Almost automorphic solutions for fractional stochastic differential equations and its optimal control

Optim Control Appl Methods

2019

Self Cite

Summary In this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.

“…Consider the Bolza problem

false(\overset{P}{true} false)

(see the works of Rajivganthi and Muthukumar, Yan and Xiumei, and Yan and Fangxia, which is to find an optimal pair

false(x^{0}, u^{0} false) \in scriptB \times U_{a d}

such that

scriptJ false(x^{0}, u^{0} false) \leq scriptJ false(x^{u}, u false), \forall u \in U_{a d}

where the cost functional

\begin{matrix} alignleft \\ align-1 J (x^{u}, u) = E \int_{0}^{t} γ (t, x_{t}^{u}, x^{u} (t), u (t)) d t + E Θ (x^{u} (T)) . & align-2 \end{matrix}

…”

Section: Existence Of Optimal Controlmentioning

confidence: 99%

“…If not means, then there is nothing to prove. Using the assumptions (H7)‐(H12), we have

\begin{matrix} alignleft \\ align-1 J (x_{u}, u) & align-2 \geq \int_{0}^{T} E \bar{w} (t) d t + c_{1} \int_{0}^{T} E ‖ x_{t u} ‖_{B} d t + c_{2} \int_{0}^{T} E ‖ x_{u} (t) ‖_{C} d t + c_{3} \int_{0}^{T} ‖ u (t) ‖_{Y}^{2} d t + E θ (x_{u} (T)) \\ align-1 & align-2 \geq - \tilde{ρ}, \end{matrix}

where

\overset{ρ}{true} \geq 0

is a constant (see the works of Rajivganthi and Muthukumar and Tamilalagan and Balasubramaniam). Hence,

ρ > - \overset{ρ}{true} > - \infty

.…”

Section: Existence Of Optimal Controlmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

\begin{matrix} alignleft \\ align-1^{c} D_{t}^{α} [x (t) + g (t, x_{t})] & align-2 = A x (t) + B u (t) + f (t, x_{t}, x (a (x (t), t))) + σ (t, x_{t}) \frac{d w (t)}{d t} + \int_{Z} h (t, x_{t}, η) \tilde{N} (d t, d η), t \in [0, T] = J, \\ align-1 x (t) & align-2 = x_{0} = ϕ \in B, t \in (- \infty, 0], \end{matrix}

where

\frac{1}{2} < α < 1

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

Durga

Optim Control Appl Methods

2019

Self Cite

“…Ren et al [30] reported the existence and stability results of time-dependent stochastic delayed differential equations with Poisson jumps. Recently, Rajivganthi and Muthukumar [31] studied the properties of almost automorphic solutions of fractional stochastic evolution equations with Poisson jumps with the help of solution operator.…”

Section: Introductionmentioning

confidence: 99%

Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

Rihan

Rajivganthi

Discrete Dynamics in Nature and Society

2017

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

Optimal control of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward system with Lévy processes

Deepa¹,

Optim Control Appl Methods

Hafayed

2020

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SummaryThis article investigates the optimal control problem of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward stochastic system with Lévy processes associated with Teugels martingales over the infinite time horizon. Based on the transversality conditions, assumption of convex control domain, infinite‐horizon version of stochastic maximum principle (Nash equilibrium), and necessary condition for optimality are established. Finally, the Nash equilibrium for the optimization problem in the financial market is considered to illustrate the observed theoretical results.