# 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

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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 and 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., 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.

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##### 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.
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