scholarly journals Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

2021 ◽  
Author(s):  
Hélène Frankowska ◽  
Nobusumi Sagara
Keyword(s):  
Infinite Horizon ◽  
Variational Problems ◽  
Necessary Condition ◽  
Value Functions ◽  
The Maximum Principle ◽  
Lipschitz Continuous ◽  
Infinite Dimensional ◽  
Set Valued Mapping ◽  
Nonconvex Variational Problems ◽  
Upper Estimate

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. First, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Second, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. The main results are applied to the derivation of the necessary optimality condition of the spatial Ramsey growth model.

CHARACTERIZATION AND APPROXIMATION OF VALUE FUNCTIONS OF DIFFERENTIAL GAMES WITH MAXIMUM COST IN INFINITE HORIZON

2005 ◽  
Vol 07 (04) ◽  
pp. 369-393
Author(s):  
S. DI MARCO ◽  
A. RAPAPORT
Keyword(s):  
Differential Games ◽  
Infinite Horizon ◽  
Integral Form ◽  
Continuous Functions ◽  
Generalized Solutions ◽  
Value Functions ◽  
Infinitesimal Operator ◽  
Isaacs Equation ◽  
Lipschitz Continuous ◽  
Regularity Properties

Value functions of differential games with L∞ criterion over infinite horizon are known to possess poor regularity. As an alternative to generalized solutions of the Isaacs equation, that usually requires some regularity properties, we propose a characterization of the value functions using the integral form of the Isaacs equation. We prove, without any regularity assumption, that value functions are the lowest super-solution and the largest element of a special set of sub-solutions, of the dynamic programming equation. We characterize also the limits of finite horizon value functions, and propose an approximation scheme in terms of iterations of an infinitesimal operator defined over the set of Lipschitz continuous functions. The images of this operator can be characterized by generalized solutions of a classical Isaacs equation. We illustrate these results on a example, whose value functions can be determined analytically.

scholarly journals Ergodic control of infinite-dimensional stochastic differential equations with degenerate noise

2019 ◽  
Vol 25 ◽  
pp. 12 ◽  
Author(s):  
Andrea Cosso ◽  
Giuseppina Guatteri ◽  
Gianmario Tessitore
Keyword(s):  
Separable Hilbert Space ◽  
Control Method ◽  
Infinite Horizon ◽  
Control Problems ◽  
Value Functions ◽  
Ergodic Control ◽  
Infinite Dimensional ◽  
State Process ◽  
Real Separable Hilbert Space ◽  
Analytical Tools

This paper is devoted to the study of the asymptotic behaviour of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relationship with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as, for instance, the so-called randomisation of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite-dimensional) real separable Hilbert space, and the diffusion coefficient is allowed to be degenerate.

On infinite-dimensional variational problems

1996 ◽  
Vol 14 (1) ◽  
pp. 47-71 ◽  
Author(s):  
Yu.L Dalecky ◽  
V.R. Steblovskaya
Keyword(s):  
Variational Problems ◽  
Infinite Dimensional

scholarly journals Singular perturbations for a subelliptic operator

2018 ◽  
Vol 24 (4) ◽  
pp. 1429-1451 ◽  
Author(s):  
Paola Mannucci ◽  
Claudio Marchi ◽  
Nicoletta Tchou
Keyword(s):  
Singular Perturbation ◽  
Singular Perturbation Problems ◽  
Value Functions ◽  
Infinitesimal Operator ◽  
Lipschitz Continuous ◽  
Subelliptic Operator ◽  
Whole Space ◽  
Perturbation Problems ◽  
Ergodic Problem ◽  
Logarithmic Growth

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

scholarly journals Near Optimality of Linear Delayed Doubly Stochastic Control Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jie Xu ◽  
Ruiqiang Lin
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Control Problem ◽  
Delay System ◽  
Necessary Condition ◽  
Linear Quadratic ◽  
The Maximum Principle ◽  
Doubly Stochastic ◽  
Convex Control ◽  
Near Optimality

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.

scholarly journals On the Convergence of Newton-like Method for Variational Inclusions under Pseudo-Lipschitz Mapping

2020 ◽  
Vol 40 (1) ◽  
pp. 43-53
Author(s):  
Mst Zamilla Khaton ◽  
MH Rashid ◽  
MI Hossain
Keyword(s):  
Differentiable Function ◽  
Divided Difference ◽  
Admissible Function ◽  
Fréchet Derivative ◽  
Variational Inclusion ◽  
Lipschitz Mapping ◽  
Lipschitz Continuous ◽  
Frechet Derivative ◽  
Set Valued Mapping ◽  
Fréchet Differentiable

In the present paper, we study a Newton-like method for solving the variational inclusion defined by the sums of a Frechet differentiable function, divided difference admissible function and a set-valued mapping with closed graph. Under some suitable assumptions on the Frechet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newton-like method and prove that the sequence generated by this method converges linearly and superlinearly to a solution of the variational inclusion. Specifically, when the Frechet derivative of the differentiable function is continuous, Lipschitz continuous, divided difference admissible function admits first order divided di_erence and the setvalued mapping is pseudo-Lipschitz continuous, we show the linear and superlinear convergence of the method. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 43-53

Statistical Inference of Second-Order Cone Programming

2019 ◽  
Vol 36 (02) ◽  
pp. 1940003
Author(s):  
Liwei Zhang ◽  
Shengzhe Gao ◽  
Saoyan Guo
Keyword(s):  
Lipschitz Continuity ◽  
Metric Regularity ◽  
Optimal Solution ◽  
Second Order ◽  
Lipschitz Continuous ◽  
Set Valued Mapping ◽  
Second Order Cone ◽  
Degeneracy Condition ◽  
Optimal Solution Set ◽  
The Stability

In this paper, we study the stability of stochastic second-order programming when the probability measure is perturbed. Under the Lipschitz continuity of the objective function and metric regularity of the feasible set-valued mapping, the outer semicontinuity of the optimal solution set and Lipschitz continuity of optimal values are demonstrated. Moreover, we prove that, if the constraint non-degeneracy condition and strong second-order sufficient condition hold at a local minimum point of the original problem, there exists a Lipschitz continuous solution path satisfying the Karush–Kuhn–Tucker conditions.

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