# American Institute of Mathematical Sciences

September  2013, 8(3): 707-726. doi: 10.3934/nhm.2013.8.707

## Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters

 1 ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

Received  April 2012 Revised  July 2013 Published  October 2013

A few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems are presented. In the considered problem the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semi-explicit form for an homogeneous lagrangian do not hold, a capture basin algorithm is proposed to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications to traffic regulation problems.
Citation: Anya Désilles. Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters. Networks and Heterogeneous Media, 2013, 8 (3) : 707-726. doi: 10.3934/nhm.2013.8.707
##### References:
 [1] J.-P. Aubin, "Viability Theory," Systems & Control: Foundations & Applications, Birkhäuser, Boston, Inc., Boston, MA, 1991. [2] J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Dirichlet problems for some hamilton-jacobi equations with inequality constraints, SIAM Journal on Control and Optimization, 47 (2008), 2348-2380. doi: 10.1137/060659569. [3] J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory: New Directions," New directions. Second edition. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6. [4] J.-P. Aubin and A. Cellina, "Differential Inclusions," Set-valued maps and viability theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [5] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745. [6] A. M. Bayen and C. G. Claudel, Solutions to switched Hamilton-Jacobi equations and conservation laws using hybrid components, Hybrid systems: Computation and control, 101–115, Lecture Notes in Computer Science, 4981, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78929-1_8. [7] C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits, In "Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference Proceedings," (2001), 5959-5964. [8] R. C. Carlson, I. Papamichail, M. Papageorgiou and A. Messmer, Optimal motorway traffic flow control involving variable speed limits and ramp metering, Transportation Science, 44 (2010), 238-253. doi: 10.1287/trsc.1090.0314. [9] C. G. Claudel and A. M. Bayen, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM Journal on Control and Optimization, 49 (2011), 383-402. doi: 10.1137/090778754. [10] C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. [11] C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods, IEEE Transactions on Automatic Control, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439. [12] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transporation Research B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003. [13] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [14] L. C. Edie, Car following and steady state theory for non-congested traffic, Operations Research, 9 (1961), 66-76. doi: 10.1287/opre.9.1.66. [15] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 31 (1993), 257-272. doi: 10.1137/0331016. [16] H. Greenberg, An analysis of traffic flow, Operations Research, 7 (1959), 79-85. doi: 10.1287/opre.7.1.79. [17] B. D. Greenshields, A study of traffic capacity, HRB Proc., 14 (1934), 448-481. [18] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Royal Society, Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [19] G. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229. doi: 10.1287/opre.9.2.209. [20] M. Papageorgiou, E. Kosmatopoulos and I. Papamichail, Effects of variable speed limits on motorway traffic flow, Transportation Research Record: Journal of the Transportation Research Board, 2047 (2008), 37-48. doi: 10.3141/2047-05. [21] P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [22] P. Saint-Pierre, Approximation of the viability kernel, Applied Mathematics and Optimisation, 29 (1994), 187-209. doi: 10.1007/BF01204182. [23] R. T. Underwood, Speed, volume and density relationships, quality and theory of traffic flow, in "Yale Bureau of Highway Traffic," 1961, 141-88.

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##### References:
 [1] J.-P. Aubin, "Viability Theory," Systems & Control: Foundations & Applications, Birkhäuser, Boston, Inc., Boston, MA, 1991. [2] J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, Dirichlet problems for some hamilton-jacobi equations with inequality constraints, SIAM Journal on Control and Optimization, 47 (2008), 2348-2380. doi: 10.1137/060659569. [3] J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory: New Directions," New directions. Second edition. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6. [4] J.-P. Aubin and A. Cellina, "Differential Inclusions," Set-valued maps and viability theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [5] E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15 (1990), 1713-1742. doi: 10.1080/03605309908820745. [6] A. M. Bayen and C. G. Claudel, Solutions to switched Hamilton-Jacobi equations and conservation laws using hybrid components, Hybrid systems: Computation and control, 101–115, Lecture Notes in Computer Science, 4981, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78929-1_8. [7] C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits, In "Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference Proceedings," (2001), 5959-5964. [8] R. C. Carlson, I. Papamichail, M. Papageorgiou and A. Messmer, Optimal motorway traffic flow control involving variable speed limits and ramp metering, Transportation Science, 44 (2010), 238-253. doi: 10.1287/trsc.1090.0314. [9] C. G. Claudel and A. M. Bayen, Convex formulations of data assimilation problems for a class of Hamilton-Jacobi equations, SIAM Journal on Control and Optimization, 49 (2011), 383-402. doi: 10.1137/090778754. [10] C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory, IEEE Transactions on Automatic Control, 55 (2010), 1142-1157. doi: 10.1109/TAC.2010.2041976. [11] C. G. Claudel and A. M. Bayen, Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods, IEEE Transactions on Automatic Control, 55 (2010), 1158-1174. doi: 10.1109/TAC.2010.2045439. [12] C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions, Transporation Research B, 39 (2005), 187-196. doi: 10.1016/j.trb.2004.04.003. [13] C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619. doi: 10.3934/nhm.2006.1.601. [14] L. C. Edie, Car following and steady state theory for non-congested traffic, Operations Research, 9 (1961), 66-76. doi: 10.1287/opre.9.1.66. [15] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 31 (1993), 257-272. doi: 10.1137/0331016. [16] H. Greenberg, An analysis of traffic flow, Operations Research, 7 (1959), 79-85. doi: 10.1287/opre.7.1.79. [17] B. D. Greenshields, A study of traffic capacity, HRB Proc., 14 (1934), 448-481. [18] M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads, Proc. Royal Society, Ser. A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [19] G. Newell, Nonlinear effects in the dynamics of car following, Operations Research, 9 (1961), 209-229. doi: 10.1287/opre.9.2.209. [20] M. Papageorgiou, E. Kosmatopoulos and I. Papamichail, Effects of variable speed limits on motorway traffic flow, Transportation Research Record: Journal of the Transportation Research Board, 2047 (2008), 37-48. doi: 10.3141/2047-05. [21] P. I. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42-51. doi: 10.1287/opre.4.1.42. [22] P. Saint-Pierre, Approximation of the viability kernel, Applied Mathematics and Optimisation, 29 (1994), 187-209. doi: 10.1007/BF01204182. [23] R. T. Underwood, Speed, volume and density relationships, quality and theory of traffic flow, in "Yale Bureau of Highway Traffic," 1961, 141-88.
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