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

Abstract

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We investigate the value function of the Bolza problem of the Calculus of Variations
 V ( t , x ) = inf 0 t L ( y ( s ) , y ' ( s ) ) d s + ϕ ( y ( t ) ) : y W 1 , 1 ( 0 , t ; n ) , y ( 0 ) = x , with a lower semicontinuous Lagrangian L and a final cost ϕ , 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.

How to cite

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Maso, 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

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