Citation: Rajesh Dhayal, Muslim Malik, Syed Abbas. Solvability and optimal controls of non-instantaneous impulsive stochastic neutral integro-differential equation driven by fractional Brownian motion[J]. AIMS Mathematics, 2019, 4(3): 663-683. doi: 10.3934/math.2019.3.663
Stochastic differential equations have been used with great success in many application areas including biology, epidemiology, mechanics, economics and finance. For the fundamental study of the theory of stochastic differential equations, we refer to [1,2,3,4]. Yang and Zhu [5] studied the existence, uniqueness, and stability of mild solutions for the stochastic differential equations with Poisson jumps by using fixed point techniques. The fBm with Hurst parameter $ {\cal H} \in (0, 1) $ is a self-similar centered Gaussian random process with stationary increments. It admits the long-range dependence properties when $ {\cal H} > 1/2. $ Many exciting applications of fBm have been established in diverse fields such as finance, economics, telecommunications, and hydrology. For more details on fBm, see [6,7,8,9] and the references cited therein. Boudaoui et al. [10] studied the existence and continuous dependence of the mild solutions for the impulsive stochastic differential equation driven by fBm.
In recent years, the differential equation with fixed moments of impulses (instantaneous impulses) has become the natural framework for modeling of many evolving processes and phenomena studied in economics, population dynamics, and physics. For more details on differential equations with instantaneous impulses, one can see the papers [11,12,13,14] and the references cited therein. Deng et al. [15] discussed the existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with non-compact semigroup. Zhu [16] obtained some sufficient conditions to ensure the $ p $th moment exponential stability of impulsive stochastic functional differential equations with Markovian switching. The action of instantaneous impulses does not describe certain dynamics of evolution processes in pharmacotherapy. For example, consider the following simplified situation concerning the hemodynamic equilibrium of a person. In the case of a decompensation (for example, high or low levels of glucose) one can prescribe some intravenous drugs (insulin). Since the introduction of the drugs into the bloodstream and the consequent absorption for the body are gradual and continuous processes, we can interpret the situation as an impulsive action that starts abruptly and remains active over a finite time interval. For these reasons, Hernández and O'Regan [17] introduced a new class of abstract differential equations with non-instantaneous impulses and they investigated the existence of mild and classical solutions. For comprehensive details on differential equation with non-instantaneous impulses, see [18,19,20]. The qualitative properties of mild solutions for differential equations with non-instantaneous impulses have been investigated in several papers [21,22,23] and the references cited therein.
On the other hand, delay differential equation has been gaining much interest and attracting the attention of several researchers, because of its wide applications in various fields of science and engineering such as control theory, heat flow, mechanics, distributed networks, and neural networks, etc. The delay depends on the state variable, is called state-dependent delay. For more details on state-dependent delay, we refer to [24,25,26,27,28]. In the neutral differential equation, the highest order derivative of the state variable appears without delay and with delay. Ezzinbi et al. [29] discussed the existence and regularity of solutions for the neutral functional integro-differential equation with delay. Vijayakumar [30] investigated the approximate controllability for integro-differential inclusions by using the resolvent operators. The optimal control problem plays an important role in many scientific fields, such as engineering, mathematics, and biomedical. When the stochastic differential equation describes the performance index and system dynamics, an optimal control problem reduces to a stochastic optimal control problem. Wei et al. [31] obtained the existence of optimal controls for the impulsive integro-differential equation of mixed type. Jiang et al. [32] discussed the existence of optimal controls for fractional evolution inclusion with Clarke subdifferential and nonlocal conditions. In particular, in [33,34], the authors analyzed the existence of optimal controls for the fractional differential equations, whereas in [35,36] the authors investigated the same type of problem for the impulsive fractional stochastic integro-differential equations with delay.
To the best of our knowledge, there is no paper discussing the solvability and optimal controls of a non-instantaneous impulsive stochastic system driven by fBm with state-dependent delay. In order to fill this gap, we consider the following non-instantaneous impulsive stochastic neutral integro-differential equation driven by fBm with state-dependent delay:
$ \begin{equation} \begin{cases} d\mathfrak{D}(t,z_{t}) = {\cal A}\Big[\mathfrak{D}(t,z_{t})+\int_{0}^{t}{\cal G}(t-s)\mathfrak{D}(s,z_{s})ds\Big]dt+{\cal C}(t)v(t)dt+{\cal F}_{2}(t,z_{\rho(t,z_{t})})dB^{\cal H}(t)\\ \qquad\qquad\qquad\quad\;t\in (p_{j},t_{j+1}],\;j = 0,1,\ldots,{\cal M}, & \text{}\\ z(t) = {\cal E}_{j}(t,z_{t}),\,\,\,t\in (t_{j},p_{j}],\;j = 1,2,\ldots,{\cal M}, & \text{}\\ z_{0} = \Omega\in \mathfrak{B},& \text{} \end{cases} \end{equation} $ | (1.1) |
where $ z(\cdot) $ takes values in a real separable Hilbert space $ Z $, $ {\cal A} $ is the generator of a $ C_{0} $-semigroup of operators $ \{\Re(t):t \geq 0\} $ on $ Z. $ $ B^{\cal H} = \{B^{\cal H}(t):t\geq 0\} $ is a fBm with Hurst index $ {\cal H} \in (1/2, 1) $, takes values in a Hilbert space $ Y $. The initial data $ \Omega = \{ \Omega(t), \; t \in (-\infty, 0]\} $ is a $ \mathfrak{B} $-valued, $ {\cal F}_{0} $-adapted random variable, which not dependent on $ B^{\cal H} $, where $ \mathfrak{B} $ abstract phase space. The history valued function $ z_t:(-\infty, 0] \rightarrow Z $ is defined as $ z_{t}(\theta) = z(t+\theta) $ for all $ \theta \in (-\infty, 0] $ belongs to $ \mathfrak{B}. $ The control function $ v $ takes value from a separable reflexive Hilbert space $ {\cal T}, $ and $ {\cal C} $ is linear operator from $ {\cal T} $ into $ Z. $ $ 0 = t_{0} = p_0 < t_1 < p_1 < \cdots < t_{\cal M} < p_{\cal M} < t_{{\cal M}+1} = b < \infty $ are prefixed numbers, $ {\cal J}_{1} = [0, b] $. Suppose that $ {\cal G}(t), \, t\in {\cal J}_{1} $ is a linear and bounded operator. The function $ \mathfrak{D}:{\cal J}_{1}\times \mathfrak{B} \rightarrow Z $ is defined by $ \mathfrak{D}(t, \psi) = \psi(0)-{\cal F}_{1}(t, \psi), \, \, \psi \in \mathfrak{B} $ and $ {\cal F}_{1}:{\cal J}_{1}\times\mathfrak{B}\rightarrow Z $, $ {\cal F}_{2}:{\cal J}_{1}\times\mathfrak{B}\rightarrow L_{2}^{0}(Y, Z) $, where $ L^{0}_{2}(Y, Z) $ is space of all $ Q $-Hilbert-Schmidt operators from $ Y $ into $ Z, $ $ {\cal E}_{j}:(t_{j}, p_{j}]\times\mathfrak{B}\rightarrow Z $, $ j = 1, 2, \ldots, {\cal M} $ and $ \rho:{\cal J}_{1}\times \mathfrak{B}\rightarrow (-\infty, b] $ are suitable functions and they will be specified later.
