scholarly journals Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions

2021 ◽  
Author(s):  
Andrzej Swiech
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Viscosity Solutions ◽  
Hilbert Spaces ◽  
Stochastic Systems ◽  
Nonlinear Dependence ◽  
Infinite Dimensional

We study a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of ``slow'' and ``fast'' components. We use the theory of viscosity solutions in Hilbert spaces to show that as the speed of the fast component goes to infinity, the value function of the optimal control problem converges to the viscosity solution of a reduced effective equation. We consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. The results of this paper generalize to the infinite dimensional case the finite dimensional results of O. Alvarez and M. Bardi and complement the results in Hilbert spaces obtained recently by G. Guatteri and G. Tessitore.

scholarly journals Backward SDEs and infinite horizon stochastic optimal control

2019 ◽  
Vol 25 ◽  
pp. 31 ◽  
Author(s):  
Fulvia Confortola ◽  
Andrea Cosso ◽  
Marco Fuhrman
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Brownian Motion ◽  
Optimal Control Problem ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Backward Stochastic Differential Equation ◽  
Infinite Horizon ◽  
Jump Process ◽  
Elliptic Type

We study an optimal control problem on infinite horizon for a controlled stochastic differential equation driven by Brownian motion, with a discounted reward functional. The equation may have memory or delay effects in the coefficients, both with respect to state and control, and the noise can be degenerate. We prove that the value, i.e. the supremum of the reward functional over all admissible controls, can be represented by the solution of an associated backward stochastic differential equation (BSDE) driven by the Brownian motion and an auxiliary independent Poisson process and having a sign constraint on jumps. In the Markovian case when the coefficients depend only on the present values of the state and the control, we prove that the BSDE can be used to construct the solution, in the sense of viscosity theory, to the corresponding Hamilton-Jacobi-Bellman partial differential equation of elliptic type on the whole space, so that it provides us with a Feynman-Kac representation in this fully nonlinear context. The method of proof consists in showing that the value of the original problem is the same as the value of an auxiliary optimal control problem (called randomized), where the control process is replaced by a fixed pure jump process and maximization is taken over a class of absolutely continuous changes of measures which affect the stochastic intensity of the jump process but leave the law of the driving Brownian motion unchanged.

A Classical Stochastic Verification Theorem for Stochastic Recursive Optimization Problems

2013 ◽  
Vol 278-280 ◽  
pp. 1742-1745
Author(s):  
Shao Lin Ji ◽  
Lin Wang ◽  
Shu Zhen Yang
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Optimization Problems ◽  
Backward Stochastic Differential Equation ◽  
Verification Theorem ◽  
Recursive Optimization

In this paper, we study a stochastic recursive optimal control problem in which the system is governed by a forward-backward stochastic differential equation. Under mild assumptions, a classical stochastic verification theorem is derived.

Optimal Control Problem for a Degenerate Fractional Differential Equation

2021 ◽  
Vol 42 (6) ◽  
pp. 1239-1247
Author(s):  
R. A. Bandaliyev ◽  
I. G. Mamedov ◽  
A. B. Abdullayeva ◽  
K. H. Safarova
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fractional Differential Equation ◽  
Fractional Differential

scholarly journals An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

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