Existence for the dynamic programming equation of control diffusion processes in Hilbert space

Havârneanu, Teodor

doi:10.1016/0362-546x(85)90045-8

Cited by 13 publications

(4 citation statements)

References 2 publications

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“…Other papers concerning more regular solutions of second order Hamilton Jacobi equations in infinite dimensions are [2], [20], [13], [4], [3], [18] and [19] for the evolution case and [9] for the stationary case. In particular the last paper studies (1.1) in the space of functions that are square integrable on X with respect to the invariant measure of the Ornstein Uhlenbeck process (see [16] for the properties of such measure).…”

Section: Introductionmentioning

confidence: 99%

Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

Gozzi

Rouy

1996

Journal of Differential Equations

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We study a second order stationary Hamilton Jacobi equation in infinite dimension. This equation is nonlinear and convex with respect to the first-order term. We use properties of the transition semigroup associated to the linear equation to write the Hamilton Jacobi equation in integral form and we prove existence, uniqueness and regularity of a solution by the theory of maximal monotone operators. We also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems. Finally, the solution is the value function of the associated optimal stochastic control problem. Some examples are given.

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Section: Introductionmentioning

confidence: 99%

Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

Gozzi

Rouy

1996

Journal of Differential Equations

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“…• The mild solution approach by means of fixed point arguments -the method used here. This method has been introduced first in [15,50] and then developed in [6,7] and in various other papers (see e.g. [39,40,45,11,13,38,41,42,60,61,62,63] 1 .…”

Section: Introductionmentioning

confidence: 99%

Mild solutions of semilinear elliptic equations in Hilbert spaces

Federico

Gozzi

2017

Journal of Differential Equations

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This paper extends the theory of regular solutions (C1 in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of G-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into G-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein–Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton–Jacobi–Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

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“…Second order Hamilton-Jacobi equations with second order terms being trace class have been studied in various papers (see, e.g., [2,15,28]). In some cases (e.g., when the second order term is linear and hypoelliptic) it is possible to prove existence and uniqueness of differentiable solutions, while in the general fully nonlinear case a theory of viscosity solutions is available (see [31,39,43]).…”

Section: Discussionmentioning

confidence: 99%

“…These equations were first studied by Barbu and Da Prato (see, e.g., [2]), setting the problem in classes of convex functions and using semigroup and perturbation methods (see also Da Prato [15] and Havarneanu [28]). Much progress has been made recently due to the introduction of the notion of viscosity solutions.…”

Section: Introductionmentioning

confidence: 99%

Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control

Gozzi¹,

Rouy²,

Święch³

2000

SIAM J. Control Optim.

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The paper is concerned with fully nonlinear second order Hamilton-Jacobi-BellmanIsaacs equations of elliptic type in separable Hilbert spaces which have unbounded first and second order terms. The viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved. A stochastic optimal control problem driven by a parabolic stochastic PDE with control of Dirichlet type on the boundary is considered. It is proved that the value function of this problem is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

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Existence for the dynamic programming equation of control diffusion processes in Hilbert space

Cited by 13 publications

References 2 publications

Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

Regular Solutions of Second-Order Stationary Hamilton–Jacobi Equations

Mild solutions of semilinear elliptic equations in Hilbert spaces

Second Order Hamilton--Jacobi Equations in Hilbert Spaces and Stochastic Boundary Control

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