Chadi Nour (2005)
ESAIM: Control, Optimisation and Calculus of Variations
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We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [, 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.
Alessandra Cutrì, Francesca Da Lio (2007)
ESAIM: Control, Optimisation and Calculus of Variations
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In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form in where the Hamiltonian may be noncoercive in the gradient As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.
Nataša Bilić (2002)
Mathematica Slovaca
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Fabio Bagagiolo (2004)
ESAIM: Control, Optimisation and Calculus of Variations
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We study a finite horizon problem for a system whose evolution is governed by a controlled ordinary differential equation, which takes also account of a hysteretic component: namely, the output of a Preisach operator of hysteresis. We derive a discontinuous infinite dimensional Hamilton–Jacobi equation and prove that, under fairly general hypotheses, the value function is the unique bounded and uniformly continuous viscosity solution of the corresponding Cauchy problem.
Arkadiusz Misztela (2010)
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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In this paper we give examples of value functions in Bolza problem that are not bilateral or viscosity solutions and an example of a smooth value function that is even not a classic solution (in particular, it can be neither the viscosity nor the bilateral solution) of Hamilton-Jacobi-Bellman equation with upper semicontinuous Hamiltonian. Good properties of value functions motivate us to introduce approximate solutions of equations with such type Hamiltonians. We show that the value...
Cyril Imbert, Régis Monneau, Hasnaa Zidani (2013)
ESAIM: Control, Optimisation and Calculus of Variations
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This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to...
Piermarco Cannarsa, Giuseppe Da Prato (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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We prove an existence and uniqueness result for a class of Hamilton-Jacobi equations in Hilbert spaces.
Bianca Stroffolini (2007)
ESAIM: Control, Optimisation and Calculus of Variations
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We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.
Hong Qiu, Jiongmin Yong (2013)
ESAIM: Control, Optimisation and Calculus of Variations
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A two-person zero-sum differential game with unbounded controls is considered. Under proper coercivity conditions, the upper and lower value functions are characterized as the unique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacs equations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upper and lower value functions coincide, leading to the existence of the value function of the differential game. Due to the unboundedness of...