Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions
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Abstract
This paper investigates the approximate controllability for Sobolev type stochastic perturbed control systems of fractional order with fractional Brownian motion and Sobolev fractional stochastic nonlocal conditions in a Hilbert space, A new set of sufficient conditions are established by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach’s fixed point theorem. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory.
Keywords
Approximate controllability Perturbed control systems Fractional stochastic system Fixed point technique Stochastic nonlocal condition Fractional Brownian motionMathematics Subject Classification
93B05 26A33 46E39 60H15 60G221 Introduction
The notion of controllability has played a central role throughout the history of modern control theory. Moreover, approximate controllable systems are more prevalent and fundamental concepts in deterministic and stochastic control theory. often approximate controllability is completely adequate in applications. Therefore, various approximate controllability problems for different kinds of nonlinear fractional dynamical systems in infinite dimensional spaces have been investigated in many publications; see [1, 2] and references therein.
Stochastic differential equations are generalization of deterministic differential equations that incorporate a “noise term.” These equations can be useful in many applications where we assume that there are deterministic changes combined with noisy fluctuations. Also, the study of stochastic differential equations has attracted great interest because of its applications in characterizing many problems in physics, biology,chemistry, mechanics, and so on. In finance and insurance, one has to deal with events such as corporate defaults, operational failures, or insured accidents, the theory and applications of stochastic differential equations in infinitedimensional spaces have received much attention, (see [3, 4, 5, 6, 7] and the references therein).
On the other hand, some real world problems in science and engineering can be modeled by stochastic differential equations driven by fractional Brownian motion (fBm, for short). In particular, many types of stochastic differential equations driven by fBm in infinite dimension received much attention, for example, Maslowski and Nualart [8] studied nonlinear stochastic evolution equations in a Hilbert space driven by cylindrical fractional Brownian motion with Hurst parameter \(H>\frac{1}{2}\) and nuclear covariance operator using techniques of fractional calculus with semigroup estimates. Boufoussi and Hajji [9] proved the existence and uniqueness of mild solutions of a neutral stochastic differential equations with nite delay, driven by a fractional Brownian motion in a Hilbert space and established some conditions ensuring the exponential decay to zero in mean square for the mild solution.
Approximate controllability for fractional stochastic systems are well investigated, we refer to [10, 11] and references therein.
Kerboua et al. [12] proved the approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces by using fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems. Kerboua et al. [13] introduced a new notion called fractional stochastic nonlocal condition for establishing approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces using Hölder’s inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems.
Fěckan et al. [14] presented the controllability results corresponding to two admissible control sets for fractional functional evolution equations of Sobolev type in Banach spaces with the help of two new characteristic solution operators and their properties, such as boundedness and compactness, the results are obtained by using Schauder fixed point theorem.
It should be mentioned that there is no work yet reported on the approximate controllability of Sobolev type perturbed control systems of fractional order. Motivated by this facts, our main objective is to study the approximate controllability for a class of Sobolev type nonlinear stochastic differential equations of fractional order (1.1). The result is obtained under the assumption that the associated linear system is approximately controllable. In particular, the controllability question is transformed to a fixed point problem for an appropriate nonlinear operator in a function space. For that we need to construct a suitable set of sufficient conditions.
A brief outline of this paper is given. In Sect. 2, we present ssome basic notations and preliminaries on the stochastic integrals with respect to fBm in Hilbert space. In Sect. 3, the approximate controllability results of stochastic perturbed system of fractional order (1.1) is investigated by means of fractional calculus, semigroup theory and control theory. The last section deal with an illustrative example and a discussion for possible future work in this direction.
2 Preliminaries
Throughout of this paper, we assume that \(H\in (\frac{1}{2},1)\) unless otherwise specified. In this section, we briefly introduce some useful results about fBm and the corresponding stochastic integral taking values in a Hilbert space.
2.1 Fractional Brownian motion
We begin by recalling the definition of a fractional Brownian motion. Let \( (\Omega ,\mathcal {F},\{\mathcal {F}_{t}\}_{t\ge 0},P)\) be a filtered complete probability space. A realvalued Gauss process \(\{\beta ^{H}(t),t\ge 0\}\) defined on \((\Omega ,\mathcal {F},\{\mathcal {F} _{t}\}_{t\ge 0},P)\) is called a fBm with Hurst parameter \(H\in (0,1)\) if \( \mathbb {E}\left[ \beta ^{H}(t)\right] =0\) and the covariance function is given by \(R_{H}(t,s)=\mathbb {E}\left[ \beta ^{H}(t)\beta ^{H}(s)\right] = \frac{1}{2}(\left t\right ^{2H}+\left s\right ^{2H}\left ts\right ^{2H})\), \(t,s\in \mathbb {R}\) (see [15]).
2.2 Stochastic integral with respect to fBm
Let \((\Omega ,\mathcal {F},\{\mathcal {F}_{t}\}_{t\ge 0},P)\) be a filtered complete probability space and and \(\mathcal {F}_{t}=\mathcal {F} _{t}^{B_{1}^{H},B_{2}^{H}}\), where \(\mathcal {F}_{t}^{B_{1}^{H},B_{2}^{H}}\) is the sigma algebra generated by \(\{(B_{1}^{H}(s),B_{2}^{H}(s)):0\le s\le t\}\).
Let \(L_{2}^{0}\left( Y,X\right) \) denote the space of all \(\psi \in \mathcal { L}\left( Y,X\right) \) such hat \(\psi Q^{\frac{1}{2}}\) is a Hilbert–Schmidt operator. The norm is defined by \(\left\ \psi \right\ _{L_{2}^{0}}^{2}=\sum _{n=1}^{\infty }\sqrt{\lambda _{n}}e_{n}\beta _{n}^{H}(t).\) Generally, \(\psi \) is called a QHilbert–Schmidt operator from Y to X.
Lemma 2.1
Lemma 2.2
(see [16]) Let M be the infinitesimal generator of an analytic semigroup \(\left\{ S(t),\ t\ge 0\right\} \) on a Hilbert space X. If \(\Delta M\) is a bounded linear operator on X then \(\left( M+\Delta M\right) \) is the infinitesimal generator of an analytic semigroup \(\left\{ \tilde{S}(t),\ t\ge 0\right\} \) on X.
 \(\left( A1\right) \)