The manuscript is structured as follows. Section 2 introduces preliminary facts and some notations. In Section 3, we discussed the solvability of the stochastic system and Section 4 is devoted to the investigation of the existence of optimal control pairs of the Lagrange problem corresponding to the proposed stochastic system. In Section 5, an example is provided to illustrate the applications of the obtained results. The last section is devoted to our conclusions.
In this section, we briefly review some basic definitions and notations that will be used in the subsequent sections. Let $ (\Omega, {\cal F}, \{{\cal F}_{t}\}_{t \geq 0}, P) $ be a filtered complete probability space, where $ {\cal F}_{t} $ the $ \sigma $-algebra is generated by $ \{B^{\cal H}(s), \, \, s \in [0, t]\}. $ By $ L(Y, Z), $ we denote the space of bounded linear operator from $ Y $ into $ Z. $ For convenience, the same notation $ \|\, .\, \| $ is used to denote the norms in $ Z $, $ Y $, $ L(Y, Z). $ The collection of all square integrable, strongly measurable, $ Z $-valued random variables, denoted by $ L^{2}(\Omega, Z) $, which is a Banach space. $ L^{2}_{{\cal F}_{0}}(\Omega, Z) = \{f \in L^{2}(\Omega, Z): f\; \mbox{is}\; {\cal F}_{0}-\mbox{measurable}\} $ is subspace of $ L^{2}(\Omega, Z). $ We denote by $ {\cal PC}([r_{1}, r_{2}], Z) $ the space formed by the normalized piecewise continuous, $ {\cal F}_{t} $-adapted measurable process from $ [r_{1}, r_{2}] $ into $ Z. $
Definition 2.1. Given $ {\cal H} \in (0, 1), $ a centered Gaussian and continuous random process $ \beta^{\cal H} = \{\beta^{\cal H}(t), \, \, t \geq 0 \} $ with covariance function
$ \mathbb{E}[\beta^{\cal H}(\varrho_{1}),\beta^{\cal H}(\varrho_{2})] = \dfrac{1}{2}(\varrho_{1}^{2{\cal H}}+\varrho_{2}^{2{\cal H}}-|\varrho_{1}-\varrho_{2}|^{2{\cal H}}), $ |
is called one dimensional fBm and $ {\cal H} $ is the Hurst parameter.
The fBm $ \beta^{\cal H}(t) $ with $ 1/2 < {\cal H} < 1 $ has the following integral representation
$ \beta^{\cal H}(t) = \int_{0}^{t}\mathfrak{K}_{\cal H}(t,\varrho)dw(\varrho), $ |
where $ w(\varrho) $ is a Wiener process or Brownian motion and the kernel $ \mathfrak{K}_{\cal H}(t, \varrho) $ is defined as
$ \mathfrak{K}_{\cal H}(t,\varrho) = \mathfrak{P}_{{\cal H}}\varrho^{1/2-{\cal H}}\int_{\varrho}^{t}(\tau-\varrho)^{{\cal H}-3/2}\tau^{{\cal H}-1/2}d\tau,\;\mbox{for}\;t \gt \varrho. $ |
We put $ \mathfrak{K}_{\cal H}(t, \varrho) = 0 $ if $ t \leq \varrho. $ Notice that $ \dfrac{\partial \mathfrak{K}_{\cal H}}{\partial t}(t, \varrho) = \mathfrak{P}_{{\cal H}}\big(t/\varrho\big)^{{\cal H}-1/2}(t-\varrho)^{{\cal H}-3/2}. $ Here, $ \mathfrak{P}_{{\cal H}} = [{\cal H}(2{\cal H}-1)/\xi(2-2{\cal H}, {\cal H}-1/2)]^{1/2} $ and $ \xi(\cdot, \cdot) $ is Beta function. For $ \Psi \in L^{2}([0, b]) $, it is well known from [37] that the Wiener-type integral of the function $ \Psi $ w.r.t fBm $ \beta^{\cal H} $ is defined by
$ \begin{eqnarray*} \int_{0}^{b} \Psi(\varrho)d\beta^{\cal H}(\varrho) = \int_{0}^{b} \mathfrak{K}_{\cal H}^{*}\Psi(\varrho)dw(\varrho), \end{eqnarray*} $ |
where $ \mathfrak{K}_{\cal H}^{*}\Psi(\varrho) = \int_{\varrho}^{b}\Psi(t)\dfrac{\partial \mathfrak{K}_{\cal H}}{\partial t}(t, \varrho)dt. $
Let the operator $ Q \in L(Y, Y) $ is defined by $ Qe_{i} = \lambda_{i}e_{i} $, where $ \{\lambda_{i} \geq 0:i = 1, 2, \ldots, \} $ are real numbers with trace $ Tr(Q) = \sum_{i = 1}^{\infty} \lambda_{i} < \infty $ and $ \{e_{i}, \; i = 1, 2, \ldots, \} $ is a complete orthonormal basis in $ Y $. Next, we define the infinite dimensional fBm $ B^{\cal H} $ on $ Y $ with covariance $ Q $ as
$ B^{\cal H}(t) = B^{\cal H}_{Q}(t) = \sum\limits_{i = 1}^{\infty}\sqrt{\lambda_{i}}e_{i}\beta^{\cal H}_{i}(t), $ |
where $ \beta^{\cal H}_{i}(t) $ are real, independent fBm. Now, we define the separable Hilbert space $ L^{0}_{2}(Y, Z) $ of all $ Q $-Hilbert-Schmidt operators from $ Y $ into $ Z $ with norm $ \|\psi\|_{L^{0}_{2}}^{2} = \sum_{i = 1}^{\infty} \|\sqrt{\lambda_{i}}\psi e_{i}\|^{2} < \infty $ and the inner product $ \langle \psi_{1}, \psi_{2}\rangle_{L^{0}_{2}} = \sum_{i = 1}^{\infty}\langle \psi_{1} e_{i}, \psi_{2} e_{i}\rangle. $ The Wiener integral of function $ \Upsilon : {\cal J}_{1} \rightarrow L^{0}_{2}(Y, Z) $ w.r.t fBm $ B^{\cal H} $ is defined by
$ \begin{eqnarray} \int_{0}^{t}\Upsilon(s)dB^{\cal H}(s) = \sum\limits_{i = 1}^{\infty} \int_{0}^{t}\sqrt{\lambda_{i}}\Upsilon(s) e_{i} d\beta^{\cal H}_{i}(s) = \sum\limits_{i = 1}^{\infty} \int_{0}^{t}\sqrt{\lambda_{i}}\mathfrak{K}_{\cal H}^{*}(\Upsilon e_{i})(s)dw_{i}(s). \end{eqnarray} $ | (2.1) |
Lemma 2.1. [6] If $ \Upsilon : {\cal J}_{1} \rightarrow L^{0}_{2}(Y, Z) $ satisfies $ \int_{0}^{b}\|\Upsilon(s)\|_{L^{0}_{2}}^{2}ds < \infty, $ then equation (2.1) is well-defined and $ Z $-valued random variable and we get
$ \begin{eqnarray} \mathbb{E}\Big\|\int_{0}^{t}\Upsilon(s)dB^{\cal H}(s)\Big\|^{2}\leq 2{\cal H}t^{2{\cal H}-1}\int_{0}^{t}\|\Upsilon(s)\|_{L^{0}_{2}}^{2}ds. \end{eqnarray} $ | (2.2) |
Now, we introduce the space $ {\cal PC}(Z) $ formed by all $ {\cal F}_{t} $-adapted measurable, $ Z $-valued stochastic processes $ \{z(t): t \in {\cal J}_{1}\} $ such that $ z $ is continuous at $ t\neq t_{j} $, $ z(t^{-}_j) = z(t_j) $ and $ z(t^{+}_j) $ exists for all $ j\, = 1, 2, \ldots, {\cal M}, $ endowed with the norm $ \|\, z\, \|_{\cal PC} = \big(\sup_{t\in {\cal J}_{1}}\mathbb{E}\|z(t)\|^{2}\big)^{1/2}. $ Then $ ({\cal PC}(Z), \|\cdot\|_{\cal PC}) $ is Banach space.
