Exact controllability of scalar conservation laws with strict convex flux

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December 2014, 4(4): 401-449. doi: 10.3934/mcrf.2014.4.401

Exact controllability of scalar conservation laws with strict convex flux

Adimurthi ^1, , Shyam Sundar Ghoshal ^1, and G. D. Veerappa Gowda ^1,

1.	Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Post Bag No 6503, Sharadanagar, Bangalore - 560065, India, India, India

Received February 2012 Revised February 2014 Published September 2014

Abstract
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We consider the scalar conservation law with strict convex flux in one space dimension. In this paper we study the exact controllability of entropy solution by using initial or boundary data control. Some partial results have been obtained in [5],[23]. Here we investigate the precise conditions under which, the exact controllability problem admits a solution. The basic ingredients in the proof of these results are, Lax-Oleinik [15] explicit formula and finer properties of the characteristics curves.

Keywords: Hamilton-Jacobi equation, controllability., scalar conservation laws, characteristics.

Mathematics Subject Classification: Primary: 35L65, 35Q9.

Citation: Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control and Related Fields, 2014, 4 (4) : 401-449. doi: 10.3934/mcrf.2014.4.401

References:

[1]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Optimal controllability for scalar conservation laws with convex flux, J. Hyperbolic Differ. Equ., 11 (2014), 477-491.
[2]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Structure of the entropy solution of a scalar conservation law with strict convex flux, J. Hyperbolic Differ. Equ., 9 (2012), 571-611. doi: 10.1142/S0219891612500191.
[3]	Adimurthi and G. D. Veerappa Gowda, Conservation Law with discontinuous flux, J.Math, Kyoto Univ, 43 (2003), 27-70.
[4]	F. Ancona, O. Glass and K. T. Nguyen, Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math, 65 (2012), 1303-1329. doi: 10.1002/cpa.21406.
[5]	F. Ancona and A. Marson, On the attainability set for scalar non linear conservation laws with boundary control, SIAM J.Control Optim, 36 (1998), 290-312. doi: 10.1137/S0363012996304407.
[6]	F. Ancona and A. Marson, Scalar non linear conservation laws with integrable boundary data, Nonlinear Anal, 35 (1999), 687-710. doi: 10.1016/S0362-546X(97)00697-4.
[7]	C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.
[8]	A. Bressan and A. Marson, A maximum principle for optimally controlled systems of conservation laws, Rend. Sem. Mat. Univ. Padova, 94 (1995), 79-94.
[9]	T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 41. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
[10]	M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim, 48 (2009), 1567-1599. doi: 10.1137/070685749.
[11]	J. M. Coron, Global asymptotic stabilization for controllable systems without drift, Mth. Control signals systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[12]	C. M. Dafermos, Characteristics in hyperbolic conservations laws, A study of the structure and the asymptotic behavior of solutions, Research notes in maths, Pitman, I (1977), 1-58.
[13]	C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $2^{nd}$ edition, Springer Verlag, Berlin, 2000.
[14]	J. I. Diaz, Obstruction and some approximate controllability results for the Burgers equation and related problems, Control of partial differential equations and applications, (Laredo, 1994) Lecture Notes in Pure and Appl. Math, 174, Dekker, New York, (1996), 63-76.
[15]	L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, vol 19, AMS 1998.
[16]	E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 229-232. doi: 10.1016/j.crma.2005.06.005.
[17]	A. Fursikov and O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow, Flow control, (Minneapolis, MN, 1992), IMA Vol. Math. Appl., 68, Springer, New York, (1995), 149-184. doi: 10.1007/978-1-4612-2526-3_7.
[18]	S. S. Ghoshal, Finer Analysis of Characteristic Curves, and Its Applications to Shock Profile, Exact and Optimal Controllability of Conservation Law with Strict convex Fluxes, Ph.D thesis, TIFR CAM, 2012.
[19]	O. Glass and S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control optim., 46 (2007), 1211-1238. doi: 10.1137/060664677.
[20]	E. Godleweski and P. A. Raviant, Hyperbolic Systems of Conservation Laws, Mathematiques and Applications, 3/4, Ellipses, Paris, 1991.
[21]	S. Guerrero and O. Y. Immunauvilov, Remarks on global controllability for the Burgers equation with two control forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 897-906. doi: 10.1016/j.anihpc.2006.06.010.
[22]	E. Hopf, The partial differential equation $u_t + u u_x = \mu u_{x x}$, Comm. Pure Appl. Math, 3 (1950), 201-230.
[23]	T. Horsin, On the controllability of the Burger equation, ESIAM, Control optimization and Calculus of variations, 3 (1998), 83-95. doi: 10.1051/cocv:1998103.
[24]	K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for the solution of Convex conservation laws with boundary condition, Duke Math.J., 62 (1991), 401-416. doi: 10.1215/S0012-7094-91-06216-2.
[25]	S. N. Kružkov, First order quasilinear equations with several independent variables, (Russian), Mat. Sb., 81 (1970), 228-255. English transl. in Math. USSR Sb., 10 (1970), 217-243.
[26]	P. D. Lax, Hyperbolic systems of conservation Laws II, comm Pure Appl. Math, 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.