L, \(\left( M+\Delta M\right) \) and V are closed linear operators.
 \(\left( A2\right) \)

\(D(V)\subset D(L)\subset D(M+\Delta M)\) and L and V are bijective.
 \(\left( A3\right) \)

\(L^{1}:X\rightarrow D(L)\subset X\) and \( V^{1}:X\rightarrow D(V)\subset X\) are linear, bounded, and compact operators.
From (A3), we deduce that \(L^{1}\) is bounded operators. Note (A3) also implies that L is closed since the fact: \(L^{1}\) is closed and injective, then its inverse is also closed. It comes from (A1)–(A3) and the closed graph theorem, we obtain the boundedness of the linear operator \((M+\Delta M)L^{1}:X\rightarrow X\). Consequently, \((M+\Delta M)L^{1}\) generates a semigroup \(\left\{ \tilde{S}(t)=e^{(M+\Delta M)L^{1}t},\ t\ge 0\right\} \). We suppose that \(K_{0}=\sup _{t\ge 0}\left\ \tilde{S}(t)\right\ <\infty \), and for short, we denote by \(C_{0}=\left \left L^{1}\right \right \ \)and \(C_{1}=\left \left V^{1}\right \right \).
Now, we recall the following known definitions on the fractional integral and derivative.
Definition 2.1
Definition 2.2
Definition 2.3
Remark 2.1
 (i)If \(f(t)\in C^{n}\left( [0,\infty )\right) \), then$$\begin{aligned} ^{C}D^{\alpha }f(t)=\frac{1}{\Gamma \left( n\alpha \right) }\int _{0}^{t} \frac{f^{\left( n\right) }(s)}{(ts)^{\alpha +1n}}ds=I^{n\alpha }f^{(n)}(s),\quad t>0. \end{aligned}$$
 (ii)
The Caputo derivative of a constant is equal to zero.
 (iii)The Riemann–Liouville derivative of a constant function is given by$$\begin{aligned} ^{L}D_{a^{+}}^{\alpha }C=\frac{C}{\Gamma \left( 1\alpha \right) }\left( xa\right) ^{\alpha }. \end{aligned}$$
Remark 2.2
 (a)
For the nonlocal condition, the function x(0) is dependent on t.
 (b)
\(^{L}D_{t}^{1q}\left[ Vx(0)\right] \) is well defined, i.e., if \(q=1\) and V is the identity, then (1.2) reduces to the usual nonlocal condition..
 (c)The function x(0) takes the formwhere \(Vx(0)_{t=0}=x_{0}.\)$$\begin{aligned} V^{1}x_{0}+\frac{1}{\Gamma (1q)} \int _{0}^{t}(ts)^{q}V^{1}g_{2}(s)dB_{2}^{H}(s), \end{aligned}$$
 (d)
The explicit and implicit integrals given in (2.1) exist (taken in Bochner’s sense).
Definition 2.4
 1.
\(x(0)\in L^{2}(\Gamma ,X),\ \)where \(x\left( 0\right) =V^{1}x_{0}+ \frac{1}{\Gamma (1q)}\int _{0}^{t}(ts)^{q}V^{1}g_{2}(s)dB_{2}^{H}(s)\) and \(Vx(0)_{t=0}=x_{0};\)
 2.
\(x(t)\in X\) has cádlág paths on \(t\in J\) almost surely and for each \(t\in J\), x(t) satisfies the integral equation
Here, \(\tilde{S}(t)\) is a \(C_{0}\)semigroup generated by a linear operator \( (M+\Delta M)L^{1}:X\rightarrow X\), \(h_{q}\) is a probability density function defined on \((0,\infty )\), that is \(h_{q}(s)\ge 0,\ s\in (0,\infty \) ) and \(\int _{0}^{\infty }h_{q}(s)ds=1.\)
The following lemma follows from the results in [16, 17, 18, 19, 20] and will be used throughout this paper.
Lemma 2.3
 1.For any fixed \(t\ge 0,\) \(\mathcal {\tilde{S}} _{q}(t)\) and \(\mathcal {\tilde{T}}_{q}(t)\) are linear and bounded operators in X,$$\begin{aligned} \text {i.e.for any}\ x\in X_{\alpha },\ \ \ \ \ \left \left \mathcal {\tilde{S}}_{q}(t)x\right \right \le C_{0}K_{0}\left \left x\right \right ,\ \ \ \ \ \ \ \left \left \mathcal {\tilde{T}}_{q}(t)x\right \right \le \frac{C_{0}K_{0}}{ \Gamma \left( q\right) }\left \left x\right \right . \end{aligned}$$
 2.
The operators \(\{\mathcal {\tilde{S}}_{q}(t):t\ge 0\}\) and \(\{\mathcal {\tilde{T}}_{q}(t):t\ge 0\}\) are strongly continuous.
 \(\left( H1\right) \)
 The functions \(f:J\times X\rightarrow X\) satisfy linear growth and Lipschitz conditions. Moreover, there exist positive constants \(N_{1},N_{2}>0\) such that$$\begin{aligned} \Vert f(t,x)f(t,y)\Vert ^{2}\le N_{1}\Vert xy\Vert ^{2},~~\Vert f(t,x)\Vert ^{2}\le N_{2}(1+\Vert x\Vert ^{2}), \end{aligned}$$
 \(\left( H2\right) \)

