# 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 -

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