Far Eastern Mathematical Journal

To content of the issue

Carleman estimates of solutions of the Neumann problem for a parabolic equation

Amosova E. V.

2015, issue 1, Ñ. 3-20

We derive a new Carleman estimates for the Neumann problem for a parabolic equation and Laplace equation.

exact controllability, estimates of Carleman type

Download the article (PDF-file)


[1] J.-L. Lions, “Are the connections between turbulence and controllability?”, Lecture Notesin Control Inform. Sci., V (1990), 144.
[2] J.-L. Lions, “Remarques sur la controllabilite approchee”, Control of Distributed Systems, 3 (1990), 77–87.
[3] A. V. Fursikov, O.Ju. Jemanuilov, “Tochnaja lokal'naja upravljaemost' dvumernyh uravnenij Nav'e-Stoksa”, Matem. sb., 187:9 (1996), 102–138.
[4] A. V. Fursikov, O.Ju. Jemanuilov, “Lokal'naja tochnaja upravljaemost' uravnenij Bussineska”, Vestn. RUDN, ser. Matem., 3:1 (1996), 177–194.
[5] A. V. Fursikov, O.Ju. Jemanuilov, “Tochnaja upravljaemost' uravnenij Nav'e-Stoksa i Bussineska”, Uspehi matematicheskih nauk, 54:3(327) (1999), 93–42.
[6] A. V. Fursikov, O.Yu. Imanuvilov, “Local exact controllability of the Navier-Stokes equations”, C. R. Acad. Sci., Paris, Serie I., 323 (1996), 275–280.
[7] A. V. Fursikov, O.Yu. Imanuvilov, “Local Exact Boundary Controllability of the Boussinesque Equations”, SIAM J. Control Optim., 36:2 (1998), 391–421.
[8] A. V. Fursikov, O.Yu. Imanuvilov, “On controllability of certain systems simulating a fluid flow”, IMA Vol.Math.Appl., 68 (1995), 149–184.
[9] O.Yu. Imanuvilov, “Local exact controllability for the 2-D Navier-Stokes equations with the Navier slip boundary conditions”, Turbulence Modeling and Vortex Dynamics. Lecture Notes in Physics., 491 (1997), 148–168.
[10] A. V. Fursikov, O.Yu. Imanuvilov, “On exact boundary zero-controllability of two-dimensional Navier-Stokes equations”, Acta Appl. Math., 37 (1994), 67–76.
[11] J. I Diaz, A. V. Fursikov, “Approximate controllability of the Stokes system on cylinders by external unidirectional forces”, J. Math. Pures Appl., 76 (1997), 353–375.
[12] C. Fabre, J.-P. Puel, E Zuazua, “Approximate controllability of the semilinear heat equation”, Proc. Roy.Edinburgh. Sect.A., 125 (1995), 31–61.
[13] L.A Fernandez, E. Zuazua, “Approximate controllability of the semilinear heat equation involving gradient terms”, J. Optim. Theory Appl., 1999.
[14] O.Ju. Jemanuilov, “Tochnaja upravljaemost' polulinejnogo parabolicheskogo uravnenija”, Vestnik Ros. Univ. Druzhby Nar. Ser. matem., 1 (1994), 109–116.
[15] O.Ju. Jemanuilov, “Granichnoe upravlenie polulinejnymi jevoljucionnymi uravnenijami”, Uspehi matematicheskih nauk, 44:6 (1988), 183–184.
[16] O.Yu. Imanuvilov, “Local exact controllability for the 2-D Navier-Stokes equations with the Navier slip boundary conditions”, Lecture Notes in Phys., 491 (1997), 148–168.
[17] A. V. Fursikov, Optimal'noe upravlenie raspredelitel'nymi sistemami. Teorija i prilozhenija, Nauchnaja kniga, N., 1999.
[18] S. Ervedoza, O. Glass O., S. Guerrero, “Local exact controllability for the 1-D compressible Navier-Stokes equation”, Seminaire Laurent Schwarts - EDP et applications, XXXIX (2011), 14.
