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 & 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, (1991).   Google Scholar

[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.  doi: 10.1137/060659569.  Google Scholar

[3]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory: New Directions,", New directions. Second edition. Springer, (2011).  doi: 10.1007/978-3-642-16684-6.  Google Scholar

[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 (1984).  doi: 10.1007/978-3-642-69512-4.  Google Scholar

[5]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Comm. Partial Differential Equations, 15 (1990), 1713.  doi: 10.1080/03605309908820745.  Google Scholar

[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, 4981 (2008).  doi: 10.1007/978-3-540-78929-1_8.  Google Scholar

[7]

C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits,, In, (2001), 5959.   Google Scholar

[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.  doi: 10.1287/trsc.1090.0314.  Google Scholar

[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.  doi: 10.1137/090778754.  Google Scholar

[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.  doi: 10.1109/TAC.2010.2041976.  Google Scholar

[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.  doi: 10.1109/TAC.2010.2045439.  Google Scholar

[12]

C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions,, Transporation Research B, 39 (2005), 187.  doi: 10.1016/j.trb.2004.04.003.  Google Scholar

[13]

C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications,, Networks and Heterogeneous Media, 1 (2006), 601.  doi: 10.3934/nhm.2006.1.601.  Google Scholar

[14]

L. C. Edie, Car following and steady state theory for non-congested traffic,, Operations Research, 9 (1961), 66.  doi: 10.1287/opre.9.1.66.  Google Scholar

[15]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 31 (1993), 257.  doi: 10.1137/0331016.  Google Scholar

[16]

H. Greenberg, An analysis of traffic flow,, Operations Research, 7 (1959), 79.  doi: 10.1287/opre.7.1.79.  Google Scholar

[17]

B. D. Greenshields, A study of traffic capacity,, HRB Proc., 14 (1934), 448.   Google Scholar

[18]

M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Royal Society, 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[19]

G. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209.  doi: 10.1287/opre.9.2.209.  Google Scholar

[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.  doi: 10.3141/2047-05.  Google Scholar

[21]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[22]

P. Saint-Pierre, Approximation of the viability kernel,, Applied Mathematics and Optimisation, 29 (1994), 187.  doi: 10.1007/BF01204182.  Google Scholar

[23]

R. T. Underwood, Speed, volume and density relationships, quality and theory of traffic flow,, in, (1961), 141.   Google Scholar

show all references

References:
[1]

J.-P. Aubin, "Viability Theory,", Systems & Control: Foundations & Applications, (1991).   Google Scholar

[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.  doi: 10.1137/060659569.  Google Scholar

[3]

J.-P. Aubin, A. M. Bayen and P. Saint-Pierre, "Viability Theory: New Directions,", New directions. Second edition. Springer, (2011).  doi: 10.1007/978-3-642-16684-6.  Google Scholar

[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 (1984).  doi: 10.1007/978-3-642-69512-4.  Google Scholar

[5]

E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians,, Comm. Partial Differential Equations, 15 (1990), 1713.  doi: 10.1080/03605309908820745.  Google Scholar

[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, 4981 (2008).  doi: 10.1007/978-3-540-78929-1_8.  Google Scholar

[7]

C. Canudas de Wit, Best-effort highway traffic congestion control via variable speed limits,, In, (2001), 5959.   Google Scholar

[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.  doi: 10.1287/trsc.1090.0314.  Google Scholar

[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.  doi: 10.1137/090778754.  Google Scholar

[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.  doi: 10.1109/TAC.2010.2041976.  Google Scholar

[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.  doi: 10.1109/TAC.2010.2045439.  Google Scholar

[12]

C. Daganzo, A variational formulation of kinematic waves: Basic theory and complex boundary conditions,, Transporation Research B, 39 (2005), 187.  doi: 10.1016/j.trb.2004.04.003.  Google Scholar

[13]

C. Daganzo, On the variational theory of traffic flow: Well-posedness, duality and applications,, Networks and Heterogeneous Media, 1 (2006), 601.  doi: 10.3934/nhm.2006.1.601.  Google Scholar

[14]

L. C. Edie, Car following and steady state theory for non-congested traffic,, Operations Research, 9 (1961), 66.  doi: 10.1287/opre.9.1.66.  Google Scholar

[15]

H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 31 (1993), 257.  doi: 10.1137/0331016.  Google Scholar

[16]

H. Greenberg, An analysis of traffic flow,, Operations Research, 7 (1959), 79.  doi: 10.1287/opre.7.1.79.  Google Scholar

[17]

B. D. Greenshields, A study of traffic capacity,, HRB Proc., 14 (1934), 448.   Google Scholar

[18]

M. J. Lighthill and G. B. Whitham, On kinematic waves: II. A theory of traffic flow on long crowded roads,, Proc. Royal Society, 229 (1955), 317.  doi: 10.1098/rspa.1955.0089.  Google Scholar

[19]

G. Newell, Nonlinear effects in the dynamics of car following,, Operations Research, 9 (1961), 209.  doi: 10.1287/opre.9.2.209.  Google Scholar

[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.  doi: 10.3141/2047-05.  Google Scholar

[21]

P. I. Richards, Shock waves on the highway,, Operations Research, 4 (1956), 42.  doi: 10.1287/opre.4.1.42.  Google Scholar

[22]

P. Saint-Pierre, Approximation of the viability kernel,, Applied Mathematics and Optimisation, 29 (1994), 187.  doi: 10.1007/BF01204182.  Google Scholar

[23]

R. T. Underwood, Speed, volume and density relationships, quality and theory of traffic flow,, in, (1961), 141.   Google Scholar

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