# American Institute of Mathematical Sciences

May  2012, 11(3): 1205-1215. doi: 10.3934/cpaa.2012.11.1205

## 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

Received  December 2010 Revised  April 2011 Published  December 2011

For Cauchy problems given by Hamilton-Jacobi evolutive type equations, we consider the variational solution proposed by Chaperon, Sikorav and Viterbo. This is a weak, Lipschitz solution constructed via a minimax procedure from the generating function quadratic at infinity of the Lagrangian manifold associated with the Cauchy problem. We state and prove a representational formula for the variational solution. This formula requires a condition on the nature of the minimax critical value of the generating function, but makes no assumption about the convexity or concavity of the Hamiltonian. We show that it generalises the well-known formula which applies when the Hamiltonian is convex or concave in the momentum variable. We then prove that the required conditions of the formula are satisfied by the non-convex Hamiltonian arising from the control-affine $H_{\infty }$ problem. Given results in the literature that the variational solution to this problem is equivalent to the lower value of the associated differential game, we therefore obtain a representational formula for this lower value.
Citation: David McCaffrey. A representational formula for variational solutions to Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1205-1215. doi: 10.3934/cpaa.2012.11.1205
##### References:
 [1] M. Bardi and L. C. Evans, On Hopf's formula for solutions of Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 1373. doi: 10.1016/0362-546X(84)90020-8. Google Scholar [2] O. Bernardi and F. Cardin, On $C^0$-variational solutions for Hamilton-Jacobi equations,, DCDS-A, 31 (2011), 385. doi: 10.3934/dcds.2011.31.385. Google Scholar [3] M. Brunella, On a theorem of Sikorav,, Ens. Math., 37 (1991), 83. Google Scholar [4] F. Cardin, On viscosity solutions and geometrical solutions of Hamilton Jacobi equations,, Nonlinear Anal., 20 (1993), 713. doi: 10.1016/0362-546X(93)90029-R. Google Scholar [5] M. Chaperon, Lois de conservation et geometrie symplectique,, C. R. Acad. Sci. Paris, 312 (1991), 345. Google Scholar [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. doi: 10.1109/9.29425. Google Scholar [7] E. Hopf, Generalized solutions of non-linear equations of first order,, J. Math. & Mech., 14 (1965), 951. Google Scholar [8] T. Joukovskaia, "Singularités de Minimax et Solutions Faibles d'Équations aux Dérivées Partielles,", Th\ese de Doctorat, (1993). Google Scholar [9] D. McCaffrey, Geometric existence theory for the control-affine $H_{\infty}$ problem,, J. Math. Anal. & Applic., 324 (2006), 682. doi: 10.1016/j.jmaa.2005.12.034. Google Scholar [10] D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM: Control, 12 (2006), 795. doi: 10.1051/cocv:2006023. Google Scholar [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. doi: 10.3929/ethz-a-004520619. Google Scholar [12] K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics,", Lecture Notes in Mathematics 1844, (1844). Google Scholar [13] J. C. Sikorav, Sur les immersions lagrangiennes dans un fibre cotangent admettant une phase generatrice globale,, C. R. Acad. Sci. Paris, 302 (1986), 119. Google Scholar [14] P. Soravia, $H_{\infty}$ control of nonlinear systems: differential games and viscosity solutions,, SIAM J. Control and Opt., 34 (1996), 1071. doi: 10.1137/S0363012994266413. Google Scholar [15] A. J. van der Schaft, On a state space approach to nonlinear $H_{\infty}$ control,, Syst. & Control Letters, 16 (1991), 1. doi: 10.1016/0167-6911(91)90022-7. Google Scholar [16] C. Viterbo, Symplectic topology as the geometry of generating functions,, Math. Ann., 292 (1992), 685. doi: 10.1007/BF01444643. Google Scholar [17] A. Ottolengi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, preprint., (). Google Scholar [18] C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, in, 217 (1992), 439. Google Scholar

show all references

##### References:
 [1] M. Bardi and L. C. Evans, On Hopf's formula for solutions of Hamilton-Jacobi equations,, Nonlinear Anal., 8 (1984), 1373. doi: 10.1016/0362-546X(84)90020-8. Google Scholar [2] O. Bernardi and F. Cardin, On $C^0$-variational solutions for Hamilton-Jacobi equations,, DCDS-A, 31 (2011), 385. doi: 10.3934/dcds.2011.31.385. Google Scholar [3] M. Brunella, On a theorem of Sikorav,, Ens. Math., 37 (1991), 83. Google Scholar [4] F. Cardin, On viscosity solutions and geometrical solutions of Hamilton Jacobi equations,, Nonlinear Anal., 20 (1993), 713. doi: 10.1016/0362-546X(93)90029-R. Google Scholar [5] M. Chaperon, Lois de conservation et geometrie symplectique,, C. R. Acad. Sci. Paris, 312 (1991), 345. Google Scholar [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. doi: 10.1109/9.29425. Google Scholar [7] E. Hopf, Generalized solutions of non-linear equations of first order,, J. Math. & Mech., 14 (1965), 951. Google Scholar [8] T. Joukovskaia, "Singularités de Minimax et Solutions Faibles d'Équations aux Dérivées Partielles,", Th\ese de Doctorat, (1993). Google Scholar [9] D. McCaffrey, Geometric existence theory for the control-affine $H_{\infty}$ problem,, J. Math. Anal. & Applic., 324 (2006), 682. doi: 10.1016/j.jmaa.2005.12.034. Google Scholar [10] D. McCaffrey, Graph selectors and viscosity solutions on Lagrangian manifolds,, ESAIM: Control, 12 (2006), 795. doi: 10.1051/cocv:2006023. Google Scholar [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. doi: 10.3929/ethz-a-004520619. Google Scholar [12] K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics,", Lecture Notes in Mathematics 1844, (1844). Google Scholar [13] J. C. Sikorav, Sur les immersions lagrangiennes dans un fibre cotangent admettant une phase generatrice globale,, C. R. Acad. Sci. Paris, 302 (1986), 119. Google Scholar [14] P. Soravia, $H_{\infty}$ control of nonlinear systems: differential games and viscosity solutions,, SIAM J. Control and Opt., 34 (1996), 1071. doi: 10.1137/S0363012994266413. Google Scholar [15] A. J. van der Schaft, On a state space approach to nonlinear $H_{\infty}$ control,, Syst. & Control Letters, 16 (1991), 1. doi: 10.1016/0167-6911(91)90022-7. Google Scholar [16] C. Viterbo, Symplectic topology as the geometry of generating functions,, Math. Ann., 292 (1992), 685. doi: 10.1007/BF01444643. Google Scholar [17] A. Ottolengi and C. Viterbo, Solutions generalisees pour l'equation de Hamilton-Jacobi dans le cas d'evolution,, preprint., (). Google Scholar [18] C. Viterbo, Symplectic topology and Hamilton-Jacobi equations,, in, 217 (1992), 439. Google Scholar
 [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] M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365 [3] 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 [4] Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513 [5] 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 [6] 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 [7] 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 [8] Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385 [9] Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291 [10] Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493 [11] Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 [12] Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231 [13] Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623 [14] Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026 [15] 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 & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 [16] 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 & Optimization, 2015, 5 (1) : 11-23. doi: 10.3934/naco.2015.5.11 [17] Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363 [18] Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461 [19] Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121 [20] Kai Zhao, Wei Cheng. On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4345-4358. doi: 10.3934/dcds.2019176

2018 Impact Factor: 0.925