Existence of almost periodic solutions to some stochastic differential equations

Bezandry, Paul H.; Diagana, Toka

doi:10.1080/00036810701397788

Cited by 89 publications

(62 citation statements)

References 6 publications

(9 reference statements)

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“…For instance among others, let us mentioned the existence, uniqueness and asymptotic stability results of almost periodic solutions, almost automorphic solutions, pseudo almost periodic solutions studied by many authors, see, e.g. ( [1]- [11]). The concept of Sasymptotically ω-periodic stochastic processes, which is the central question to be treated in this paper, was first introduced in the literature by Henriquez, Pierri et al in ( [12,13]).…”

Section: Dx(t) = A(t)x(t)dt + F (T X(t))dt +G(t X(t))dw (T)mentioning

confidence: 99%

S-asymptotically w-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion

Manou-Abi¹,

Dimbour²

2017

Adv. sci. technol. eng. syst. j.

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Section: Dx(t) = A(t)x(t)dt + F (T X(t))dt +G(t X(t))dw (T)mentioning

confidence: 99%

S-asymptotically w-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion

Manou-Abi¹,

Dimbour²

2017

Adv. sci. technol. eng. syst. j.

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“…See [7]. For a hyperbolic analytic semigroup (T (t)) t≥0 there exists constants C(α) > 0, δ > 0, M (α) > 0 and γ > 0 such that…”

Section: Not Necessarily Densely Defined) Is Said To Be Sectorial If mentioning

confidence: 99%

“…See [7]. If the operator A is sectorial, then it generates an analytic semigroup (T (t)) t≥0 , which maps (0, ∞) into L(Y ) and such that there exists constants M 0 , M 1 > 0 such that…”

Section: Not Necessarily Densely Defined) Is Said To Be Sectorial If mentioning

confidence: 99%

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Almost Automorphy and Riccati Equation

Mishra

2016

Differ Equ Dyn Syst

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Abstract. In this paper we first consider a linear time invariant systems with almost periodic forcing term. We propose a new deterministic quadratic control problem, motivated by Da-Prato. With the help of associated degenerate Riccati equation we study the existence and uniqueness of an almost automorphic solutions. IntroductionThe study of periodic systems, periodic optimization problems and the existence of almost periodic solutions to quadratic control problems have been quite interesting topic to work for many Mathematicians for e.g. [1, 4, 5, 8, 11] and [14]. In [5], Da-Prato has considered the quadratic control problem for the periodic systems in infinite dimension and showed that optimal control is given by feedback control which involves periodic solution of Riccati equation.In [1], Da-Prato and Ichikawa considered the following systemwhere u ∈ L 2 ap (U ), U and Y are real separable Hilbert spaces, B ∈ L(U, Y ) and f ∈ L 2 ap (Y ). Under the assumptions that the pair (A, B) is stabilizable and (M, A) is detectable, they studied the existence of almost periodic solutions for (1.1) and minimized the cost functional:over the set of all controls u such that (1.1) has an almost periodic mild solution.In this paper we study the same problem as considered in [1], by weakening their hypothesis. The conditions (A, B) stabilizable (M, A) detectable is somehow more demanding, whereas similar results as in [1] can be obtained with the assumption on only one pair that the pair (−A, B) is exactly controllable. We generalize their results [1] for the case of almost automorphic solutions that is we wish to minimize the cost functional considered over the set of all controls such that equation (1.1), with almost automorphic forcing term has an almost automorphic solution.Almost automorphy is the generalization of almost periodicity. The notion of almost automorphic functions were introduced by S. Bochner [2] in relation to some aspects of differential geometry.2000 Mathematics Subject Classification. 34 K06, 34 A12, 37 L05.

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“…This concept has been applied to study the existence and uniqueness of square-mean almost periodic mild solutions to different classes of non-autonomous semilinear stochastic differential equations driven by two-sided Wiener processes (see for instance Bezandry and Diagana [1,2]). …”

Section: Introductionmentioning

confidence: 99%

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

Bonaccorsi

Ziglio

2014

Random Operators and Stochastic Equations

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Abstract. Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with ascillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the mean square norm for all times.

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Existence of almost periodic solutions to some stochastic differential equations

Abstract: The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of squaremean almost periodic solutions to some semilinear stochastic equations.

Cited by 89 publications

References 6 publications

S-asymptotically w-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion

S-asymptotically w-periodic solutions in the p-th mean for a Stochastic Evolution Equation driven by Q-Brownian motion

Almost Automorphy and Riccati Equation

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

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