# American Institute of Mathematical Sciences

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

## Exact controllability of scalar conservation laws with strict convex flux

 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

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.
Citation: Adimurthi , Shyam Sundar Ghoshal, G. D. Veerappa Gowda. Exact controllability of scalar conservation laws with strict convex flux. Mathematical Control & 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. Google Scholar [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.  Google Scholar [3] Adimurthi and G. D. Veerappa Gowda, Conservation Law with discontinuous flux, J.Math, Kyoto Univ, 43 (2003), 27-70.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim, 48 (2009), 1567-1599. doi: 10.1137/070685749.  Google Scholar [11] J. M. Coron, Global asymptotic stabilization for controllable systems without drift, Mth. Control signals systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.  Google Scholar [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.  Google Scholar [13] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $2^{nd}$ edition, Springer Verlag, Berlin, 2000.  Google Scholar [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.  Google Scholar [15] L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, vol 19, AMS 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [20] E. Godleweski and P. A. Raviant, Hyperbolic Systems of Conservation Laws, Mathematiques and Applications, 3/4, Ellipses, Paris, 1991.  Google Scholar [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.  Google Scholar [22] E. Hopf, The partial differential equation $u_t + u u_x = \mu u_{x x}$, Comm. Pure Appl. Math, 3 (1950), 201-230.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] P. D. Lax, Hyperbolic systems of conservation Laws II, comm Pure Appl. Math, 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

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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. Google Scholar [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.  Google Scholar [3] Adimurthi and G. D. Veerappa Gowda, Conservation Law with discontinuous flux, J.Math, Kyoto Univ, 43 (2003), 27-70.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Chapouly, Global controllability of nonviscous and viscous Burgers-type equations, SIAM J. Control Optim, 48 (2009), 1567-1599. doi: 10.1137/070685749.  Google Scholar [11] J. M. Coron, Global asymptotic stabilization for controllable systems without drift, Mth. Control signals systems, 5 (1992), 295-312. doi: 10.1007/BF01211563.  Google Scholar [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.  Google Scholar [13] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, $2^{nd}$ edition, Springer Verlag, Berlin, 2000.  Google Scholar [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.  Google Scholar [15] L. C. Evans, Partial Differential Equations, Graduate studies in Mathematics, vol 19, AMS 1998.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [20] E. Godleweski and P. A. Raviant, Hyperbolic Systems of Conservation Laws, Mathematiques and Applications, 3/4, Ellipses, Paris, 1991.  Google Scholar [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.  Google Scholar [22] E. Hopf, The partial differential equation $u_t + u u_x = \mu u_{x x}$, Comm. Pure Appl. Math, 3 (1950), 201-230.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] P. D. Lax, Hyperbolic systems of conservation Laws II, comm Pure Appl. Math, 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar
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