In the following, let $ {\cal T} $ is a separable reflexive Hilbert space from which the controls $ v $ take the values. Operator $ {\cal C}\in L^{\infty}({\cal J}_{1}, L({\cal T}, Z)) $, where $ L^{\infty}({\cal J}_{1}, L({\cal T}, Z)) $ denote the space of operator-valued functions which are measurable in the strong operator topology and uniformly bounded on the interval $ {\cal J}_{1}, $ endowed with the norm $ \|\, \cdot\, \|_{\infty} $. Let $ L^{2}_{\cal F}({\cal J}_{1}, {\cal T}) $ denote the space of all measurable and $ {\cal F}_{t} $-adapted, $ {\cal T} $-valued stochastic processes satisfying the condition $ \mathbb{E}\int_{0}^{b}\|v(t)\|^{2}_{\cal T}dt < \infty, $ and endowed with the norm $ \|\, v\, \|_{L^{2}_{\cal F}} = \big(\mathbb{E}\int_{0}^{b}\|v(t)\|^{2}_{\cal T}dt\big)^{1/2}. $ Let $ {\cal U} $ be a non-empty closed bounded convex subset of $ {\cal T}. $ We define the admissible control set
$ {\cal U}_{ad} = \{v \in L^{2}_{\cal F}({\cal J}_{1},{\cal T}) \;|\; v(t)\in {\cal U}\;a.e.\;t\,\in\,{\cal J}_{1}\}. $ |
Then, $ {\cal C}v \in L^{2}({\cal J}_{1}, Z) $ for all $ v \in {\cal U}_{ad}. $
In this paper, we assume that the phase space $ (\mathfrak{B}, \|\cdot\|_{\mathfrak{B}}) $ is a seminormed linear space of functions mapping $ (-\infty, 0] $ into $ Z $ and subsequent conditions are satisfied.
[A1]: If $ z : (-\infty, e+b] \rightarrow Z, \; b > 0 $ is such that $ z|_{[e, e+b]} \in {\cal PC}([e, e+b], Z) $ and $ z_{e} \in \mathfrak{B} $, then for each $ t \in [e, e+b] $ the subsequent conditions are satisfied:
1. $ z_{t} \in \mathfrak{B} $.
2. $ \|z(t)\| \leq {\cal K}_{1}\|z_{t}\|_{\mathfrak{B}} $.
3. $ \|z_{t}\|_{\mathfrak{B}} \leq {\cal K}_{2}(t-e)\, \sup\{\|z(s)\| : e \leq s \leq t\} + {\cal K}_{3}(t-e)\, \|z_{e}\|_{\mathfrak{B}} $, where $ {\cal K}_{1} $ is a positive constant, $ {\cal K}_{2}, {\cal K}_{3} : [0, +\infty) \rightarrow [1, +\infty) $, $ {\cal K}_{2} $ is a continuous function, $ {\cal K}_{3} $ is a locally bounded function and $ {\cal K}_{1}, {\cal K}_{2}, {\cal K}_{3} $ are independent of $ z(\cdot) $.
[A2]: For the function $ z(\cdot) $ in [A1], the function $ t \rightarrow z_{t} $ is continuous from $ [e, e+b] $ into $ \mathfrak{B}. $
[A3]: The phase space $ \mathfrak{B} $ is complete.
For more details on phase space, we refer to [38,39].
Lemma 2.2. [21] Let $ z : (-\infty, b] \rightarrow Z $ be an $ {\cal F}_{t} $-adapted measurable process such that the $ {\cal F}_{0} $-adapted process $ z_{0} $ = $ \Omega(t)\in L^{2}_{{\cal F}_{0}}(\Omega, \mathfrak{B}) $ and $ z|_{{\cal J}_{1}} \in {\cal PC}(Z) $, then
$ \begin{eqnarray*} \|z_{s}\|_{\mathfrak{B}}\leq {\cal K}_{3}^{*}\,\mathbb{E}\|\Omega\|_{\mathfrak{B}}+{\cal K}_{2}^{*}\,\sup\limits_{s \in {\cal J}_{1}}\mathbb{E}\|z(s)\|, \end{eqnarray*} $ |
where $ {\cal K}_{2}^{*} = \sup_{t \in {\cal J}_{1}}{\cal K}_{2}(t), $ $ {\cal K}_{3}^{*} = \sup_{t \in {\cal J}_{1}}{\cal K}_{3}(t) $.
Definition 2.2. A one parameter family $ \{\Re(t):t \geq 0\} $ of bounded linear operators, is called resolvent operator for
$ \begin{equation} \dfrac{dz}{dt} = {\cal A}\Big[z(t)+\int_{0}^{t}{\cal G}(t-\kappa)z(\kappa)d\kappa\Big], \end{equation} $ | (2.3) |
if
1. $ \Re(0) = I $ and $ \|\Re(t)\| \leq Ne^{\beta\, t} $ for some constants $ \beta $ and $ N \geq 1. $
2. For all $ z \in Z, $ $ \Re(t)z $ is strongly continuously for $ t \in {\cal J}_{1}. $
3. For all $ t \in {\cal J}_{1}, $ $ \Re(t)\in L(X). $ For all $ x \in X, $ $ \Re(\cdot)x \in C^{1}({\cal J}_{1}, Z) \cap C({\cal J}_{1}, X) $ and
$ \begin{eqnarray*} \dfrac{d}{dt}\Re(t)x & = &{\cal A}\Big[\Re(t)x+\int_{0}^{t}{\cal G}(t-\kappa)\Re(\kappa)xd\kappa\Big]\\& = &\Re(t){\cal A}x+\int_{0}^{t}\Re(t-\kappa){\cal A}{\cal G}(\kappa)xd\kappa,\;\;t \in {\cal J}_{1}. \end{eqnarray*} $ |
For more details on the resolvent operator, we refer to [40,41].
Definition 2.3. A $ Z- $valued stochastic process $ \{z(t), t\in(-\infty, b]\} $ is called a mild solution of the stochastic system (1.1) if $ z_{0} = \Omega, $ $ z_{\rho(s, z_{s})}\in {\mathfrak{B}}, $ $ z|_{[0, b]} \in {\cal PC}(Z) $ and
1. $ z(t) $ is measurable and adapted to $ {\cal F}_{t}, $ $ t \geq 0 $.
