Graph selectors and viscosity solutions on lagrangian manifolds

McCaffrey, David

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 795-815.

Let $\Lambda$ be a lagrangian submanifold of $T^{*}X$ for some closed manifold $X.$ Let $S(x,\xi )$ be a generating function for $\Lambda$ which is quadratic at infinity, and let $W\left(x\right)$ be the corresponding graph selector for $\Lambda ,$ in the sense of Chaperon-Sikorav-Viterbo, so that there exists a subset $X_0\subset X$ of measure zero such that $W$ is Lipschitz continuous on $X,$ smooth on $X\setminus X_0$ and $(x,\partial W/\partial x(x\left)\right)\in \Lambda$ for $X\setminus X_0.$ Let $H(x,p)=0$ for $(x,p)\in \Lambda$. Then $W$ is a classical solution to $H(x,\partial W/\partial x(x\left)\right)=0$ on $X\setminus X_0$ and extends to a Lipschitz function on the whole of $X.$ Viterbo refers to $W$ as a variational solution. We prove that $W$ is also a viscosity solution under some simple and natural conditions. We also prove that these conditions are satisfied in many cases, including certain commonly occuring cases where $H(x,p)$ is not convex in $p$.

Classification : 49L25, 53D12
Mots clés : viscosity solution, lagrangian manifold, graph selector

[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. | Zbl

[2] F. Cardin, On viscosity solutions and geometrical solutions of Hamilton-Jacobi equations. Nonlinear Anal. Th. Meth. Appl. 20 (1993) 713-719. | Zbl

[3] M. Chaperon, Lois de conservation et geometrie symplectique. C.R. Acad. Sci. Paris Ser. I Math., 312 (1991) 345-348. | Zbl

[4] F.H. Clarke, Optimization and Nonsmooth Analysis. J. Wiley, New York (1983). | MR | Zbl

[5] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS 277 (1983) 1-42. | Zbl

[6] M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. AMS 282 (1984) 487-502. | Zbl

[7] M.V. Day, On Lagrange manifolds and viscosity solutions. J. Math. Syst. Estim. Contr. 8 (1998) http://www.math.vt.edu/people/day/research/LMVS.pdf | MR | Zbl

[8] S.Yu. Dobrokhotov, V.N. Kolokoltsov and V.P. Maslov, Quantization of the Bellman equation, exponential asymptotics and tunneling, in Advances in Soviet Mathematics, V.P. Maslov and S.N. Samborskii, Eds., American Mathematical Society, Providence, Rhode Island 13 (1992) 1-46 . | Zbl

[9] W.H Fleming and H.M. Soner, Controlled markov processes and viscosity solutions. Springer-Verlag, New York (1993). | MR | Zbl

[10] H. Frankowska, Hamilton-Jacobi equations: viscosity solutions and generalised gradients. J. Math. Anal. Appl. 141 (1989) 21-26. | Zbl

[11] E. Hopf, Generalized solutions of non-linear equations of first order. J. Math. Mech. 14 (1965) 951-973. | Zbl

[12] T. Joukovskaia, Thèse de Doctorat, Université de Paris VII, Denis Diderot (1993).

[13] F. Laudenbach and J.C. Sikorav, Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibre cotangent. Invent. Math. 82 (1985) 349-357. | Zbl

[14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences Series 74, Springer-Verlag, Berlin (1989). | MR | Zbl

[15] D. Mccaffrey and S.P. Banks, Lagrangian Manifolds, Viscosity Solutions and Maslov Index. J. Convex Anal. 9 (2002) 185-224. | Zbl

[16] D. Mccaffrey, Viscosity Solutions on Lagrangian Manifolds and Connections with Tunnelling Operators, in Idempotent Mathematics and Mathematical Physics, V.P. Maslov and G.L. Litvinov Eds., Contemp. Math. 377, American Mathematical Society, Providence, Rhode Island (2005). | MR | Zbl

[17] D. Mccaffrey, Geometric existence theory for the control-affine $H_{\infty }$ problem, to appear in J. Math. Anal & Appl. (August 2005). | MR | Zbl

[18] 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. | Zbl

[19] 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. | Zbl

[20] P. Soravia, $H_{\infty }$ control of nonlinear systems: differential games and viscosity solutions. SIAM J. Contr. Opt. 34 (1996) 1071-1097. | Zbl

[21] A.J. Van Der Schaft, On a state space approach to nonlinear $H_{\infty }$ control. Syst. Contr. Lett. 16 (1991) 1-8. | Zbl

[22] A.J. Van Der Schaft, $L_2$ gain analysis of nonlinear systems and nonlinear state feedback $H_{\infty }$ control. IEEE Trans. Automatic Control AC-37 (1992) 770-784. | Zbl

[23] C. Viterbo, Symplectic topology as the geometry of generating functions. Math. Ann. 292 (1992) 685-710. | Zbl

[24] C. Viterbo, Solutions d'equations d'Hamilton-Jacobi et geometrie symplectique, Addendum to: Séminaire sur les équations aux Dérivés Partielles 1994-1995, École Polytech., Palaiseau (1996). | Numdam | Zbl

[25] A. Ottolengi and C. Viterbo, Solutions généralisées pour l'équation de Hamilton-Jacobi dans le cas d'évolution, unpublished.

[26] A. Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics 29, Conference Board of the Mathematical Sciences, AMS, Providence, Rhode Island (1977). | MR | Zbl