-
Previous Article
Limits of anisotropic and degenerate elliptic problems
- CPAA Home
- This Issue
-
Next Article
Analysis of a contact problem for electro-elastic-visco-plastic materials
A representational formula for variational solutions to Hamilton-Jacobi equations
1. | KSS Ltd., St. James's Buildings, 79 Oxford St., Manchester, M1 6SS, United Kingdom |
References:
[1] |
M. Bardi and L. C. Evans, On Hopf's formula for solutions of Hamilton-Jacobi equations, Nonlinear Anal., Th., Meth. & Appl., 8 (1984), 1373-1381.
doi: 10.1016/0362-546X(84)90020-8. |
[2] |
O. Bernardi and F. Cardin, On $C^0$-variational solutions for Hamilton-Jacobi equations, DCDS-A, 31 (2011), 385-406.
doi: 10.3934/dcds.2011.31.385. |
[3] |
M. Brunella, On a theorem of Sikorav, Ens. Math., 37 (1991), 83-87. |
[4] |
F. Cardin, On viscosity solutions and geometrical solutions of Hamilton Jacobi equations, Nonlinear Anal., Th., Meth. & Appl., 20 (1993), 713-719.
doi: 10.1016/0362-546X(93)90029-R. |
[5] |
M. Chaperon, Lois de conservation et geometrie symplectique, C. R. Acad. Sci. Paris, Ser. I Math., 312 (1991), 345-348. |
[6] |
J. C. Doyle, K. Glover, P. Pramod and B. A. Francis, State space solutions to standard $H_2$ and $H_{\infty}$ control problems, IEEE Trans. Automatic Control, AC-34 (1989), 831-847.
doi: 10.1109/9.29425. |
[7] |
E. Hopf, Generalized solutions of non-linear equations of first order, J. Math. & Mech., 14 (1965), 951-973. |
[8] |
T. Joukovskaia, "Singularités de Minimax et Solutions Faibles d'Équations aux Dérivées Partielles," Thèse de Doctorat, Université de Paris VII, Denis Diderot, 1993. |
[9] |
D. McCaffrey, Geometric existence theory for the control-affine $H_{\infty}$ problem, J. Math. Anal. & Applic., 324 (2006), 682-695.
doi: 10.1016/j.jmaa.2005.12.034. |
[10] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds, ESAIM: Control, Opt. & Calc. of Variations, 12 (2006), 795-815.
doi: 10.1051/cocv:2006023. |
[11] |
G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections and Aubry-Mather theory, Moscow Math. J., 3 (2003), 593-619.
doi: 10.3929/ethz-a-004520619. |
[12] |
K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics 1844, Springer-Verlag, Berlin, 2003. |
[13] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibre cotangent admettant une phase generatrice globale, C. R. Acad. Sci. Paris, Ser. I Math., 302 (1986), 119-122. |
[14] |
P. Soravia, $H_{\infty}$ control of nonlinear systems: differential games and viscosity solutions, SIAM J. Control and Opt., 34 (1996), 1071-1097.
doi: 10.1137/S0363012994266413. |
[15] |
A. J. van der Schaft, On a state space approach to nonlinear $H_{\infty}$ control, Syst. & Control Letters, 16 (1991), 1-8.
doi: 10.1016/0167-6911(91)90022-7. |
[16] |
C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710.
doi: 10.1007/BF01444643. |
[17] |
A. Ottolengi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution, preprint. |
[18] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, in "Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology," NATO Sci. Ser., 217, Springer, Dordrecht (1992), 439-459. |
show all references
References:
[1] |
M. Bardi and L. C. Evans, On Hopf's formula for solutions of Hamilton-Jacobi equations, Nonlinear Anal., Th., Meth. & Appl., 8 (1984), 1373-1381.