2. $ z(t)\in Z $ has càdlàg paths on $ [0, b] $ almost everywhere and for every $ t \in [0, b] $, $ z(t) $ satisfies $ z(t) $ = $ {\cal E}_{j}(t, z_{t}) $ for all $ t \in (t_{j}, p_{j}] $, $ j = 1, 2, \ldots, {\cal M}, $ and
$ \begin{eqnarray*} z(t)& = & \Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,z_{t})\\&&+\int_{0}^{t}\Re(t-s){\cal C}(s)v(s)ds+\int_{0}^{t}\Re(t-s){\cal F}_{2}(s,z_{\rho(s,z_{s})})dB^{\cal H}(s) \end{eqnarray*} $ |
for all $ t \in [0, t_{1}] $ and
$ \begin{eqnarray*} z(t)& = & \Re(t-p_{j})[{\cal E}_{j}(p_{j},z_{p_{j}})-{\cal F}_{1}(p_{j},z_{p_{j}})]+{\cal F}_{1}(t,z_{t})\\&&+\int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v(s)ds+\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,z_{\rho(s,z_{s})})dB^{\cal H}(s) \label{mild1} \end{eqnarray*} $ |
for all $ t \in (p_{j}, t_{j+1}], \; j = 1, 2, \ldots, {\cal M}. $
In this section, we prove the existence of mild solutions for the stochastic system (1.1). Let $ \rho:{\cal J}_{1}\times {\mathfrak{B}}\rightarrow (-\infty, b] $ be a continuous function. To prove our main results, we need the following hypotheses:
[H1]: $ \Re(t), \; t > 0 $ is compact and there exists a constant $ N > 0 $ such that $ \|\Re(t)\| \leq N $ for every $ t\in{\cal J}_{1}. $
[H2]: The function $ t \rightarrow \Omega_{t} $ is continuous from the set $ {\cal S}(\rho^{-}) = \{ \rho(t, \psi)\leq 0:(t, \psi) \in {\cal J}_{1} \times {\mathfrak{B}}\} $ into $ {\mathfrak{B}} $ and there exists a bounded and continuous function $ {\cal L}_{\Omega}:{\cal S}(\rho^{-})\rightarrow (0, \infty) $ to ensure that $ \|\Omega_{t}\|_{\mathfrak{B}} \leq {\cal L}_{\Omega}(t)\|\Omega\|_{\mathfrak{B}} $ for all $ t\, \in\; {\cal S}(\rho^{-}). $
[H3]: There exists a constant $ L_{{\cal F}_{1}} > 0 $ such that the function $ {\cal F}_{1}:{\cal J}_{1}\times{\mathfrak{B}}\rightarrow Z $ satisfies the following conditions
$ \begin{eqnarray*} &&\mathbb{E}\|{\cal F}_{1}(t,\psi)\|^{2} \leq L_{{\cal F}_{1}}(\|\psi\|^{2}_{\mathfrak{B}}+1),\,\forall\,\psi\,\in {\mathfrak{B}},\,t \in {\cal J}_{1},\\&&\mathbb{E}\|{\cal F}_{1}(t,\psi_{1})-{\cal F}_{1}(t,\psi_{2})\|^{2} \leq L_{{\cal F}_{1}}\;\|\psi_{1}-\psi_{2}\|^{2}_{\mathfrak{B}},\;\forall\,\psi_{1},\psi_{2}\,\in {\mathfrak{B}},\,t \in {\cal J}_{1}. \end{eqnarray*} $ |
[H4]: There exist constants $ L_{{\cal E}_{j}} > 0, $ $ j = 1, 2, \ldots, {\cal M}, $ such that the functions $ {\cal E}_{j}:(t_{j}, p_{j}]\times{\mathfrak{B}}\rightarrow Z $, $ j = 1, 2, \ldots, {\cal M}, $ satisfies the following conditions
$ \begin{eqnarray*} &&\mathbb{E}\|{\cal E}_{j}(t,\psi)\|^{2} \leq L_{{\cal E}_{j}}(\|\psi\|^{2}_{\mathfrak{B}}+1),\,\forall\,\psi\in {\mathfrak{B}},\\&&\mathbb{E}\|{\cal E}_{j}(t,\psi_{1})-{\cal E}_{j}(t,\psi_{2})\|^{2}\leq L_{{\cal E}_{j}}\|\psi_{1}-\psi_{2}\|^{2}_{\mathfrak{B}},\,\forall\,\psi_{1},\psi_{2}\,\in\,{\mathfrak{B}}. \end{eqnarray*} $ |
[H5]: The function $ {\cal F}_{2}:{\cal J}_{1}\times \mathfrak{B}\rightarrow L_{2}^{0}(Y, Z) $ satisfies the conditions
(a) The function $ {\cal F}_{2}(t, \, \cdot\, ):\mathfrak{B} \rightarrow L_{2}^{0}(Y, Z) $ is continuous for a.e $ t \in {\cal J}_{1}, $ and $ t \rightarrow {\cal F}_{2}(t, \psi) $ is measurable for all $ \psi \in \mathfrak{B} $.
(b) There exists a continuous function $ \eta:{\cal J}_{1}\rightarrow [0, \infty) $ and a continuous nondecreasing function $ \Theta:[0, \infty)\rightarrow (0, \infty) $ to ensure that for all $ (t, \psi)\; \in\; {\cal J}_{1}\times \mathfrak{B} $
$ \begin{eqnarray*} \mathbb{E}\|{\cal F}_{2}(t,\psi)\|^{2}_{L_{2}^{0}} \leq \eta(t)\Theta_{{\cal F}_{2}}(\|\psi\|^{2}_{\mathfrak{B}}),\;\;\;\liminf\limits_{w\rightarrow \infty}\dfrac{\Theta_{{\cal F}_{2}}(w)}{w} = \Theta_{1}. \end{eqnarray*} $ |
[H6]: The following inequality holds
$ \begin{eqnarray*} \max\limits_{1\leq j \leq {\cal M}} 2[{\cal K}_{2}^{*}]^{2}\big[L_{{\cal E}_{j}}+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})+4L_{{\cal F}_{1}}+8{\cal H}N^{2}b^{2{\cal H}-1}\Theta_{1}\int_{0}^{b} \eta(s)ds\big] \lt 1. \end{eqnarray*} $ |
Lemma 3.1. [28] Let $ z : (-\infty, b] \rightarrow Z $ such that $ z_{0} = \Omega $ and $ z|_{{\cal J}_{1}} \in {\cal PC}(Z) $. If [H2] be hold, then