[19] E. V. Amosova, “Exact Local Controllability for the Equations of Viscous Gas Dynamics”, Differential Equations, 47:12 (2011), 1776–1795.
[20] J.-M. Coron, “On the controllability of 2-D incompressible perfect fluids”, J. Math. Pures et Appl., 75 (1996), 155–188.
[21] J.-M. Coron, “Controlabilite exacte frontiere de l’equation d’Euler des fluides parfaits incompressibles bidimensionnels”, C. R. Acad. Sci. Paris. Ser. I., 317 (1993), 271–276.
[22] J.-M. Coron, A. V. Fursikov, “Global exact controllability of the 2D Navier-Stokes equations on manifold without boundary”, Russian Journal of Math. Physics., 4:3 (1996), 1–20.
[23] V. Barbu, “Exact controllability of superlinear heat equation”, Applied Mathematics and Optimization, 42 (2000), 73–89.
[24] V. Barbu, “Controllability of parabolic and Navier-Stokes equations”, Scientiae Mathematicae Japonicae, 56 (2002), 143-211.
[25] C. Bardos, G. Lebeau, J. Rauch, “Controle et stabilisation de l’equation des ondes”, SIAM Journal on Control and Optimization, 30 (1992), 1024–1065.
[26] E. Fernandez-Cara, “Null controllability of the semilinear heat equation”, ESAIM: Control Optimization and Calculus of Variations, 2 (1997), 87–103.
[27] G. Aniculaesei, S. Anita, “Null controllability of a nonlinear heat equation”, Abstract and Applied Analysis, 7 (2002), 375–383.
[28] J. Klamka, “Constrained controllability of semilinear systems with multiple delays in control”, Bulletin of the Polish Academy of Sciences. Technical Sciences, 52 (2004), 25–30.
[29] K. Sakthivel, K. Balachandran, B. R. Nagaraj, “On a class of nonlinear parabolic control systems with memory effects”, International Journal of Control, 81 (2008), 764–777.
[30] K. Sakthivel, K. Balachandran, S. S. Sritharan, “Exact controllability of nonlinear diffusion equations arising in reactor dynamics”, Nonlinear Analysis:Real World Applications, 9 (2008), 2029–2054.
[31] A. Kazemi, M. Klibanov, “Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities”, Appl. Anal., 50 (1993), 93–102.
[32] I. Lasiecka I, R. Triggiani, “Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations”, Lecture Notes in Pure Appl. Math., 188 (1997), 215–243.
[33] C. Fabre, “Uniqueness results for Stokes equations and theirconsequences in linear and nonlinear control problems”, ESAIM, Control Optim. Caic. Var., 1 (1996), 267–302.
[34] O.Yu. Imanuvilov, M. Yamamoto, “On Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations”, Publ. Res. Inst. Math. Sci., 39 (2003), 227–274.
[35] A. Ruiz, “Unique continuation for weak solutions of the wave equation plus a potential”, J. Math. Pures Appl., 71 (1992), 455–467.
[36] D. Tatary, “A prior estimates of Carleman’s type in domains with boundary”, J. Math. Pures Appl., 73 (1994), 355–387.
[37] D. Chae, O.Yu. Imanuvilov, S. M. Kim, “Exact controllability for semilinear parabolic equations with Neumann boundary conditions”, J. of Dynamical and Control Systems, 2 (1996), 449–483.
[38] K. Sakthivel, G. Devipriya, K. Balachandran, J.-H. Kim, “Exact null controllability of a semilinear parabolic equation arising in finance”, Nonlinear Analysis: Hybrid Systems, 3 (2009), 565–577.
[39] Zh.-L. Lions, Je. Madzhenes, Neodnorodnye granichnye zadachi i ih prilozhenija, Mir, M., 1971.

To content of the issue