Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

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April 2012, 32(4): 1169-1208. doi: 10.3934/dcds.2012.32.1169

Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

Mehdi Badra ^1,

1.	Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex

Received October 2010 Revised May 2011 Published October 2011

Abstract
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Let $-A: \mathcal{D}(A)\to H$ be the generator of an analytic semigroup and $B : U \to [\mathcal{D}(A^*)]'$ a relatively bounded control operator such that $(A-\sigma,B)$ is stabilizable for some $\sigma>0$. In this paper, we consider the stabilization of the nonlinear system $y'+Ay+G(y,u)=Bu$ by means of a feedback or a dynamical control $u$. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.

Keywords: Dirichlet control., Riccati equation, Navier-Stokes, Feedback stabilization, Neumann control, Boussinesq, Lyapunov function.

Mathematics Subject Classification: Primary: 93D15, 93C20; Secondary: 76D05, 76D55, 35Q30, 35Q3.

Citation: Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169

References:

[1]	C. Amrouche and V. Girault, Problèmes généralisés de Stokes, Portugal. Math., 49 (1992), 463-503.
[2]	C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.
[3]	G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, In "Fluids and Waves," 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007.
[4]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9.
[5]	G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behaviour of the resolvent operator on the imaginary axis, Appl. Anal., 88 (2009), 1357-1396. doi: 10.1080/00036810903278513.
[6]	G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.
[7]	M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system, In "Systems, Control, Modeling and Optimization," 13-24, IFIP Int. Fed. Inf. Process., 202, Springer, New York, 2006.
[8]	M. Badra, Local stabilization of the Navier-Stokes equations with a feedback controller localized in an open subset of the domain, Numer. Funct. Anal. Optim., 28 (2007), 559-589. doi: 10.1080/01630560701348434.
[9]	M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968. doi: 10.1051/cocv:2008059.
[10]	M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48 (2009), 1797-1830. doi: 10.1137/070682630.
[11]	M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463. doi: 10.1137/090778146.
[12]	V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 9 (2003), 197-206 (electronic). doi: 10.1051/cocv:2003009.
[13]	V. Barbu, "Stabilization of Navier-Stokes Flow," Communications and Control Engineering, Springer Verlag, Berlin-London, 2010.
[14]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, In "Fluids and Waves," 55-82, Contemp. Math, 440, Amer. Math. Soc., Providence, RI, 2007.
[15]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284.
[16]	V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746. doi: 10.1016/j.na.2005.09.012.
[17]	V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), :x+128 pp.
[18]	V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d=2,3$, via feedback stabilization of its linearization, In "Control of Coupled Partial Differential Equations," 13-46, Internat. Ser. Numer. Math., 155, Birkhäuser, Basel, 2007.
[19]	V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls, 2010. Available from: http://hal.archives-ouvertes.fr/hal-00455250/fr/.
[20]	V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445.
[21]	F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006.
[22]	F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 (2010), 217-235.
[23]	F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505-521. doi: 10.3934/dcdss.2011.4.505.
[24]	P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[25]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp.
[26]	H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad., 46 (1970), 1141-1143. doi: 10.3792/pja/1195526510.
[27]	A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary, In "Partial Differential Equations," 95-123, Amer. Math. Soc. Transl. Ser. 2, 206, Amer. Math. Soc., Providence, RI, 2002.
[28]	A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289.
[29]	P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63.
[30]	G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Math. Scand., 69 (1991), 217-290.
[31]	S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61.
[32]	E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957.
[33]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005.
[34]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254.
[35]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008.
[36]	I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach, Appl. Anal., 42 (1991), 243-261. doi: 10.1080/00036819108840045.
[37]	I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati's feedback synthesis and regularity of optimal solution, Appl. Math. Optim., 16 (1987), 147-168. doi: 10.1007/BF01442189.
[38]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.
[39]	I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction, In "Optimal Control of Coupled Systems of Partial Differential Equations," 171-190, Internat. Ser. Numer. Math., 158, Birkhäuser Verlag, Basel, 2009.
[40]	I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems Control Lett., 58 (2009), 499-509. doi: 10.1016/j.sysconle.2009.02.010.
[41]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[42]	J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I, Dunod, Paris, 1968.
[43]	J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic). doi: 10.1137/050628726.
[44]	J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002.
[45]	J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
[46]	J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.
[47]	J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete and Continuous Dynamical Systems A, 27 (2010), 1159-1187.
[48]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Second edition, Johann Ambrosius Barth, Heidelberg, 1995.
[49]	R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation, Nonlinear Anal., 71 (2009), 4967-4976. doi: 10.1016/j.na.2009.03.073.
[50]	R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645-677. doi: 10.3934/dcdss.2009.2.645.

show all references

References:

