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Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

  • Gianni Dal Maso (a1) and Hélène Frankowska (a2)


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.



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[1] 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.
[2] 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.
[3] 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.
[4] Aubin, J.-P., A survey of viability theory. SIAM J. Control Optim. 28 (1990) 749-788.
[5] J.-P. Aubin, Viability Theory. Birkhäuser, Boston (1991).
[6] J.-P. Aubin, Optima and Equilibria. Springer-Verlag, Berlin, Grad. Texts in Math. 140 (1993).
[7] J.-P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 264 (1984).
[8] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis. Wiley & Sons, New York (1984).
[9] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
[10] Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonian. Comm. Partial Differential Equations 15 (1990) 1713-1742.
[11] Bebernes, J.W. and Schuur, J.D., The Wazewski topological method for contingent equations. Ann. Mat. Pura Appl. 87 (1970) 271-280.
[12] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problems in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. (1989).
[13] L. Cesari, Optimization Theory and Applications. Problems with Ordinary Differential Equations. Springer-Verlag, Berlin, Appl. Math. 17 (1983).
[14] B. Cornet, Regular properties of tangent and normal cones. Cahiers de Maths. de la Décision No. 8130 (1981).
[15] Crandall, M.G., Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42.
[16] Dal Maso, G. and Modica, L., Integral functionals determined by their minima. Rend. Sem. Mat. Univ. Padova 76 (1986) 255-267.
[17] C. Dellacherie, P.-A. Meyer, Probabilités et potentiel. Hermann, Paris (1975).
[18] Frankowska, H., L'équation d'Hamilton-Jacobi contingente. C. R. Acad. Sci. Paris Sér. I Math. 304 (1987) 295-298.
[19] Frankowska, H., Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations. Appl. Math. Optim. 19 (1989) 291-311.
[20] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, in Proc. of IEEE CDC Conference. Brighton, England (1991).
[21] Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272.
[22] Frankowska, H., Plaskacz, S. and Rzezuchowski, T., Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differential Equations 116 (1995) 265-305.
[23] G.N. Galbraith, Extended Hamilton-Jacobi characterization of value functions in optimal control. Preprint Washington University, Seattle (1998).
[24] 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.
[25] Ioffe, A.D., On lower semicontinuity of integral functionals. SIAM J. Control Optim. 15 (1977) 521-521 and 991-1000.
[26] Olech, C., Weak lower semicontinuity of integral functionals. J. Optim. Theory Appl. 19 (1976) 3-16.
[27] Rockafellar, T., Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization. Math. Oper. Res. 6 (1981) 424-436.
[28] T. Rockafellar and R. Wets, Variational Analysis. Springer-Verlag, Berlin, Grundlehren Math. Wiss. 317 (1998).
[29] Subbotin, A.I., A generalization of the basic equation of the theory of the differential games. Soviet. Math. Dokl. 22 (1980) 358-362.


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Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

  • Gianni Dal Maso (a1) and Hélène Frankowska (a2)


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