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Displaying similar documents to “Limiting average cost control problems in a class of discrete-time stochastic systems”

Synthesis of optimal control for nonlinear third order systems

Wiesław Szwiec

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The paper is devoted to a certain problem in the theory of synthesis of optimal control. The fundamental results of research in the synthesis of optimal control for second order systems are given in [1], [2], [5], [10]. In particular, in [2] and [5] some specific nonlinear systems are investigated, namely the systems related with the differential equations of the forms̈ + F(s,ṡ) = u or s̈ + F(s,ṡ,u) = 0where u is a control parameter in the interval [—1,1]. Some results concerning third...

Time-varying Markov decision processes with state-action-dependent discount factors and unbounded costs

Beatris A. Escobedo-Trujillo, Carmen G. Higuera-Chan (2019)

Kybernetika

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In this paper we are concerned with a class of time-varying discounted Markov decision models n with unbounded costs c n and state-action dependent discount factors. Specifically we study controlled systems whose state process evolves according to the equation x n + 1 = G n ( x n , a n , ξ n ) , n = 0 , 1 , ... , with state-action dependent discount factors of the form α n ( x n , a n ) , where a n and ξ n are the control and the random disturbance at time n , respectively. Assuming that the sequences of functions { α n } , { c n } and { G n } converge, in certain sense, to α ,...

Stability estimating in optimal stopping problem

Elena Zaitseva (2008)

Kybernetika

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We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space X . It is supposed that an unknown transition probability p ( · | x ) , x X , is approximated by the transition probability p ˜ ( · | x ) , x X , and the stopping rule τ ˜ * , optimal for p ˜ , is applied to the process governed by p . We found an upper bound for the difference between the total expected cost, resulting when applying τ ˜ * , and the minimal total expected cost. The bound given is a constant times sup x X p ( · | x ) - p ˜ ( · | x ) , where · is the...

Linearization techniques for 𝕃 See PDF-control problems and dynamic programming principles in classical and 𝕃 See PDF-control problems

Dan Goreac, Oana-Silvia Serea (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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The aim of the paper is to provide a linearization approach to the 𝕃 See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the 𝕃 p See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating 𝕃 See PDF problems in continuous and...

Systems of Bellman Equations to Stochastic Differential Games with Discount Control

Alain Bensoussan, Jens Frehse (2008)

Bollettino dell'Unione Matematica Italiana

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We consider two dimensional diagonal elliptic systems Δ u + a u = H ( x , u , u ) which arise from stochastic differential games with discount control. The Hamiltonians H have quadratic growth in u and a special structure which has notyet been covered by regularity theory. Without smallness condition on H , the existence of a regular solution is established.

A continuous mapping theorem for the argmin-set functional with applications to convex stochastic processes

Dietmar Ferger (2021)

Kybernetika

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For lower-semicontinuous and convex stochastic processes Z n and nonnegative random variables ϵ n we investigate the pertaining random sets A ( Z n , ϵ n ) of all ϵ n -approximating minimizers of Z n . It is shown that, if the finite dimensional distributions of the Z n converge to some Z and if the ϵ n converge in probability to some constant c , then the A ( Z n , ϵ n ) converge in distribution to A ( Z , c ) in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular,...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

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In this paper we establish a decoupling feature of the random interlacement process u d at level u , d 3 . Roughly speaking, we show that observations of u restricted to two disjoint subsets A 1 and A 2 of d are approximately independent, once we add a sprinkling to the process u by slightly increasing the parameter u . Our results differ from previous ones in that we allow the mutual distance between the sets A 1 and A 2 to be much smaller than their diameters. We then provide an important application...

On reliability analysis of consecutive k -out-of- n systems with arbitrarily dependent components

Ebrahim Salehi (2016)

Applications of Mathematics

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In this paper, we consider the linear and circular consecutive k -out-of- n systems consisting of arbitrarily dependent components. Under the condition that at least n - r + 1 components ( r n ) of the system are working at time t , we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k -out-of- n systems. In the following, we investigate the inactivity time of the component with lifetime...

Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian (2010)

Studia Mathematica

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Recently it was proved for 1 < p < ∞ that ω m ( f , t ) p , a modulus of smoothness on the unit sphere, and K ̃ ( f , t m ) p , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence ω m ( f , t ) p K ̃ ( f , t r ) p does not hold either for p = ∞ or for p = 1.