doi: 10.1016/0362-546X(84)90020-8. |
[2] |
O. Bernardi and F. Cardin, On $C^0$-variational solutions for Hamilton-Jacobi equations, DCDS-A, 31 (2011), 385-406.
doi: 10.3934/dcds.2011.31.385. |
[3] |
M. Brunella, On a theorem of Sikorav, Ens. Math., 37 (1991), 83-87. |
[4] |
F. Cardin, On viscosity solutions and geometrical solutions of Hamilton Jacobi equations, Nonlinear Anal., Th., Meth. & Appl., 20 (1993), 713-719.
doi: 10.1016/0362-546X(93)90029-R. |
[5] |
M. Chaperon, Lois de conservation et geometrie symplectique, C. R. Acad. Sci. Paris, Ser. I Math., 312 (1991), 345-348. |
[6] |
J. C. Doyle, K. Glover, P. Pramod and B. A. Francis, State space solutions to standard $H_2$ and $H_{\infty}$ control problems, IEEE Trans. Automatic Control, AC-34 (1989), 831-847.
doi: 10.1109/9.29425. |
[7] |
E. Hopf, Generalized solutions of non-linear equations of first order, J. Math. & Mech., 14 (1965), 951-973. |
[8] |
T. Joukovskaia, "Singularités de Minimax et Solutions Faibles d'Équations aux Dérivées Partielles," Thèse de Doctorat, Université de Paris VII, Denis Diderot, 1993. |
[9] |
D. McCaffrey, Geometric existence theory for the control-affine $H_{\infty}$ problem, J. Math. Anal. & Applic., 324 (2006), 682-695.
doi: 10.1016/j.jmaa.2005.12.034. |
[10] |
D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds, ESAIM: Control, Opt. & Calc. of Variations, 12 (2006), 795-815.
doi: 10.1051/cocv:2006023. |
[11] |
G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections and Aubry-Mather theory, Moscow Math. J., 3 (2003), 593-619.
doi: 10.3929/ethz-a-004520619. |
[12] |
K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics," Lecture Notes in Mathematics 1844, Springer-Verlag, Berlin, 2003. |
[13] |
J. C. Sikorav, Sur les immersions lagrangiennes dans un fibre cotangent admettant une phase generatrice globale, C. R. Acad. Sci. Paris, Ser. I Math., 302 (1986), 119-122. |
[14] |
P. Soravia, $H_{\infty}$ control of nonlinear systems: differential games and viscosity solutions, SIAM J. Control and Opt., 34 (1996), 1071-1097.
doi: 10.1137/S0363012994266413. |
[15] |
A. J. van der Schaft, On a state space approach to nonlinear $H_{\infty}$ control, Syst. & Control Letters, 16 (1991), 1-8.
doi: 10.1016/0167-6911(91)90022-7. |
[16] |
C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685-710.
doi: 10.1007/BF01444643. |
[17] |
A. Ottolengi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution, preprint. |
[18] |
C. Viterbo, Symplectic topology and Hamilton-Jacobi equations, in "Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology," NATO Sci. Ser., 217, Springer, Dordrecht (1992), 439-459. |
[1] |
Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295 |
[2] |
Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461 |
[3] |
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 |
[4] |
María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207 |
[5] |
Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441 |
[6] |
Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 |
[7] |
M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365 |
[8] |
Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $ H_{\infty} $ control design for active suspension systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 197-212. doi: 10.3934/dcdss.2021036 |
[9] |
Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385 |
[10] |
Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 |
[11] |
Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493 |
[12] |
Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations and Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 |
[13] |
Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 |
[14] |
Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 |
[15] |
Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623 |
[16] |
Zhen-Zhen Tao, Bing Sun. A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation. Electronic Research Archive, 2021, 29 (5) : 3429-3447. doi: 10.3934/era.2021046 |
[17] |
Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 |
[18] |
Li-Min Wang, Jing-Xian Yu, Jia Shi, Fu-Rong Gao. Delay-range dependent $H_\infty$ control for uncertain 2D-delayed systems. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 11-23. doi: 10.3934/naco.2015.5.11 |
[19] |
Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3273-3293. doi: 10.3934/dcds.2020405 |
[20] |
Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]