$ \begin{eqnarray*} \|z_{t}\|_{\mathfrak{B}}\leq ({\cal K}_{3}^{*}+{\cal L}_{\Omega}^{*})\|\,\Omega\,\|_{\mathfrak{B}}+{\cal K}_{2}^{*}\;\sup\big\{\mathbb{E}\|z(\omega)\| : \omega \in [0,\max\{0,t\}]\big\},\;t \in {\cal S}(\rho^{-}) \cup {\cal J}_{1}, \end{eqnarray*} $ |
where $ {\cal K}_{2}^{*} = \sup_{t \in {\cal J}_{1}}{\cal K}_{2}(t), $ $ {\cal K}_{3}^{*} = \sup_{t \in {\cal J}_{1}}{\cal K}_{3}(t) $, and $ {\cal L}_{\Omega}^{*} = \sup_{t \in {\cal S}(\rho^{-})}{\cal L}_{\Omega}(t). $
Theorem 3.1. If the hypotheses [H1]–[H6] are fulfilled. Then for each $ v \in {\cal U}_{ad} $, the stochastic system (1.1) has at least one mild solution on $ {\cal J}_{1}, $ provided that
$ \begin{eqnarray} \max\limits_{1\leq j \leq {\cal M}}2[{\cal K}_{2}^{*}]^{2}\big(L_{{\cal E}_{j}}+4N^{2}L_{{\cal E}_{j}}+2(2N^{2}+1)L_{{\cal F}_{1}}\big) \lt 1. \end{eqnarray} $ | (3.1) |
Proof. On the space $ {\cal BPC} $ = $ \{z \in {\cal PC}(Z):z(0) = \Omega(0)\} $ endowed with the uniform convergence topology. For each $ l > 0, $ let
$ \begin{eqnarray*} \overline{\cal B}_{l} = \{z \in {\cal BPC}\,:\,\|z\|^{2}_{\cal PC} \leq l\}. \end{eqnarray*} $ |
Let the operator $ {\mathfrak{F}}:\overline{\cal B}_{l} $ $ \rightarrow $ $ {\cal BPC} $ be specified by
$ \begin{equation*} ({\mathfrak{F}}z)(t) = \begin{cases} \mathfrak{R}(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,\overline{z}_{t})\\+\int_{0}^{t}\mathfrak{R}(t-s){\cal C}(s)v(s)ds+\int_{0}^{t}\mathfrak{R}(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{}\; t \in [0,t_{1}],\,j = 0,\\ {\cal E}_{j}(t,\overline{z}_{t}), & \text{}\; t \in (t_{j},p_{j}],\,j \geq 1,\\ \mathfrak{R}(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z}_{p_{j}})]+{\cal F}_{1}(t,\overline{z}_{t})\\+\int_{p_{j}}^{t} \mathfrak{R}(t-s){\cal C}(s)v(s)ds+\int_{p_{j}}^{t}\mathfrak{R}(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{}\; t \in (p_{j},t_{j+1}],\,j \geq 1, \end{cases} \end{equation*} $ |
where $ \overline{z} : (-\infty, b] \rightarrow Z $ is such that $ \overline{z}_{0} = \Omega $ and $ \overline{z} = z $ on $ {\cal J}_{1}. $ For $ z \in \overline{\cal B}_{l} $, from Lemma 3, we have
$ \begin{eqnarray*} \|\overline{z}_{\rho(s,\overline{z}_{s})}\|_{\mathfrak{B}}^{2} &\leq & 2({\cal K}_{3}^{*}+{\cal L}_{\Omega}^{*})^{2}\|\,\Omega\,\|_{\mathfrak{B}}^{2}+2[{\cal K}_{2}^{*}]^{2}l = l^{*}. \end{eqnarray*} $ |
From [H1] and Hölder's inequality, we have
$ \begin{eqnarray*} \mathbb{E}\bigg\|\int_{p_{j}}^{t} \Re(t-s){\cal C}(s)v(s)ds\bigg\|^{2} &\leq & \mathbb{E}\bigg[\int_{p_{j}}^{t} \|\Re(t-s)\| \|{\cal C}(s)v(s)\|ds\bigg]^{2}\\ &\leq & N^{2}\|{\cal C}\|^{2}_{\infty}(t_{j+1}-p_{j})\mathbb{E}\int_{p_{j}}^{t}\|v(s)\|^{2}_{\cal T}ds\\& \leq & N^{2}\|{\cal C}\|^{2}_{\infty}(t_{j+1}-p_{j})\|v\|^{2}_{L^{2}_{\cal F}}. \end{eqnarray*} $ |
By Bochner theorem, it follows that $ \Re(t-s){\cal C}(s)v(s) $ are integrable on $ (p_{j}, t), \; j = 0, 1, \ldots, {\cal M}. $ Therefore $ \mathfrak{F} $ is well defined on $ \overline{\cal B}_{l} $. Now, we split $ {\mathfrak{F}} $ as $ {\mathfrak{F}}_{1}+{\mathfrak{F}}_{2} $, where
$ \begin{equation*} ({\mathfrak{F}}_{1}z)(t) = \begin{cases} \Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,\overline{z}_{t}),& \text{}\qquad\qquad\;\;\;\;\;\; t \in [0,t_{1}],\,j = 0,\\ {\cal E}_{j}(t,\overline{z}_{t}), & \text{}\qquad\qquad\;\;\;\;\;\;t \in (t_{j},p_{j}],\,j \geq 1,\\ \Re(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z}_{p_{j}})]+{\cal F}_{1}(t,\overline{z}_{t}),& \text{}\qquad\qquad\;\;\;\;\;\; t \in (p_{j},t_{j+1}],\,j \geq 1, \end{cases} \end{equation*} $ |
and
$ \begin{equation*} ({\mathfrak{F}}_{2}z)(t) = \begin{cases} \int_{0}^{t}\Re(t-s){\cal C}(s)v(s)ds+\int_{0}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{} \quad t \in [0,t_{1}],\,j = 0,\\ 0, & \text{}\quad t \in (t_{j},p_{j}],\,j \geq 1,\\ \int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v(s)ds+\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{}\quad t \in (p_{j},t_{j+1}],\,j \geq 1. \end{cases} \end{equation*} $ |
For the sake of convenience, we break the proof into a sequence of steps.
Step 1. There exists $ l > 0 $ such that $ {\mathfrak{F}}(\overline{\cal B}_{l}) \subset \overline{\cal B}_{l} $.
