Piermarco Cannarsa, Giuseppe Da Prato (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.
Fabio Bagagiolo (2010)
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.
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.
Piermarco Cannarsa, Giuseppe Da Prato (1988)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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We prove an existence and uniqueness result for a class of Hamilton-Jacobi equations in Hilbert spaces.
Giuseppe Da Prato, Jerzy Zabczyk (1997)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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The Hamilton-Jacobi-Bellman equation corresponding to a large class of distributed control problems is reduced to a linear parabolic equation having a regular solution. A formula for the first derivative is obtained.
Di Marco, Silvia C., González, Roberto L.V. (2003)
International Journal of Mathematics and Mathematical Sciences
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Sophie Gombao, Jean-Pierre Raymond (2006)
ESAIM: Control, Optimisation and Calculus of Variations
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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 – where is an unbounded operator in a Hilbert space – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already...
Albert Altarovici, Olivier Bokanowski, Hasnaa Zidani (2013)
ESAIM: Control, Optimisation and Calculus of Variations
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The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can...
P. L. Lions (1985)
Banach Center Publications
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Fabio Bagagiolo (2004)
ESAIM: Control, Optimisation and Calculus of Variations
Similarity:
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.
Djivede Kelome, Andrzej Święch (2006)
Studia Mathematica
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We prove that Perron's method and the method of half-relaxed limits of Barles-Perthame works for the so called B-continuous viscosity solutions of a large class of fully nonlinear unbounded partial differential equations in Hilbert spaces. Perron's method extends the existence of B-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible...