On the controllability of the Burger equation. (English) Zbl 0897.93034

The author considers the existence of weak solutions to the problem of controllability for the equation: \[ u_t+ (u^2/2)_x=0 \tag{1} \] in the following sense: Suppose that there exists a weak solution \(\widehat u:\mathbb{R}^+ \times\mathbb{R} \to\mathbb{R}\in L^\infty ((0,\infty), X)\) such that for any \(z_0\), \(z_1\in X\), and for any \(T\in \mathbb{R}^+\), \(\widehat u(0,.) =z_0\), \(\widehat u(T,.) =z_1\), then equation (1) is said to be controllable in \(X\). If the exact equality is replaced by the requirement that \(\| u(T,.)-z_1 \|< \varepsilon\) for any given \(\varepsilon \), then it is said to be approximately controllable in \(X\). Such solutions are sought in a subset of the class of entropic functions of bounded variation. A function \(f(x)\) is said to be entropic if for any \(x\) in its domain (which is a subset of \(\mathbb{R})\) the limit from the left \(f(x-)\) is greater or equal than the limit from the right \(f(x+)\). The author uses J. M. Coron’s idea of considering the linearization of the operator not around the zero solution (which does not work for this system), but instead around trajectories that vanish at times 0 and \(T\). This is closely related to the “extension technique” originally introduced by D. L. Russell around 1973, which included the idea of variation of the domain.
Reviewer: V.Komkov (Roswell)


93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
35Q30 Navier-Stokes equations
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI EuDML


[1] F. Ancona, A. Marson: On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control, SIAM J. of control, to appear. Zbl0919.35082 MR1616586 · Zbl 0919.35082
[2] J-M. Coron: Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5, 1992, 295-312. Zbl0760.93067 MR1164379 · Zbl 0760.93067
[3] J-M. Coron: Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C.-R. Acad. Sci. Paris, 317, Série 1, 1993, 271-276. Zbl0781.76013 MR1233425 · Zbl 0781.76013
[4] J-M. Coron: On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM: Control, Optimisation and Caleulus of Variations, http://www.emath.fr/cocv/, 1, 1996, 35-75. Zbl0872.93040 MR1393067 · Zbl 0872.93040
[5] J.I. Diaz: Sobre la controlabilidad aproximada de problemas no lineales disipativos, proceedings of Jornadas Hispano-Francsas sobre Control de sistemas distribuidos, A. Valle ed., Univ. de Malága, 1990, 41-48. Zbl0752.49002 MR1108869 · Zbl 0752.49002
[6] C. Fabre, J-P. Puel, E. Zuazua: Contrôlabilité approchée de l’équation de la chaleur semilinaire, C.-R. Acad. Sci. Paris, 315, Série 1, 1992, 807-812. Zbl0770.35009 MR1184907 · Zbl 0770.35009
[7] C. Fabre, J-P. Puel, E. Zuazua: Approximate Controllability of the semilinear heat equation, Proc. of the Royal Soc. of Edinburgh, 125A, 1995, 31-61. Zbl0818.93032 MR1318622 · Zbl 0818.93032
[8] A. Fursikov, O. Yu. Imanuvilov: On controllability of certain systems simulating a fluid flow, IMA vol. in Math. and its Appl. Flow Control, M.D. Gunzburger ed., Springer Verlag, New York, 68, 1994. Zbl0922.93006 MR1348646 · Zbl 0922.93006
[9] A. Fursikov, O. Yu. Imanuvilov: Controllability of evolution equations, Lecture Notes Series 34, Res. Imst., Math. GARC, Seoul National University, 1996. Zbl0862.49004 MR1406566 · Zbl 0862.49004
[10] M. Gisclon: Etude des conditions aux limites pour des systèmes strictement hyperboliques, via l’approximation parabolique, Thèse de l’université Lyon I, 1996. Zbl0869.35061 · Zbl 0869.35061
[11] G. Godlevski, P.A. Raviart: Hyperbolic systems of conservation laws, Ellipses, 1990. Zbl0768.35059 · Zbl 0768.35059
[12] P.D. Lax: Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series in Applied Mathematics, 11, SIAM: Philadelphia, 1973. Zbl0268.35062 MR350216 · Zbl 0268.35062
[13] P. Le Floch: Explicit Formula for Scalar Nonlinear Conservation Laws with boundary condition, Math. Methods Appl. Sci., 10, 1988, 265-287. Zbl0679.35065 MR949657 · Zbl 0679.35065
[14] B.J. Lucier: Regularity through approximation for scalar conservation laws, SIAM J. Math. Anal., 19, 1988, 763-773. MR946641
[15] O.A. Oleinik: Discontinuous solutions of nonlinear differential equations, Usp. Math. Nauk (N.S.), 12, 1957, 3-73; English translation: Amer. Math. Soc. Transl., 26, Ser. 2, 95-172. Zbl0131.31803 MR151737 · Zbl 0131.31803
[16] L. Rosier: Exact Boundary Controllability for the Korteveg-de Vries Equation on a Bounded Domain, ESAIM: Control, Optimisation and Calculus of Variations, http://www.emath.fr/cocv/, 2, 1997, 33-55. Zbl0873.93008 MR1440078 · Zbl 0873.93008
[17] D.L. Russell: Exact boundary value controllability theorems for wave and heat processes in starcomplemented regions, in Differential Games and Control Theory, Roxin, Liu and Sternberg Eds., Marcel Dekker, New York, 1974, 291-319. Zbl0308.93007 MR467472 · Zbl 0308.93007
[18] A.I. Vol’pert: The spaces BV and quasilinear equations, Math. USSR Sbornik, 2, 1967, 225-267. Zbl0168.07402 MR216338 · Zbl 0168.07402
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