# On ergodic problem for Hamilton-Jacobi-Isaacs equations

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

- Volume: 11, Issue: 4, page 522-541
- ISSN: 1292-8119

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topBettiol, Piernicola. "On ergodic problem for Hamilton-Jacobi-Isaacs equations." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 522-541. <http://eudml.org/doc/90776>.

@article{Bettiol2010,

abstract = {
We study the asymptotic behavior of $\lambda v_\lambda$ as
$\lambda\rightarrow 0^+$, where $v_\lambda$
is the viscosity solution of the following Hamilton-Jacobi-Isaacs
equation (infinite horizon case)
\[
\lambda v\_\lambda + H(x,Dv\_\lambda)=0,
\]
with
\[
H(x,p):=\min\_\{b\in B\}\max\_\{a \in A\} \\{-f(x,a,b)\cdot p -l(x,a,b)\\}.
\]
We discuss the cases in which the state of the system is required to stay in an
n-dimensional torus, called periodic boundary conditions,
or in the closure
of a bounded connected domain $\Omega\subset\{\xR\}^n$ with sufficiently smooth boundary.
As far as the latter is concerned, we treat
both
the case of the Neumann boundary conditions
(reflection on the boundary) and
the case of state constraints boundary conditions.
Under the uniform approximate controllability
assumption of one player, we extend
the uniform convergence result of the value function to a constant as
$\lambda\rightarrow 0^+$ to differential games.
As far as state constraints boundary conditions are concerned,
we give an example where the value function is Hölder continuous.
},

author = {Bettiol, Piernicola},

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

keywords = {Hamilton-Jacobi-Isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity.; ergodicity; Abelian-Tauberian theorem; Skorokhod problem},

language = {eng},

month = {3},

number = {4},

pages = {522-541},

publisher = {EDP Sciences},

title = {On ergodic problem for Hamilton-Jacobi-Isaacs equations},

url = {http://eudml.org/doc/90776},

volume = {11},

year = {2010},

}

TY - JOUR

AU - Bettiol, Piernicola

TI - On ergodic problem for Hamilton-Jacobi-Isaacs equations

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 11

IS - 4

SP - 522

EP - 541

AB -
We study the asymptotic behavior of $\lambda v_\lambda$ as
$\lambda\rightarrow 0^+$, where $v_\lambda$
is the viscosity solution of the following Hamilton-Jacobi-Isaacs
equation (infinite horizon case)
\[
\lambda v_\lambda + H(x,Dv_\lambda)=0,
\]
with
\[
H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}.
\]
We discuss the cases in which the state of the system is required to stay in an
n-dimensional torus, called periodic boundary conditions,
or in the closure
of a bounded connected domain $\Omega\subset{\xR}^n$ with sufficiently smooth boundary.
As far as the latter is concerned, we treat
both
the case of the Neumann boundary conditions
(reflection on the boundary) and
the case of state constraints boundary conditions.
Under the uniform approximate controllability
assumption of one player, we extend
the uniform convergence result of the value function to a constant as
$\lambda\rightarrow 0^+$ to differential games.
As far as state constraints boundary conditions are concerned,
we give an example where the value function is Hölder continuous.

LA - eng

KW - Hamilton-Jacobi-Isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity.; ergodicity; Abelian-Tauberian theorem; Skorokhod problem

UR - http://eudml.org/doc/90776

ER -

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