June  2021, 11(2): 237-251. 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  June 2021 Early access  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, 2021, 11 (2) : 237-251. 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

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