If we assume that this assertion is false, then for any $ l > 0 $, we can choose $ z^{l} \in \overline{\cal B}_{l} $ and $ t\in {\cal J}_{1} $ such that $ \mathbb{E}\|{\mathfrak{F}}(z^{l})(t)\|^{2} > l $. By [H1], [H3]–[H6] and Hölder's inequality, we have for $ t\in [0, t_{1}], $
$ \begin{eqnarray*} l \lt \mathbb{E}\|{\mathfrak{F}}(z^{l})(t)\|^{2} &\leq & 4\mathbb{E}\|\Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]\|^{2}+4\mathbb{E}\|{\cal F}_{1}(t,\overline{z^{l}}_{t})\|^{2}\\&&+4\mathbb{E}\bigg\|\int_{0}^{t}\Re(t-s){\cal C}(s) v(s)ds\bigg\|^{2}+4\mathbb{E}\bigg\|\int_{0}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})})dB^{\cal H}(s)\bigg\|^{2}\qquad\quad \\ \qquad\qquad\;\;\;\; &\leq & 8N^{2}[{\cal K}_{1}^{2}\|\Omega\|_{\mathfrak{B}}^{2}+\mathbb{E}\|{\cal F}_{1}(0,\Omega)\|^{2}]+4\mathbb{E}\|{\cal F}_{1}(t,\overline{z^{l}}_{t})\|^{2}\\&&+4\mathbb{E}\bigg[\int_{0}^{t}\|\Re(t-s)\|\|{\cal C}(s) v(s)\|ds\bigg]^{2}+8{\cal H}N^{2}t_{1}^{2{\cal H}-1}\int_{0}^{t}\mathbb{E}\|{\cal F}_{2}(s,\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})})\|^{2}_{L^{0}_{2}}ds \\ \qquad\qquad\, &\leq & 8N^{2}[{\cal K}_{1}^{2}\|\Omega\|_{\mathfrak{B}}^{2}+L_{{\cal F}_{1}}(\|\Omega\|^{2}_{\mathfrak{B}}+1)]+4L_{{\cal F}_{1}}(\|\overline{z^{l}}_{t}\|^{2}_{\mathfrak{B}}+1)\\&&+4N^{2}\|{\cal C}\|^{2}_{\infty}t_{1}\|v\|^{2}_{L^{2}_{\cal F}}+8{\cal H}N^{2}{t}_{1}^{2{\cal H}-1}\int_{0}^{t} \eta(s)\Theta_{{\cal F}_{2}}(\|\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})}\|^{2}_{\mathfrak{B}})ds.\qquad\qquad\;\;\;\, \end{eqnarray*} $ |
For any $ t \in (t_{j}, p_{j}] $, $ j = 1, 2, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} l \lt \mathbb{E}\|{\mathfrak{F}}(z^{l})(t)\|^{2} \leq L_{{\cal E}_{j}}(\|\overline{z^{l}}_{t}\|^{2}_{\mathfrak{ B}}+1). \end{eqnarray*} $ |
Similarly, for any $ t \in (p_{j}, t_{j+1}] $, $ j = 1, 2, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} l \lt \mathbb{E}\|{\mathfrak{F}}(z^{l})(t)\|^{2} &\leq & 4\mathbb{E}\|\Re(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z^{l}}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z^{l}}_{p_{j}})]\|^{2}+4\mathbb{E}\|{\cal F}_{1}(t,\overline{z^{l}}_{t})\|^{2}\\&&+4\mathbb{E}\bigg\|\int_{p_{j}}^{t} \Re(t-s){\cal C}(s)v(s)ds\bigg\|^{2}+4\mathbb{E}\bigg\|\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})})dB^{\cal H}(s)\bigg\|^{2}\qquad\quad\,\, \\\qquad\qquad\;\;\; &\leq & 8N^{2}[\mathbb{E}\|{\cal E}_{j}(p_{j},\overline{z^{l}}_{p_{j}})\|^{2}+\mathbb{E}\|{\cal F}_{1}(p_{j},\overline{z^{l}}_{p_{j}})\|^{2}]+4\mathbb{E}\|{\cal F}_{1}(t,\overline{z^{l}}_{t})\|^{2}\\&&+4\mathbb{E}\bigg[\int_{p_{j}}^{t}\|\Re(t-s)\|\|{\cal C}(s)v(s)\|ds\bigg]^{2}+8{\cal H}N^{2}t_{j+1}^{2{\cal H}-1}\int_{p_{j}}^{t}\mathbb{E}\|{\cal F}_{2}(s,\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})})\|^{2}_{L^{0}_{2}}ds\\ \qquad\; &\leq & 8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})\|\overline{z^{l}}_{p_{j}}\|^{2}_{\mathfrak{B}}+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})+4L_{{\cal F}_{1}}(\|\overline{z^{l}}_{t}\|^{2}_{\mathfrak{ B}}+1)\\&&+4N^{2}\|{\cal C}\|^{2}_{\infty}t_{j+1}\|v\|^{2}_{L^{2}_{\cal F}}+8{\cal H}N^{2}{t}_{j+1}^{2{\cal H}-1}\int_{p_{j}}^{t}\eta(s)\Theta_{{\cal F}_{2}}(\|\overline{z^{l}}_{\rho(s,\overline{z^{l}}_{s})}\|^{2}_{\mathfrak{B}})ds.\qquad\;\; \end{eqnarray*} $ |
For any $ t \in [0, b] $, we have
$ \begin{eqnarray*} l \lt \mathbb{E}\|{\mathfrak{F}}(z^{l})(t)\|^{2} &\leq & W^{*}+L_{{\cal E}_{j}}l^{*}+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})l^{*}+4L_{{\cal F}_{1}}l^{*}+8{\cal H}N^{2}b^{2{\cal H}-1}\Theta_{{\cal F}_{2}}(l^{*})\int_{0}^{t} \eta(s)ds, \end{eqnarray*} $ |
and hence,
$ \begin{eqnarray*} l^{*} & \lt & 2({\cal K}_{3}^{*}+{\cal L}_{\Omega}^{*})^{2}\|\,\Omega\,\|_{\mathfrak{B}}^{2}+2[{\cal K}_{2}^{*}]^{2}\big[W^{*}\\&&+L_{{\cal E}_{j}}l^{*}+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})l^{*}+4L_{{\cal F}_{1}}l^{*}+8{\cal H}N^{2}b^{2{\cal H}-1}\Theta_{{\cal F}_{2}}(l^{*})\int_{0}^{b} \eta(s)ds\big], \end{eqnarray*} $ |
where
$ \begin{eqnarray*} W^{*} & = & \max\limits_{1\leq j \leq {\cal M}}\Big\{ 8N^{2}[{\cal K}_{1}^{2}\|\Omega\|_{\mathfrak{B}}^{2}+L_{{\cal F}_{1}}(\|\Omega\|^{2}_{\mathfrak{B}}+1)]+L_{{\cal E}_{j}}\\&&+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})+4L_{{\cal F}_{1}}+4N^{2}\|{\cal C}\|^{2}_{\infty}b\|v\|^{2}_{L^{2}_{\cal F}}\Big\}. \end{eqnarray*} $ |
Dividing both sides by $ l^{*} $ and taking the limit as $ l^{*} \rightarrow \infty $, we have
$ \begin{eqnarray*} 1 \lt 2[{\cal K}_{2}^{*}]^{2}\big[L_{{\cal E}_{j}}+8N^{2}(L_{{\cal E}_{j}}+L_{{\cal F}_{1}})+4L_{{\cal F}_{1}}+8{\cal H}N^{2}b^{2{\cal H}-1}\Theta_{1}\int_{0}^{b} \eta(s)ds\big], \end{eqnarray*} $ |
which is contrary to our assumption [H6]. Hence, for some $ l > 0 $, $ {\mathfrak{F}}(\overline{\cal B}_{l}) \subset \overline{\cal B}_{l} $.
Step 2. $ {\mathfrak{F}}_{1} $ is a contraction map on $ \overline{\cal B}_{l} $.
