doi: 10.3934/mcrf.2020035

Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension

Université Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de Versailles, 78000, Versailles, France, Centre de Recherches Mathématiques, CNRS UMI 3457, Université de Montréal, Montréal, QC, H3C 3J7, Canada

Received  December 2019 Revised  June 2020 Published  August 2020

In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.

Citation: Vahagn Nersesyan. Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020035
References:
[1]

D. R. Adams, On the existence of capacitary strong type estimates in $R^{n}$, Arkiv för Matematik, 14 (1976), 125-140.  doi: 10.1007/BF02385830.  Google Scholar

[2]

A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.  doi: 10.1007/s00021-004-0110-1.  Google Scholar

[3]

A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Comm. Math. Phys., 265 (2006), 673-697.  doi: 10.1007/s00220-006-0002-8.  Google Scholar

[4]

A. Agrachev and A. Sarychev, Solid controllability in fluid dynamics, Int. Math. Ser. (N.Y.), Springer, New York, 6 (2008), 1-35.  doi: 10.1007/978-0-387-75217-4_1.  Google Scholar

[5]

P.-M. Boulvard, P. Gao and V. Nersesyan, Controllability and ergodicity of 3D primitive equations driven by a finite-dimensional force, In preparation, (2020). Google Scholar

[6]

N. E. Glatt-Holtz, D. P. Herzog and J. C. Mattingly, Scaling and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations, Ann. PDE, 4 (2018), Art. 16, 103 pp. doi: 10.1007/s40818-018-0052-1.  Google Scholar

[7]

N. Jacobson, Basic Algebra. I, W. H. Freeman and Company, New York, Second edition, 1985.  Google Scholar

[8]

V. Jurdjevic and I. Kupka, Polynomial control systems, Math. Ann., 272 (1985), 361-368.  doi: 10.1007/BF01455564.  Google Scholar

[9]

V. Jurdjevic, Geometric Control Theory, volume 52 of Cambridge Studies in Advanced Mathematics, 1997. Google Scholar

[10]

S. KuksinV. Nersesyan and A. Shirikyan, Mixing via controllability for randomly forced nonlinear dissipative PDEs, J. Éc. Polytech. Math., 7 (2020), 871-896.  doi: 10.5802/jep.130.  Google Scholar

[11]

H. Nersisyan, Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var., 16 (2010), 677-694.  doi: 10.1051/cocv/2009017.  Google Scholar

[12]

H. Nersisyan, Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.  doi: 10.1080/03605302.2011.596605.  Google Scholar

[13]

V. Nersesyan, Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force, Nonlinearity, 28 (2015), 825-848.  doi: 10.1088/0951-7715/28/3/825.  Google Scholar

[14]

D. Phan and S. S. Rodrigues, Approximate controllability for Navier–Stokes equations in 3D rectangles under Lions boundary conditions, J. Dyn. Control Syst., 25 (2019), 351-376.  doi: 10.1007/s10883-018-9412-0.  Google Scholar

[15]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[16]

S. S. Rodrigues, Navier–Stokes equation on the rectangle: Controllability by means of low mode forcing, J. Dyn. Control Syst., 12 (2006), 517-562.  doi: 10.1007/s10883-006-0004-z.  Google Scholar

[17]

A. Sarychev, Controllability of the cubic Schrödinger equation via a low-dimensional source term, Math. Control Relat. Fields, 2 (2012), 247-270.  doi: 10.3934/mcrf.2012.2.247.  Google Scholar

[18]

A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys., 266 (2006), 123-151.  doi: 10.1007/s00220-006-0007-3.  Google Scholar

[19]

A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.  doi: 10.1016/j.anihpc.2006.04.002.  Google Scholar

[20]

A. Shirikyan, Approximate controllability of the viscous Burgers equation on the real line, In Geometric Control Theory and Sub-Riemannian Geometry, Springer, Cham, 5 (2014), 351–370. doi: 10.1007/978-3-319-02132-4_20.  Google Scholar

[21]

A. Shirikyan, Control theory for the Burgers equation: Agrachev-Sarychev approach, Pure Appl. Funct. Anal., 3 (2018), 219-240.   Google Scholar

show all references

References:
[1]

D. R. Adams, On the existence of capacitary strong type estimates in $R^{n}$, Arkiv för Matematik, 14 (1976), 125-140.  doi: 10.1007/BF02385830.  Google Scholar

[2]

A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: controllability by means of low modes forcing, J. Math. Fluid Mech., 7 (2005), 108-152.  doi: 10.1007/s00021-004-0110-1.  Google Scholar

[3]

A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Comm. Math. Phys., 265 (2006), 673-697.  doi: 10.1007/s00220-006-0002-8.  Google Scholar

[4]

A. Agrachev and A. Sarychev, Solid controllability in fluid dynamics, Int. Math. Ser. (N.Y.), Springer, New York, 6 (2008), 1-35.  doi: 10.1007/978-0-387-75217-4_1.  Google Scholar