The function \(g_{i}:J\rightarrow L_{2}^{0}\) for \( i=1,2\), satisfies \(\int _{0}^{T}\left\ g_{i}\left( s\right) \right\ _{L_{2}^{0}}^{2}ds<\infty .\)
 \(\left( H3\right) \)

The linear stochastic system is approximately controllable on J.
Definition 2.5
System (1.1)–(1.2) is approximately controllable on J if \(\overline{\mathfrak {R}(T)}=L^{2}(\Omega ,\Gamma _{T},\)
The following lemma is required to define the control function [23].
Lemma 2.4
Lemma 2.5
3 Approximate controllability
In this section, we formulate and prove conditions for the existence and approximate controllability results of the nonlocal fractional stochastic perturbed control system of Sobolev type (1.1)–(1.2) using the contraction mapping principle.
Theorem 3.1
Assume assumptions (H1)–(H3) are satisfied. Then, for all \(T>0\), the system (1.1)–(1.2) has a mild solution on [0, T].
Proof

Step 1 For arbitrary \(x\in H_{2}\), let us prove that \(t\rightarrow F_{\lambda }\left( x\right) (t)\) is continuous on the interval J in \(L^{2}\) sense.Let \(0<t<t+h<T\), where \(t,t+h\in [0,T]\), and let \(\left h\right \) be sufficiently small.