For any $ y, z \in \overline{\cal B}_{l}, $ if $ t \in [0, t_{1}], $ then we have
$ \begin{eqnarray*} \mathbb{E}\|({\mathfrak{F}}_{1}y)(t)-({\mathfrak{F}}_{1}z)(t)\|^{2} & \leq & L_{{\cal F}_{1}}\|\overline{y}_{t}-\overline{z}_{t}\|^{2}_{\mathfrak{B}}\\ & \leq & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}} \sup\{\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}: 0 \lt s \lt t\}\\ &\leq & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\sup\limits_{s \in [0,b]}\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}\\ & = & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\sup\limits_{s \in [0,b]}\mathbb{E}\|y(s)-z(s)\|^{2},\;\;(\mbox{since}\;\overline{z} = z\;\mbox{in}\;[0,b])\\ & = & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\|y-z\|^{2}_{\cal PC}. \label{zz} \end{eqnarray*} $ |
If $ t \in (t_{j}, p_{j}] $, $ j = 1, 2, \ldots, {\cal M}, $ then we have
$ \begin{eqnarray*} \mathbb{E}\|({\mathfrak{F}}_{1}y)(t)-({\mathfrak{F}}_{1}z)(t)\|^{2} &\leq & L_{{\cal E}_{j}}\|\overline{y}_{t}-\overline{z}_{t}\|^{2}_{\mathfrak{B}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;\\&\leq & 2[{\cal K}_{2}^{*}]^{2} L_{{\cal E}_{j}} \sup\limits_{s \in [0,b]}\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}\\& = & 2[{\cal K}_{2}^{*}]^{2} L_{{\cal E}_{j}}\|y-z\|^{2}_{\cal PC}. \end{eqnarray*} $ |
Similarly, if $ t \in (p_{j}, t_{j+1}] $, $ j = 1, 2, \ldots, {\cal M}, $ then we have
$ \begin{eqnarray*} \;\;\;\;\mathbb{E}\|({\mathfrak{F}}_{1}y)(t)-({\mathfrak{F}}_{1}z)(t)\|^{2} &\leq & 2N^{2}[2\mathbb{E}\|{\cal E}_{j}(p_{j},\overline{y}_{p_{j}})-{\cal E}_{j}(p_{j},\overline{z}_{p_{j}})\|^{2}+2\mathbb{E}\|{\cal F}_{1}(p_{j},\overline{y}_{p_{j}})\qquad\qquad\\&&-{\cal F}_{1}(p_{j},\overline{z}_{p_{j}})\|^{2}]+2\mathbb{E}\|{\cal F}_{1}(t,\overline{y}_{t})-{\cal F}_{1}(t,\overline{z}_{t})\|^{2}\\\qquad\qquad\qquad\qquad\;\;\,\, & \leq & 8N^{2}[{\cal K}_{2}^{*}]^{2}L_{{\cal E}_{j}}\sup\{\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}: 0 \lt s \lt t\}\\&&+4[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}(2N^{2}+1)\sup\{\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}: 0 \lt s \lt t\} \\ &\leq & 4[{\cal K}_{2}^{*}]^{2}[2N^{2}L_{{\cal E}_{j}}+(2N^{2}+1)L_{{\cal F}_{1}}]\sup\limits_{s \in [0,b]}\mathbb{E}\|\overline{y}(s)-\overline{z}(s)\|^{2}\\ & = & 4[{\cal K}_{2}^{*}]^{2}[2N^{2}L_{{\cal E}_{j}}+(2N^{2}+1)L_{{\cal F}_{1}}]\sup\limits_{s \in [0,b]}\mathbb{E}\|y(s)-z(s)\|^{2}\\ & = & 4[{\cal K}_{2}^{*}]^{2}[2N^{2}L_{{\cal E}_{j}}+(2N^{2}+1)L_{{\cal F}_{1}}]\|y-z\|^{2}_{\cal PC}. \end{eqnarray*} $ |
For any $ t \in [0, b], $ we have
$ \begin{eqnarray*} \mathbb{E}\|({\mathfrak{F}}_{1}y)(t)-({\mathfrak{F}}_{1}z)(t)\|^{2} \leq L_{\mathfrak{F}_{1}}\|y-z\|^{2}_{\cal PC}. \end{eqnarray*} $ |
Taking supremum over $ t $
$ \begin{eqnarray*} \|{\mathfrak{F}}_{1}y-{\mathfrak{F}}_{1}z\|^{2}_{\cal PC} \leq L_{\mathfrak{F}_{1}}\|y-z\|^{2}_{\cal PC}, \end{eqnarray*} $ |
where $ L_{\mathfrak{F}_{1}} = 2[{\cal K}_{2}^{*}]^{2}\big(L_{{\cal E}_{j}}+4N^{2}L_{{\cal E}_{j}}+2(2N^{2}+1)L_{{\cal F}_{1}}\big). $ By Eq. (3.1), we see that $ L_{\mathfrak{F}_{1}} < 1. $ Hence, $ {\mathfrak{F}}_{1} $ is a contraction map on $ \overline{\cal B}_{l} $.
Step 3. We show that $ {\mathfrak{F}}_{2} $ is continuous on $ \overline{\cal B}_{l}. $
Let $ \{z^{m}\}_{m = 1}^{\infty} \subseteq \overline{\cal B}_{l} $ be a sequence such that $ z^{m} \rightarrow z $ in $ \overline{\cal B}_{l} $ as $ m \rightarrow \infty $. From axiom [A1], we have that $ (\overline{z^{m}})_{s} \rightarrow \overline{z}_{s} $ uniformly for $ s \in (-\infty, b] $ as $ m \rightarrow \infty $. By hypothese [H5] and [42, Theorem 2.2, Step-3], we have
$ \begin{eqnarray*} {\cal F}_{2}(s,\overline{z^{m}}_{\rho(s,(\overline{z^{m}})_{s})}) \rightarrow {\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})}), \end{eqnarray*} $ |
for any $ s \in [0, t] $, and since
$ \begin{eqnarray*} \mathbb{E}\|{\cal F}_{2}(s,\overline{z^{m}}_{\rho(s,(\overline{z^{m}})_{s})}) - {\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})\|^{2}_{L_{2}^{0}} \leq 2\Theta_{{\cal F}_{2}}(l^{*})\eta(s). \end{eqnarray*} $ |
For any $ t \in (p_{j}, t_{j+1}], \; j = 0, 1, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} \mathbb{E}\|({\mathfrak{F}_{2}}z^{m})(t) - ({\mathfrak{F}_{2}}z)(t)\|^{2}& = & \mathbb{E}\bigg\|\int_{p_{j}}^{t}\Re(t-s)[{\cal F}_{2}(s,\overline{z^{m}}_{\rho(s,\overline{z^{m}}_{s})})-{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})]dB^{\cal H}(s)\bigg\|^{2}\\& \leq & 2{\cal H}N^{2}t_{j+1}^{2{\cal H}-1}\int_{p_{j}}^{t}\mathbb{E}\|{\cal F}_{2}(s,\overline{z^{m}}_{\rho(s,(\overline{z^{m}})_{s})})-{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})\|^{2}_{L_{2}^{0}}ds\\& \leq & 2{\cal H}N^{2}b^{2{\cal H}-1}\int_{0}^{t}\mathbb{E}\|{\cal F}_{2}(s,\overline{z^{m}}_{\rho(s,(\overline{z^{m}})_{s})})-{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})\|^{2}_{L_{2}^{0}}ds. \end{eqnarray*} $ |
By the Lebesgue dominated convergence theorem, we have
$ \begin{eqnarray*} \|{\mathfrak{F}_{2}}z^{m} - {\mathfrak{F}_{2}}z\|^{2}_{\cal PC} \rightarrow\;\;0\;\;as\;\;m\;\;\rightarrow\;\;\infty. \end{eqnarray*} $ |
Thus, $ {\mathfrak{F}}_{2} $ is continuous.
Step 4. We show that $ \{{\mathfrak{F}}_{2}z: z \in \overline{\cal B}_{l}\} $ is equicontinuous.
