Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

Bonaccorsi, Stefano; Confortola, Fulvia; Mastrogiacomo, Elisa

doi:10.1137/100782875

Cited by 22 publications

(26 citation statements)

References 40 publications

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“…These results were extended by [10] (deterministic) and [2] (stochastic) to more general operators in Hilbert spaces. The state space approach has already proved to be useful for treating control problems [1,5]. In the present paper we will provide simplified proofs and generalize the theory to reflexive Banach spaces.…”

Section: In a Banach Space (E | · |)mentioning

confidence: 98%

Volterra equations in Banach spaces with completely monotone kernels

Bonaccorsi

Desch

2012

Nonlinear Differ. Equ. Appl.

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Abstract. We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.Mathematics Subject Classification. 60H15, 60H20, 45K05.

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Section: In a Banach Space (E | · |)mentioning

confidence: 98%

Volterra equations in Banach spaces with completely monotone kernels

Bonaccorsi

Desch

2012

Nonlinear Differ. Equ. Appl.

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“…As a preliminary step for the sequel, we state two results of existence and uniqueness for (a special case of) the original Volterra equation. The proofs can be found in [5,Section 2].…”

Section: The Controlled Stochastic Volterra Equationmentioning

confidence: 99%

Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

Mastrogiacomo

2019

Journal of Mathematical Analysis and Applications

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The aim of the paper is to study an optimal control problem on infinite horizon for an infinite dimensional integro-differential equation with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provide other interesting result and require a precise description of the properties of the generated semigroup. The main tools consist in studying the differentiability of the forward-backward system with infinite horizon corresponding with the reformulated problem and the proof of existence and uniqueness of of mild solutions to the corresponding HJB equation.

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“…Øksendal-Zhang [19], Agram-Øksendal [1] present some investigations with open set U by means of Malliavin calculus. Some related studies include Bonaccorsi et al [4], Shi et al [22], Wang-Zhang [26] and so on. However, none of above papers can treat the case when both U := {0, 1} and diffusion depends on control.…”

Section: Introductionmentioning

confidence: 99%

Linear quadratic control problems of stochastic Volterra integral equations

Wang

2018

ESAIM: COCV

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Optimal control problems of forward stochastic Volterra integral equations (SVIEs) are formulated and studied. When control region is arbitrary subset of Euclidean space and control enters into the diffusion, necessary conditions of Pontryagin's type for optimal controls are established via spike variation. Our conclusions naturally cover the analogue of stochastic differential equations (SDEs), and our developed methodology drops the reliance on Itô formula and second-order adjoint equations. Some new features, that are concealed in the SDEs framework, are revealed in our situation. For example, instead of using second-order adjoint equations, it is more appropriate to introduce second-order adjoint processes. Moreover, the conventional way of using one second-order adjoint equation is inadequate here. In other words, two adjoint processes, which just merge into the solution of second-order adjoint equation in SDEs situation, are actually required and proposed in our setting.

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Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

Cited by 22 publications

References 40 publications

Volterra equations in Banach spaces with completely monotone kernels

Volterra equations in Banach spaces with completely monotone kernels

Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

Linear quadratic control problems of stochastic Volterra integral equations

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