Controllability and Optimal Control for a Class of Time-Delayed Fractional Stochastic Integro-Differential Systems
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Controllability and Optimal Control for a Class of Time-Delayed Fractional Stochastic Integro-Differential Systems T. Sathiyaraj1 · JinRong Wang1,2
· P. Balasubramaniam3
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study the controllability and optimal control for a class of timedelayed fractional stochastic integro-differential system with Poisson jumps. A set of sufficient conditions is established for complete and approximate controllability by assuming non-Lipschitz conditions and pth mean square norm. We also give an existence of optimal control for Bolza problem. Our result is valid for fractional order α > p−1 p , p ≥ 2. Finally, an example is provided to illustrate the efficiency of the obtained theoretical results. Keyword Fractional stochastic systems · Controllability · Optimal control Mathematics Subject Classification 93B05 · 26A33 · 34A08 · 34K50 · 60J65 · 49J15
This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012) and Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016).
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JinRong Wang [email protected] T. Sathiyaraj [email protected] P. Balasubramaniam [email protected]
1
Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China
3
Department of Mathematics, The Gandhigram Rural Institute, Gandhigram 624302, India
123
Applied Mathematics & Optimization
1 Introduction Initially, the following Volterra type model for population growth have been represented by integro-differential equation in [25] t d X (t) X (s)ds X (t), (α, β, γ > 0) = α − β X (t) − γ dt 0
(1)
where α is the coefficient of self-increase and X is population size (for more details, see [25]). Equation (1) has been solved by successive approximations method. Further, (1) is characterized by fractional nonlinear integro-differential equation in the following form [27] d q x(t) = Ax(t) − Bx 2 (t) − C x(t) dt q x(0) = x0 ,
t
x(s)ds, 0 < q < 1,
0
(2)
where x(t) is the scaled population of identical individuals at time t, and q is a constant describing the order q ∈ (0, 1) of time-fractional derivative A > 0 and B > 0 are the birth rate coefficient and crowding coefficient respectively. Toxicity coefficient C > 0 denotes the essential behavior of the population evolution before its level falls to zero in the long run (for more details, one can refer [27]). Furthermore, one of the most significant models is the following Volttera integro-differential equation established in [6] for the population growth d x(t) = A(t)x(t) + dt
0
t
K (t, s)x(s)ds + f (t, x(t)) + σ (t, x(t))
dw(t) dt
(3)
in which A(t) = diag[a1 (t), a2 (t), . . . , an (t)], ai (t) ∈ C(R, R). The integral term of Eq. (3) denotes spe
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