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References:

[1]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Optimal controllability for scalar conservation laws with convex flux, J. Hyperbolic Differ. Equ., 11 (2014), 477-491.
[2]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Structure of the entropy solution of a scalar conservation law with strict convex flux, J. Hyperbolic Differ. Equ., 9 (2012), 571-611. doi: 10.1142/S0219891612500191.
[3]	Adimurthi and G. D. Veerappa Gowda, Conservation Law with discontinuous flux, J.Math, Kyoto Univ, 43 (2003), 27-70.
[4]	F. Ancona, O. Glass and K. T. Nguyen, Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math, 65 (2012), 1303-1329. doi: 10.1002/cpa.21406.
[5]	F. Ancona and A. Marson, On the attainability set for scalar non linear conservation laws with boundary control, SIAM J.Control Optim, 36 (1998), 290-312. doi: 10.1137/S0363012996304407.
[6]	F. Ancona and A. Marson, Scalar non linear conservation laws with integrable boundary data, Nonlinear Anal, 35 (1999), 687-710. doi: 10.1016/S0362-546X(97)00697-4.
[7]	C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117.
[8]	A. Bressan and A. Marson, A maximum principle for optimally controlled systems of conservation laws, Rend. Sem. Mat. Univ. Padova, 94 (1995), 79-94.
[9]	T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gas Dynamics, Pitman Monographs and Surveys in Pure and Applied Mathematics, 41. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
[10]	M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim, 48 (2009), 1567-1599. doi: 10.1137/070685749.
[11]	J. M. Coron, Global asymptotic stabilization for controllable systems without drift, Mth. Control signals systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.
[12]	C. M. Dafermos, Characteristics in hyperbolic conservations laws, A study of the structure and the asymptotic behavior of solutions, Research notes in maths, Pitman, I (1977), 1-58.
[13]	C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $2^{nd}$ edition, Springer Verlag, Berlin, 2000.
[14]	J. I. Diaz, Obstruction and some approximate controllability results for the Burgers equation and related problems, Control of partial differential equations and applications, (Laredo, 1994) Lecture Notes in Pure and Appl. Math, 174, Dekker, New York, (1996), 63-76.
[15]	L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, vol 19, AMS 1998.
[16]	E. Fernández-Cara and S. Guerrero, Remarks on the null controllability of the Burgers equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 229-232. doi: 10.1016/j.crma.2005.06.005.
[17]	A. Fursikov and O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow, Flow control, (Minneapolis, MN, 1992), IMA Vol. Math. Appl., 68, Springer, New York, (1995), 149-184. doi: 10.1007/978-1-4612-2526-3_7.
[18]	S. S. Ghoshal, Finer Analysis of Characteristic Curves, and Its Applications to Shock Profile, Exact and Optimal Controllability of Conservation Law with Strict convex Fluxes, Ph.D thesis, TIFR CAM, 2012.
[19]	O. Glass and S. Guerrero, On the uniform controllability of the Burgers equation, SIAM J. Control optim., 46 (2007), 1211-1238. doi: 10.1137/060664677.
[20]	E. Godleweski and P. A. Raviant, Hyperbolic Systems of Conservation Laws, Mathematiques and Applications, 3/4, Ellipses, Paris, 1991.
[21]	S. Guerrero and O. Y. Immunauvilov, Remarks on global controllability for the Burgers equation with two control forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 897-906. doi: 10.1016/j.anihpc.2006.06.010.
[22]	E. Hopf, The partial differential equation $u_t + u u_x = \mu u_{x x}$, Comm. Pure Appl. Math, 3 (1950), 201-230.
[23]	T. Horsin, On the controllability of the Burger equation, ESIAM, Control optimization and Calculus of variations, 3 (1998), 83-95. doi: 10.1051/cocv:1998103.
[24]	K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for the solution of Convex conservation laws with boundary condition, Duke Math.J., 62 (1991), 401-416. doi: 10.1215/S0012-7094-91-06216-2.
[25]	S. N. Kružkov, First order quasilinear equations with several independent variables, (Russian), Mat. Sb., 81 (1970), 228-255. English transl. in Math. USSR Sb., 10 (1970), 217-243.
[26]	P. D. Lax, Hyperbolic systems of conservation Laws II, comm Pure Appl. Math, 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.

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