smyshlyaev, Boundary controllers and observers for Schrödinger equation, in 2007 46th IEEE Conference on Decision and Control, (2007), 4149–4154.

[20]

M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718607.

[21]

C. Laurey, The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.  doi: 10.1016/S0362-546X(96)00081-8.

[22]

W. Liu, Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim., 42 (2003), 1033-1043.  doi: 10.1137/S0363012902402414.

[23]

S. Marx and E. Cerpa, Output feedback stabilization of the Korteweg–de Vries equation, Automatica J. IFAC, 87 (2018), 210-217.  doi: 10.1016/j.automatica.2017.07.057.

[24]

T. Özsarı and E. Arabacı, Boosting the decay of solutions of the linearised Korteweg–de Vries–Burgers equation to a predetermined rate from the boundary, Internat. J. Control, 92 (2019), 1753-1763.  doi: 10.1080/00207179.2017.1408923.

[25]

T. Özsarı and A. Batal, Pseudo-backstepping and its application to the control of Korteweg–de Vries equation from the right endpoint on a finite domain, SIAM J. Control Optim., 57 (2019), 1255-1283.  doi: 10.1137/18M1211933.

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

A. Smyshlyaev and M. Krstic, Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. Automat. Control, 49 (2004), 2185-2202.  doi: 10.1109/TAC.2004.838495.

[28]

A. Smyshlyaev and M. Krstic, Backstepping observers for a class of parabolic PDEs, Systems Control Lett., 54 (2005), 613-625.  doi: 10.1016/j.sysconle.2004.11.001.

[29]

A. Smyshlyaev and M. Krstic, Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary, Systems Control Lett., 58 (2009), 617-623.  doi: 10.1016/j.sysconle.2009.04.005.

[30]

G. Staffilani, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations, 10 (1997), 777-796. 

[31]

H. Takaoka, Well-posedness for the higher order nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 10 (2000), 149-171. 

[32]

S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries systems with anti-diffusion, in IEEE 2013 American Control Conference, Washington, DC, (2013), 3302–3307.

[33]

S. Tang and M. Krstic, Stabilization of linearized Korteweg-de Vries with anti-diffusion by boundary feedback with non-collocated observation, in IEEE 2015 American Control Conference, 2015. doi: 10.1109/ACC.2015.7171020.

[34]

Z. XuL. LiZ. Li and G. Zhou, Soliton interaction under the influence of higher-order effects, Optics Communications, 210 (2002), 375-384.  doi: 10.1016/S0030-4018(02)01803-5.

Figure 1.  Backstepping
Figure 2.  Triangular regions
Figure 3.  Backstepping with an imperfect kernel
Figure 4.  $ H(k, l) : [1, \infty) \times [1, \infty) \to (-1, 1) $
Figure 5.  Integration path for the case $ \alpha^2 + 3\beta\delta > 0 $
Figure 6.  Plot of transformation $ \Im(s) = \omega(\xi) $ when $ \alpha^2 + 3\beta\delta > 0 $
Figure 7.  Integration path for the case $ \alpha^2 + 3\beta\delta = 0 $
Figure 8.  Plot of transformation $ \Im(s) = \omega(\xi) $ when $ \alpha^2 + 3\beta\delta = 0 $
Figure 9.  Integration path for the case $ \alpha^2 + 3\beta\delta < 0 $
Figure 10.  Plot of transformation $ \Im(s) = \omega(\xi) $ when $ \alpha^2 + 3\beta \delta < 0 $
Figure 11.  Up: Plots of $ k(x, y) $ on $ \Delta_{x, y} $. Down: Controller gains for Dirichlet and Neumann boundary conditions. $ L = \pi $, $ \beta = 1 $, $ \alpha = 2 $, $ \delta = 8 $ and $ r = 0.05 $
Figure 12.  $ p(x, y) $ defined on $ \Delta_{x, y} $ for $ L = \pi $, $ \beta = 1 $, $ \alpha = 2 $, $ \delta = 8 $ and $ r = 0.05 $
Figure 13.  Observer gains for $ L = \pi $, $ \beta = 1 $, $ \alpha = 2 $, $ \delta = 8 $ and $ r = 0.05 $
Figure 14.  Numerical results in the presence of controllers. Left: Time evolution of $ |u(x, t)| $. Right: Time evolution of $ \|u(\cdot, t)\|_2 $
Figure 15.  Numerical results. Left: Time evolution of $ |u(x, t)| $. Right: Time evolution of $ \|u(\cdot, t)\|_2 $, $ \|\hat u(\cdot, t)\|_2 $ and $ \|\tilde u(\cdot, t)\|_2 $
Table 1.  Some numerical values for the decay rate $ \lambda $ corresponding to various values of $ r $
$r$ $\lambda = \beta\left(\displaystyle\frac{r}{\beta} - \frac{\|k_y(\cdot, 0;r)\|_2^2}{2}\right)$
$0.001$ $0.001981$
$0.01$ $0.018054$
$0.02$ $0.032221$
$0.03$ $0.042507$
$0.04$ $0.048918$
$0.05$ $0.051463$
$0.1$ $0.006407$
$0.11$ $-0.014113$
$0.5$ $-3.729586$
$1$ $-16.379897$
$r$ $\lambda = \beta\left(\displaystyle\frac{r}{\beta} - \frac{\|k_y(\cdot, 0;r)\|_2^2}{2}\right)$
$0.001$ $0.001981$
$0.01$ $0.018054$
$0.02$ $0.032221$
$0.03$ $0.042507$
$0.04$ $0.048918$
$0.05$ $0.051463$
$0.1$ $0.006407$
$0.11$ $-0.014113$
$0.5$ $-3.729586$
$1$ $-16.379897$
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