[1]	C. Amrouche and V. Girault, Problèmes généralisés de Stokes, Portugal. Math., 49 (1992), 463-503.
[2]	C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.
[3]	G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties, In "Fluids and Waves," 15-54, Contemp. Math., 440, Amer. Math. Soc., Providence, RI, 2007.
[4]	G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Equ., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9.
[5]	G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behaviour of the resolvent operator on the imaginary axis, Appl. Anal., 88 (2009), 1357-1396. doi: 10.1080/00036810903278513.
[6]	G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.
[7]	M. Badra, Feedback stabilization of the 3-D Navier-Stokes equations based on an extended system, In "Systems, Control, Modeling and Optimization," 13-24, IFIP Int. Fed. Inf. Process., 202, Springer, New York, 2006.
[8]	M. Badra, Local stabilization of the Navier-Stokes equations with a feedback controller localized in an open subset of the domain, Numer. Funct. Anal. Optim., 28 (2007), 559-589. doi: 10.1080/01630560701348434.
[9]	M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968. doi: 10.1051/cocv:2008059.
[10]	M. Badra, Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations, SIAM J. Control Optim., 48 (2009), 1797-1830. doi: 10.1137/070682630.
[11]	M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers: Application to the Navier-Stokes system, SIAM J. Control Optim., 49 (2011), 420-463. doi: 10.1137/090778146.
[12]	V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 9 (2003), 197-206 (electronic). doi: 10.1051/cocv:2003009.
[13]	V. Barbu, "Stabilization of Navier-Stokes Flow," Communications and Control Engineering, Springer Verlag, Berlin-London, 2010.
[14]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, In "Fluids and Waves," 55-82, Contemp. Math, 440, Amer. Math. Soc., Providence, RI, 2007.
[15]	V. Barbu, Z. Grujić , I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-1207. doi: 10.1512/iumj.2008.57.3284.
[16]	V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006), 2704-2746. doi: 10.1016/j.na.2005.09.012.
[17]	V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), :x+128 pp.
[18]	V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d=2,3$, via feedback stabilization of its linearization, In "Control of Coupled Partial Differential Equations," 13-46, Internat. Ser. Numer. Math., 155, Birkhäuser, Basel, 2007.
[19]	V. Barbu, S. S. Rodrigues and A. Shirikyan, Internal exponential stabilization for Navier-Stokes equations by means of finite-dimensional distributed controls, 2010. Available from: http://hal.archives-ouvertes.fr/hal-00455250/fr/.
[20]	V. Barbu and R Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494. doi: 10.1512/iumj.2004.53.2445.
[21]	F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006.
[22]	F. Bucci and I. Lasiecka, Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions, Calc. Var. Partial Differential Equations, 37 (2010), 217-235.
[23]	F. Bucci and I. Lasiecka, Regularity of boundary traces for a fluid-solid interaction model, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 505-521. doi: 10.3934/dcdss.2011.4.505.
[24]	P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.
[25]	R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114 pp.
[26]	H. Fujita and H. Morimoto, On fractional powers of the Stokes operator, Proc. Japan Acad., 46 (1970), 1141-1143. doi: 10.3792/pja/1195526510.
[27]	A. V. Fursikov, Real process corresponding to the 3D Navier-Stokes system, and its feedback stabilization from the boundary, In "Partial Differential Equations," 95-123, Amer. Math. Soc. Transl. Ser. 2, 206, Amer. Math. Soc., Providence, RI, 2002.
[28]	A. V. Fursikov, Stabilization for the 3D Navier-Stokes system by feedback boundary control. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 289-314. doi: 10.3934/dcds.2004.10.289.
[29]	P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63.
[30]	G. Grubb and V. A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Math. Scand., 69 (1991), 217-290.
[31]	S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 29-61.
[32]	E. Hille and R. S. Phillips, "Functional Analysis and Semi-Groups," rev. ed., American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957.
[33]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005.
[34]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254.
[35]	I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008.
[36]	I. Lasiecka, Exponential stabilization of hyperbolic systems with nonlinear, unbounded perturbations--Riccati operator approach, Appl. Anal., 42 (1991), 243-261. doi: 10.1080/00036819108840045.
[37]	I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati's feedback synthesis and regularity of optimal solution, Appl. Math. Optim., 16 (1987), 147-168. doi: 10.1007/BF01442189.
[38]	I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems," Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.
[39]	I. Lasiecka and A. Tuffaha, Optimal feedback synthesis for Bolza control problem arising in linearized fluid structure interaction, In "Optimal Control of Coupled Systems of Partial Differential Equations," 171-190, Internat. Ser. Numer. Math., 158, Birkhäuser Verlag, Basel, 2009.
[40]	I. Lasiecka and A. Tuffaha, Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction, Systems Control Lett., 58 (2009), 499-509. doi: 10.1016/j.sysconle.2009.02.010.
[41]	J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[42]	J.-L. Lions and E. Magenes, "Problemes aux Limites Non Homognes et Applications," Vol. I, Dunod, Paris, 1968.
[43]	J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM J. Control Optim., 45 (2006), 790-828 (electronic). doi: 10.1137/050628726.
[44]	J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pures Appl. (9), 87 (2007), 627-669. doi: 10.1016/j.matpur.2007.04.002.
[45]	J.-P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 921-951.
[46]	J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM J. Control Optim., 48 (2010), 5398-5443.
[47]	J.-P. Raymond and L. Thevenet, Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers, Discrete and Continuous Dynamical Systems A, 27 (2010), 1159-1187.
[48]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Second edition, Johann Ambrosius Barth, Heidelberg, 1995.
[49]	R. Triggiani, Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation, Nonlinear Anal., 71 (2009), 4967-4976. doi: 10.1016/j.na.2009.03.073.
[50]	R. Triggiani, Unique continuation of boundary over-determined Stokes and Oseen eigenproblems, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 645-677. doi: 10.3934/dcdss.2009.2.645.

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