Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Masiero, Federica

doi:10.1007/s00245-004-0810-6

Cited by 33 publications

(81 citation statements)

References 41 publications

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“…Since the constant L is independent of t, the case of generic T > 0 follows by dividing the interval [0, T ] into a finite number of subintervals of length δ sufficiently small, or equivalently, as done in [32], by taking an equivalent norm with an adequate exponential weight, such as (4.15). This allows to perform the fixed point, exactly as done in the first part of the proof, in C 0,1,B b ([0, T ] × H) and to prove estimate (6.5).…”

Section: Existence and Uniqueness Of Mild Solutionsmentioning

confidence: 99%

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Gozzi¹,

Masiero²

2017

SIAM J. Control Optim.

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Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than the the ones when the delay appears only in the state. This is particularly true when we look at the associated Hamilton-Jacobi-Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong [40] for the deterministic case) the HJB equation is an infinite dimensional second order semilinear Partial Differential Equation (PDE) that does not satisfy the so-called "structure condition" which substantially means that the control can act on the system modifying its dynamics at most along the same directions along which the noise acts. The absence of such condition, together with the lack of smoothing properties which is a common feature of problems with delay, prevents the use of the known techniques (based on Backward Stochastic Differential Equations (BSDEs) or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation and so no results on this direction have been proved till now.In this paper we provide a result on existence of regular solutions of such kind of HJB equations. This opens the road to prove existence of optimal feedback controls, a task that will be accomplished in the companion paper [26]. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specific class of equations and data which arises naturally in many applied problems. Key words:Optimal control of stochastic delay equations; Delay in the control; Lack of structure condition; Second order Hamilton-Jacobi-Bellman equations in infinite dimension; Smoothing properties of transition semigroups.

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Section: Existence and Uniqueness Of Mild Solutionsmentioning

confidence: 99%

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Gozzi¹,

Masiero²

2017

SIAM J. Control Optim.

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“…It should be emphasized that our proof does not present any significant innovation as we follow closely the arguments in the proof for the Hilbert-space case in (Masiero 2005, Theorem 2.9). However, to the best of our knowledge, this is the first paper that deals with infinite-dimensional non-autonomous semi-linear HJ equations in the general Banach-space framework, particularly in Lesbesgue spaces L p (O) with p ≥ 2.…”

Section: T T P(t R)h(r · D X V(r ·)) (X) Dr (T X) ∈ [0 T ] × Ementioning

confidence: 69%

Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces

Serrano¹

2014

JMR

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In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients.

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“…In Hilbert spaces, the problem of solving in mild sense Equation (4.1) has already been treated in many articles; we cite [13] where G has to satisfy some nondegeneracy assumptions, which imply regularizing properties of the transition semigroup. We also cite [16], where the case of G not necessary constant is treated, and moreover less restrictive regularizing properties on the associated transition semigroup are asked. In [12], Equation (4.1) is studied in the more general case of G not necessary constant and without any nondegeneracy assumptions, via the backward stochastic differential equations approach, by requiring and differentiable with respect to x.…”

Section: The Associated Hamilton Jacobi Bellman Equationmentioning

confidence: 99%

“…The transition semigroup related to a linear heat equation with noise on a subdomain does not possess such regularizing properties. Indeed, see, for example, [7], Section 9.4.1, [16], Section 3.1 and [18], Section 2, in the case of the heat equation with noise on a subdomain, such regularizing properties are related to null controllability of a linear heat equation with control on a subinterval and to the behaviour of the minimal energy steering to 0 an initial condition. Deterministic linear heat equations with control on a subinterval are null controllable, see, for example, [14] and [22], but the minimal energy blows up too fast, see, for example, [10] and [21].…”

Section: A Controlled Semilinear Heat Equation With Noise and Controlmentioning

confidence: 99%

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Masiero

2008

Stochastic Analysis and Applications

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We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.

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Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Cited by 33 publications

References 41 publications

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

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