Hamilton-Jacobi equations for control problems of parabolic equations

Sophie Gombao; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 311-349
  • ISSN: 1292-8119

Abstract

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We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ( - A ) β – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power ( - A ) β appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space Y ↪ X . To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.

How to cite

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Gombao, Sophie, and Raymond, Jean-Pierre. "Hamilton-Jacobi equations for control problems of parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 311-349. <http://eudml.org/doc/249677>.

@article{Gombao2006,
abstract = { We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power $(-A)^\beta$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power $(-A)^\beta$ appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space $Y\hookrightarrow X$. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable. },
author = {Gombao, Sophie, Raymond, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations.; semilinear parabolic equations},
language = {eng},
month = {3},
number = {2},
pages = {311-349},
publisher = {EDP Sciences},
title = {Hamilton-Jacobi equations for control problems of parabolic equations},
url = {http://eudml.org/doc/249677},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Gombao, Sophie
AU - Raymond, Jean-Pierre
TI - Hamilton-Jacobi equations for control problems of parabolic equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 311
EP - 349
AB - We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power $(-A)^\beta$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power $(-A)^\beta$ appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space $Y\hookrightarrow X$. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.
LA - eng
KW - Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations.; semilinear parabolic equations
UR - http://eudml.org/doc/249677
ER -

References

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