Step 2 Now, we are going to show that \(F_{\lambda }\) is a contraction mapping in \(H_{2}.\)
Theorem 3.2
Assume that the assumptions (H1)–(H3) hold. Further, if the functions \(f,g_{1}\) and \(g_{2}\) are uniformly bounded and \(\left\{ \mathcal {\tilde{T}}_{q}:t\ge 0\right\} \) is compact, then the system (1.1)–(1.2) is approximately controllable on J.
Proof
4 Example
In this section, we present an example to illustrate our main result.
5 Conclusion
Sufficient conditions for the approximate controllability of a class of control systems described by Sobolev type nonlocal nonlinear fractional stochastic perturbed equations with fractional Brownian motion in Hilbert spaces are considered. Using fixed point technique, fractional calculations, stochastic integrals for fractional Brownian motion, and methods adopted directly from deterministic control problems. In particular, conditions are formulated and proved under the assumption that the approximate controllability of the fractional stochastic control nonlinear perturbed system is implied by the approximate controllability of its corresponding linear part. More precisely, the controllability problem is transformed into a fixed point problem for an appropriate nonlinear operator in a function space. The main used tools are the above required conditions, we guarantee the existence of a fixed point of this operator and study controllability of the considered systems.
Our future work will be focused on investigating the approximate controllability for fractional stochastic dynamical systems of Sobolev type with Lévy process and impulsive effects. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a class of Sobolev type nonlocal nonlinear fractional stochastic dynamical systems with Lé vy process and impulsive effects in Hilbert spaces.
Notes
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous reviewers for their careful reading of the manuscript, as well as their comments that lead to a considerable improvement of the original manuscript.
Compliance with ethical standards
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
References
 1.Sakthivel, R., Ganesh, R., Suganya, R.: Approximate controllability of fractional neutral stochastic system with infinite delay. Rep. Math. Phys. 70, 291–311 (2012)MathSciNetCrossRefMATHGoogle Scholar
 2.Sakthivel, R., Ganesh, R., Ren, Y., Anthoni, S.M.: Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18, 3498–3508 (2013)MathSciNetCrossRefMATHGoogle Scholar
 3.Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for linear deterministic and stochastic systems. SIAM J. Control Optim. 37, 1808–1821 (1999)MathSciNetCrossRefMATHGoogle Scholar
 4.Cao, J., Yang, Q., Huang, Z.: On almost periodic mild solutions for stochastic functional differential equations. Nonlinear Anal. Real World Appl. 13, 275–286 (2012)MathSciNetCrossRefMATHGoogle Scholar
 5.Chang, Y.K., Zhao, Z.H., N’Guérékata, G.M., Ma, R.: Stepanovlike almost automorphy for stochastic processes and applications to stochastic differential equations. Nonlinear Anal. Real World Appl. 12, 130–1139 (2011)MathSciNetCrossRefMATHGoogle Scholar
 6.Mao, X.: Stochastic Differential Equations and Their Applications. Horwood, Chichester (1997)MATHGoogle Scholar
 7.Sakthivel, R., Revathi, P., Renc, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)MathSciNetCrossRefMATHGoogle Scholar
 8.Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202, 277–305 (2003)MathSciNetCrossRefMATHGoogle Scholar
 9.Boufoussi, B., Hajji, S.: Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space. Stat. Probab. Lett. 82, 1549–1558 (2012)MathSciNetCrossRefMATHGoogle Scholar
 10.Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42, 1604–1622 (2003)MathSciNetCrossRefMATHGoogle Scholar
 11.Li, F., Liang, J., Xu, H.K.: Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, 510–525 (2012)CrossRefMATHGoogle Scholar
 12.Kerboua, M., Debbouche, A., Baleanu, D.: Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. Abstr. Appl. Anal. Art. ID 262191 (2013)Google Scholar
 13.Kerboua, M., Debbouche, A., Baleanu, D.: Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces. Electron. J. Qual. Theory Differ. Equ. 58, 1–16 (2014)MathSciNetCrossRefMATHGoogle Scholar
 14.Fěckan, M., Wang, J.R., Zhou, Y.: Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J. Optim. Theory. Appl. 156, 79–95 (2013)MathSciNetCrossRefMATHGoogle Scholar
 15.Mandelbrot, B.B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefMATHGoogle Scholar
 16.ElBorai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14(3), 433–440 (2002)MathSciNetCrossRefMATHGoogle Scholar
 17.ElBorai, M.M.: On some stochastic fractional integrodifferential equations. Adv/. Dyn. Syst. Appl. 1(1), 49–57 (2006)MathSciNetMATHGoogle Scholar
 18.Liu, H., Chang, J.C.: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal. Theory Methods Appl. 70(9), 3076–3083 (2009)MathSciNetCrossRefMATHGoogle Scholar
 19.Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12(1), 262–272 (2011)MathSciNetCrossRefMATHGoogle Scholar
 20.Wang, R.N., Xiao, T.J., Liang, J.: A note on the fractional Cauchy problems with nonlocal initial conditions. Appl. Math. Lett. 24(8), 1435–1442 (2011)MathSciNetCrossRefMATHGoogle Scholar
 21.Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations (Applied Mathematical Sciences), vol. 44. Springer, New York (1983)CrossRefMATHGoogle Scholar
 22.Yan, Z., Yan, X.: Existence of solutions for a impulsive nonlocal stochastic functional integrodifferential inclusion in Hilbert spaces. Zeitschrift furangewandte Mathematik und Physik 64(3), 573–590 (2013)MathSciNetCrossRefMATHGoogle Scholar
 23.Debbouche, A., Torres, D.F.M.: Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces. Int. J. Control 86, 1577–1585 (2013)MathSciNetCrossRefMATHGoogle Scholar
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