[5]

P.-M. Boulvard, P. Gao and V. Nersesyan, Controllability and ergodicity of 3D primitive equations driven by a finite-dimensional force, In preparation, (2020). Google Scholar

[6]

N. E. Glatt-Holtz, D. P. Herzog and J. C. Mattingly, Scaling and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations, Ann. PDE, 4 (2018), Art. 16, 103 pp. doi: 10.1007/s40818-018-0052-1.  Google Scholar

[7]

N. Jacobson, Basic Algebra. I, W. H. Freeman and Company, New York, Second edition, 1985.  Google Scholar

[8]

V. Jurdjevic and I. Kupka, Polynomial control systems, Math. Ann., 272 (1985), 361-368.  doi: 10.1007/BF01455564.  Google Scholar

[9]

V. Jurdjevic, Geometric Control Theory, volume 52 of Cambridge Studies in Advanced Mathematics, 1997. Google Scholar

[10]

S. KuksinV. Nersesyan and A. Shirikyan, Mixing via controllability for randomly forced nonlinear dissipative PDEs, J. Éc. Polytech. Math., 7 (2020), 871-896.  doi: 10.5802/jep.130.  Google Scholar

[11]

H. Nersisyan, Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var., 16 (2010), 677-694.  doi: 10.1051/cocv/2009017.  Google Scholar

[12]

H. Nersisyan, Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564.  doi: 10.1080/03605302.2011.596605.  Google Scholar

[13]

V. Nersesyan, Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force, Nonlinearity, 28 (2015), 825-848.  doi: 10.1088/0951-7715/28/3/825.  Google Scholar

[14]

D. Phan and S. S. Rodrigues, Approximate controllability for Navier–Stokes equations in 3D rectangles under Lions boundary conditions, J. Dyn. Control Syst., 25 (2019), 351-376.  doi: 10.1007/s10883-018-9412-0.  Google Scholar

[15]

P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[16]

S. S. Rodrigues, Navier–Stokes equation on the rectangle: Controllability by means of low mode forcing, J. Dyn. Control Syst., 12 (2006), 517-562.  doi: 10.1007/s10883-006-0004-z.  Google Scholar

[17]

A. Sarychev, Controllability of the cubic Schrödinger equation via a low-dimensional source term, Math. Control Relat. Fields, 2 (2012), 247-270.  doi: 10.3934/mcrf.2012.2.247.  Google Scholar

[18]

A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys., 266 (2006), 123-151.  doi: 10.1007/s00220-006-0007-3.  Google Scholar

[19]

A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537.  doi: 10.1016/j.anihpc.2006.04.002.  Google Scholar

[20]

A. Shirikyan, Approximate controllability of the viscous Burgers equation on the real line, In Geometric Control Theory and Sub-Riemannian Geometry, Springer, Cham, 5 (2014), 351–370. doi: 10.1007/978-3-319-02132-4_20.  Google Scholar

[21]

A. Shirikyan, Control theory for the Burgers equation: Agrachev-Sarychev approach, Pure Appl. Funct. Anal., 3 (2018), 219-240.   Google Scholar

[1]

Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167

[2]

T. Gallouët, J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2371-2391. doi: 10.3934/cpaa.2012.11.2371

[3]

Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305

[4]

Lorena Bociu, Barbara Kaltenbacher, Petronela Radu. Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications. Evolution Equations & Control Theory, 2013, 2 (2) : i-ii. doi: 10.3934/eect.2013.2.2i

[5]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953

[6]

Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control & Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375

[7]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control & Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263

[8]

Mu-Ming Zhang, Tian-Yuan Xu, Jing-Xue Yin. Controllability properties of degenerate pseudo-parabolic boundary control problems. Mathematical Control & Related Fields, 2020, 10 (1) : 157-169. doi: 10.3934/mcrf.2019034

[9]

Enrico Valdinoci. Contemporary PDEs between theory and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : i-i. doi: 10.3934/dcds.2015.35.12i

[10]

Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741

[11]

Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431

[12]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397

[13]

Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784

[14]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[15]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171

[16]

Moncef Aouadi, Taoufik Moulahi. Approximate controllability of abstract nonsimple thermoelastic problem. Evolution Equations & Control Theory, 2015, 4 (4) : 373-389. doi: 10.3934/eect.2015.4.373

[17]

Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023

[18]

Assia Benabdallah, Michel Cristofol, Patricia Gaitan, Luz de Teresa. Controllability to trajectories for some parabolic systems of three and two equations by one control force. Mathematical Control & Related Fields, 2014, 4 (1) : 17-44. doi: 10.3934/mcrf.2014.4.17

[19]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

[20]

Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (34)
  • HTML views (49)
  • Cited by (0)

Other articles
by authors

[Back to Top]