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

Gianni Dal Maso; Hélène Frankowska

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 5, page 369-393
- ISSN: 1292-8119

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topMaso, Gianni Dal, and Frankowska, Hélène. "Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 369-393. <http://eudml.org/doc/197304>.

@article{Maso2010,

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.
},

author = {Maso, Gianni Dal, Frankowska, Hélène},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Discontinuous Lagrangians;
Hamilton-Jacobi equations; viability theory; viscosity solutions.; discontinuous Lagrangians; Hamilton-Jacobi equations; viscosity solutions; Bolza problem},

language = {eng},

month = {3},

pages = {369-393},

publisher = {EDP Sciences},

title = {Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities},

url = {http://eudml.org/doc/197304},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Maso, Gianni Dal

AU - Frankowska, Hélène

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 369

EP - 393

AB -
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.

LA - eng

KW - Discontinuous Lagrangians;
Hamilton-Jacobi equations; viability theory; viscosity solutions.; discontinuous Lagrangians; Hamilton-Jacobi equations; viscosity solutions; Bolza problem

UR - http://eudml.org/doc/197304

ER -

## References

top- M. Amar, G. Bellettini and S. Venturini, Integral representation of functionals defined on curves of W1,p. Proc. Roy. Soc. Edinburgh Sect. A128 (1998) 193-217.
- L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl.142 (1989) 301-316.
- 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.
- J.-P. Aubin, A survey of viability theory. SIAM J. Control Optim.28 (1990) 749-788.
- 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).
- E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonian. Comm. Partial Differential Equations15 (1990) 1713-1742.
- J.W. Bebernes and J.D. Schuur, The Wazewski topological method for contingent equations. Ann. Mat. Pura Appl.87 (1970) 271-280.
- 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).
- M.G. Crandall, P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1-42.
- G. Dal Maso and L. Modica, Integral functionals determined by their minima. Rend. Sem. Mat. Univ. Padova76 (1986) 255-267.
- C. Dellacherie, P.-A. Meyer, Probabilités et potentiel. Hermann, Paris (1975).
- H. Frankowska, L'équation d'Hamilton-Jacobi contingente. C. R. Acad. Sci. Paris Sér. I Math.304 (1987) 295-298.
- H. Frankowska, Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations. Appl. Math. Optim.19 (1989) 291-311.
- H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, in Proc. of IEEE CDC Conference. Brighton, England (1991).
- H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim.31 (1993) 257-272.
- H. Frankowska, S. Plaskacz and T. Rzezuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differential Equations116 (1995) 265-305.
- G.N. Galbraith, Extended Hamilton-Jacobi characterization of value functions in optimal control. Preprint Washington University, Seattle (1998).
- H.G. Guseinov, A.I. Subbotin and. V.N. Ushakov, Derivatives for multivalued mappings with application to game-theoretical problems of control. Problems Control Inform.14 (1985) 155-168.
- A.D. Ioffe, On lower semicontinuity of integral functionals. SIAM J. Control Optim.15 (1977) 521-521 and 991-1000.
- C. Olech, Weak lower semicontinuity of integral functionals. J. Optim. Theory Appl.19 (1976) 3-16.
- T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization. Math. Oper. Res.6 (1981) 424-436.
- T. Rockafellar and R. Wets, Variational Analysis. Springer-Verlag, Berlin, Grundlehren Math. Wiss.317 (1998).
- A.I. Subbotin, A generalization of the basic equation of the theory of the differential games. Soviet. Math. Dokl.22 (1980) 358-362.

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