Transportation flow problems with Radon measure variables

Marcus Wagner (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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For a multidimensional control problem ( P ) K involving controls u L , we construct a dual problem ( D ) K in which the variables ν to be paired with u are taken from the measure space rca (Ω,) instead of ( L ) * . For this purpose, we add to ( P ) K a Baire class restriction for the representatives of the controls u. As main results, we prove a strong duality theorem and saddle-point conditions.

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain D d and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by 𝒫 . We associate to Problem 𝒫 an optimal control problem, denoted by 𝒬 . Then, using appropriate Tykhonov triples, governed by a nonlinear operator G and a convex K ˜ , we provide results concerning the well-posedness...

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b , σ . When α ( 1 , 2 ) , we investigate pathwise uniqueness for this equation. When α ( 0 , 1 ) , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α ( 0 , 1 ) or α ( 1 , 2 ) and on whether the driving stable process is symmetric or not. Our...

Covariance structure of wide-sense Markov processes of order k ≥ 1

Arkadiusz Kasprzyk, Władysław Szczotka (2006)

Applicationes Mathematicae

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A notion of a wide-sense Markov process X t of order k ≥ 1, X t W M ( k ) , is introduced as a direct generalization of Doob’s notion of wide-sense Markov process (of order k=1 in our terminology). A base for investigation of the covariance structure of X t is the k-dimensional process x t = ( X t - k + 1 , . . . , X t ) . The covariance structure of X t W M ( k ) is considered in the general case and in the periodic case. In the general case it is shown that X t W M ( k ) iff x t is a k-dimensional WM(1) process and iff the covariance function of x t has the triangular...

Optimal Constants in Khintchine Type Inequalities for Fermions, Rademachers and q-Gaussian Operators

Artur Buchholz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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For ( P k ) being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants C 2 n , n∈ ℕ, in the inequality k = 1 N A k P k 2 n [ C 2 n ] 1 / 2 n m a x ( k = 1 N A * k A k 1 / 2 L 2 n , ( k = 1 N A k A * k 1/2∥L2n , valid for all finite sequences of operators ( A k ) in the non-commutative L 2 n space related to a semifinite von Neumann algebra with trace. In particular, C 2 n = ( 2 n r - 1 ) ! ! for the Rademacher and Fermion sequences.

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form d X ε ( t ) d t = ε B ( X ε ( t ) , 𝛯 ( q 1 ( t ) ) , 𝛯 ( q 2 ( t ) ) , ... , 𝛯 ( q ( t ) ) ) where 𝛯 ( t ) , t 0 is either a stochastic process or a dynamical system with sufficiently fast mixing while q j ( t ) = α j t , α 1 l t ; α 2 l t ; l t ; α k and q j , j = k + 1 , ... , grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Giant component and vacant set for random walk on a discrete torus

Itai Benjamini, Alain-Sol Sznitman (2008)

Journal of the European Mathematical Society

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We consider random walk on a discrete torus E of side-length N , in sufficiently high dimension d . We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time u N d . We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const log N . Moreover, this connected component occupies a...

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the 2 -dimensional case, our main contribution is a method for determining the optimal coefficients c 1 , ... , c m in the inequality δ P ( E ) k = 1 m c k α ( E ) k + o ( α ( E ) m ) , valid for each Borel set E with positive and finite area, with δ P ( E ) and α ( E ) being, respectively, the 𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡 and the 𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 of E . In n dimensions, besides proving existence and regularity properties of minimizers for a wide class of 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠 including the lower semicontinuous extension of δ P ( E ) α ( E ) 2 , we...

Spectral condition, hitting times and Nash inequality

Eva Löcherbach, Oleg Loukianov, Dasha Loukianova (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let X be a μ -symmetric Hunt process on a LCCB space 𝙴 . For an open set 𝙶 𝙴 , let τ 𝙶 be the exit time of X from 𝙶 and A 𝙶 be the generator of the process killed when it leaves 𝙶 . Let r : [ 0 , [ [ 0 , [ and R ( t ) = 0 t r ( s ) d s . We give necessary and sufficient conditions for 𝔼 μ R ( τ 𝙶 ) l t ; in terms of the behavior near the origin of the spectral measure of - A 𝙶 . When r ( t ) = t l , l 0 , by means of this condition we derive the Nash inequality for the killed process. In the diffusion case this permits to show that the existence of moments of order l + 1 for τ 𝙶 ...