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Value functions for Bolza problems with discontinuous LagrangiansandHamilton-Jacobi inequalities

Published online by Cambridge University Press:  15 August 2002

Gianni Dal Maso  and
Hélène Frankowska
Gianni Dal Maso
Affiliation:
SISSA, via Beirut 2, 34014 Trieste, Italy.
Hélène Frankowska
Affiliation:
CNRS, ERS2064, Centre de Recherche Viabilité, Jeux, Contrôle, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; frankows@viab.dauphine.fr.
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Abstract

We investigate the value function of the Bolza problem of the Calculus of Variations
$$ V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\},$$ with a lower semicontinuous Lagrangian L and a final cost $ \varphi $, and show that it is locally Lipschitz for t>0 whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities.

Keywords

Discontinuous LagrangiansHamilton-Jacobi equationsviability theoryviscosity solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 369 - 393
Copyright
© EDP Sciences, SMAI, 2000

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References

Amar, M., Bellettini, G. and Venturini, S., Integral representation of functionals defined on curves of W 1,p . Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 193-217. CrossRef
Ambrosio, L., Ascenzi, O. and Buttazzo, G., Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl. 142 (1989) 301-316. CrossRef
J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. Advances in Mathematics, Supplementary Studies, edited by L. Nachbin (1981) 160-232.
Aubin, J.-P., A survey of viability theory. SIAM J. Control Optim. 28 (1990) 749-788. CrossRef
J.-P. Aubin, Viability Theory. Birkhäuser, Boston (1991).
J.-P. Aubin, Optima and Equilibria. Springer-Verlag, Berlin, Grad. Texts in Math. 140 (1993).
J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 264 (1984).
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis. Wiley & Sons, New York (1984).
J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonian. Comm. Partial Differential Equations 15 (1990) 1713-1742. CrossRef
Bebernes, J.W. and Schuur, J.D., The Wazewski topological method for contingent equations. Ann. Mat. Pura Appl. 87 (1970) 271-280. CrossRef
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. (1989).
L. Cesari, Optimization Theory and Applications. Problems with Ordinary Differential Equations. Springer-Verlag, Berlin, Appl. Math. 17 (1983).
B. Cornet, Regular properties of tangent and normal cones. Cahiers de Maths. de la Décision No. 8130 (1981).
Crandall, M.G., Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. CrossRef
Dal Maso, G. and Modica, L., Integral functionals determined by their minima. Rend. Sem. Mat. Univ. Padova 76 (1986) 255-267.
C. Dellacherie, P.-A. Meyer, Probabilités et potentiel. Hermann, Paris (1975).
Frankowska, H., L'équation d'Hamilton-Jacobi contingente. C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 295-298.
Frankowska, H., Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations. Appl. Math. Optim. 19 (1989) 291-311. CrossRef
H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, in Proc. of IEEE CDC Conference. Brighton, England (1991).
Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. CrossRef
Frankowska, H., Plaskacz, S. and Rzezuchowski, T., Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differential Equations 116 (1995) 265-305. CrossRef
G.N. Galbraith, Extended Hamilton-Jacobi characterization of value functions in optimal control. Preprint Washington University, Seattle (1998).
Guseinov, H.G., Subbotin, A.I. and. V.N. Ushakov, Derivatives for multivalued mappings with application to game-theoretical problems of control. Problems Control Inform. 14 (1985) 155-168.
Ioffe, A.D., On lower semicontinuity of integral functionals. SIAM J. Control Optim. 15 (1977) 521-521 and 991-1000. CrossRef
Olech, C., Weak lower semicontinuity of integral functionals. J. Optim. Theory Appl. 19 (1976) 3-16. CrossRef
Rockafellar, T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization. Math. Oper. Res. 6 (1981) 424-436. CrossRef
T. Rockafellar and R. Wets, Variational Analysis. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 317 (1998).
Subbotin, A.I., A generalization of the basic equation of the theory of the differential games. Soviet. Math. Dokl. 22 (1980) 358-362.