Since $ \Re(t) $ is compact, which implies that the continuity of $ \Re(t) $ in $ (0, b] $. Let $ p_{j} < \epsilon < t \leq t_{j+1} $, $ j = 0, 1, \ldots, {\cal M}, $ and $ \omega > 0 $ such that $ \|\Re(\xi_{1}-s)- \Re(\xi_{2}-s) \|^{2} < \epsilon $ for every $ \xi_{1}, \xi_{2} \in (p_{j}, t_{j+1}] $ with $ |\xi_{1}-\xi_{2}| < \omega $. For each $ \; z \in \overline{\cal B}_{l} $, $ 0 < |\kappa| < \omega $ with $ t, t+\kappa \in(p_{j}, t_{j+1}] $, $ j = 0, 1, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} \mathbb{E}\|({\mathfrak{F}}_{2}z)(t+\kappa) - ({\mathfrak{F}}_{2}z)(t)\|^{2} & \leq & 4\mathbb{E}\bigg\|\int_{t}^{t+\kappa}\Re(t+\kappa-s){\cal C}(s)v(s)ds\bigg\|^{2}\\&&+4\mathbb{E}\bigg\|\int_{p_{j}}^{t}[\Re(t+\kappa-s)-\Re(t-s)]{\cal C}(s)v(s)ds\bigg\|^{2}\\ &&+4\mathbb{E}\bigg\|\int_{t}^{t+\kappa}\Re(t+\kappa-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s)\bigg\|^{2}\\&&+4\mathbb{E}\bigg\|\int_{p_{j}}^{t}[\Re(t+\kappa-s)-\Re(t-s)]{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s)\bigg\|^{2}\\ & = & 4[\chi_{1}+\chi_{2}], \end{eqnarray*} $ |
where
$ \begin{eqnarray*} \chi_{1} &\leq & \mathbb{E}\bigg[\int_{t}^{t+\kappa}\|\Re(t+\kappa-s)\|\|{\cal C}(s)v(s)\|ds\bigg]^{2}\\&&+\mathbb{E}\bigg[\int_{p_{j}}^{t}\|\Re(t+\kappa-s)-\Re(t-s)\|\|{\cal C}(s)v(s)\|ds\bigg]^{2}\\ &\leq & N^{2}\kappa\|{\cal C}\|_{\infty}^{2}\mathbb{E}\int_{t}^{t+\kappa}\|v(s)\|^{2}_{\cal T}ds\\&&+\|{\cal C}\|_{\infty}^{2}t_{j+1}\mathbb{E}\int_{p_{j}}^{t} \|\Re(t+\kappa-s)-\Re(t-s)\|^{2}\|v(s)\|^{2}_{\cal T}ds\\&\leq & N^{2}\kappa\|{\cal C}\|_{\infty}^{2}\mathbb{E}\int_{t}^{t+\kappa}\|v(s)\|^{2}_{\cal T}ds+\epsilon\|{\cal C}\|_{\infty}^{2}t_{j+1}\mathbb{E}\int_{p_{j}}^{t}\|v(s)\|^{2}_{\cal T}ds,\qquad\; \\\chi_{2} & \leq & 2{\cal H}N^{2}t_{j+1}^{2{\cal H}-1}\int_{t}^{t+\kappa}\mathbb{E}\|{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})\|^{2}_{L_{2}^{0}}ds\\&&+2{\cal H}t_{j+1}^{2{\cal H}-1}\int_{p_{j}}^{t}\|\Re(t+\kappa-s)-\Re(t-s)\|^{2}\mathbb{E}\|{\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})\|^{2}_{L_{2}^{0}}ds\\ &\leq & 2{\cal H}N^{2}t_{j+1}^{2{\cal H}-1}\Theta_{{\cal F}_{2}}(l^{*})\int_{t}^{t+\kappa}\eta(s)ds+2\epsilon{\cal H}t_{j+1}^{2{\cal H}-1}\Theta_{{\cal F}_{2}}(l^{*})\int_{p_{j}}^{t}\eta(s)ds. \end{eqnarray*} $ |
We conclude that $ \mathbb{E}\|({\mathfrak{F}}_{2}z)(t+\kappa) - ({\mathfrak{F}}_{2}z)(t)\|^{2}\rightarrow\; 0 $ as $ \kappa \rightarrow 0 $ and $ \epsilon $ is sufficiently small. Hence, $ \{{\mathfrak{F}}_{2}z: z \in \overline{\cal B}_{l}\} $ is equicontinuous. Also, clearly $ \{{\mathfrak{F}}_{2}z: z \in \overline{\cal B}_{l}\} $ is uniformly bounded.
Step 5. The set $ {\cal Q}(t) = \{({\mathfrak{F}}_{2}z)(t): z \in \overline{\cal B}_{l}\}, \; t \in {\cal J}_{1} $ is relatively compact in $ Z. $
Clearly, $ {\cal Q}(0) = \{0\} $ is compact. Let $ \xi $ is real number and $ t \in (p_{j}, t_{j+1}], \; j = 0, 1, \ldots, {\cal M}, $ be fixed with $ 0 < \xi < t. $ For $ z \in \overline{\cal B}_{l} $, we define
$ \begin{equation*} ({\mathfrak{F}}_{2}^{\xi}z)(t) = \begin{cases} \int_{0}^{t-\xi}\Re(t-s){\cal C}(s)v(s)ds+\int_{0}^{t-\xi}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{}\quad t \in [0,t_{1}],\,j = 0,\\ 0, & \text{}\quad t \in (t_{j},p_{j}],\,j \geq 1,\\ \int_{p_{j}}^{t-\xi}\Re(t-s){\cal C}(s)v(s)ds+\int_{p_{j}}^{t-\xi}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s),& \text{}\quad t \in (p_{j},t_{j+1}],\,j \geq 1. \end{cases} \end{equation*} $ |
Since $ \Re(t) $ is compact, the set $ {\cal Q}^{\xi}(t) = \{({\mathfrak{F}}_{2}^{\xi}z)(t): z \in \overline{\cal B}_{l}\} $ is relatively compact in $ Z $ for every $ \xi $. For $ t \in (p_{j}, t_{j+1}], \; j = 0, 1, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} &&\mathbb{E}\|({\mathfrak{F}}_{2}z)(t)-({\mathfrak{F}}_{2}^{\xi}z)(t)\|^{2}\\&&\qquad\qquad \leq 2\mathbb{E}\bigg\|\int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v(s)ds-\int_{p_{j}}^{t-\xi}\Re(t-s){\cal C}(s)v(s)ds\bigg\|^{2}\\&&\qquad\qquad\;\;\; + 2\mathbb{E}\bigg\|\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s)-\int_{p_{j}}^{t-\xi}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s)\bigg\|^{2}\qquad\, \\ &&\qquad\quad\; \leq 2\mathbb{E}\bigg\|\int_{t-\xi}^{t}\Re(t-s){\cal C}(s)v(s)ds\bigg\|^{2}+2\mathbb{E}\bigg\|\int_{t-\xi}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z}_{\rho(s,\overline{z}_{s})})dB^{\cal H}(s)\bigg\|^{2}\\&&\qquad\quad\; \leq 2N^{2}\xi\|{\cal C}\|_{\infty}^{2}\mathbb{E}\int_{t-\xi}^{t}\|v(s)\|^{2}_{\cal T}ds+4{\cal H}N^{2}t_{j+1}^{2{\cal H}-1}\Theta_{{\cal F}_{2}}(l^{*})\int_{t-\xi}^{t}\eta(s)ds\;\;\rightarrow\;\;0\;\;\mbox{as}\;\;\xi \rightarrow 0.\;\;\; \end{eqnarray*} $ |
The relatively compact set $ {\cal Q}^{\xi}(t) $ and set $ {\cal Q}(t) $ are arbitrarily close. Hence, $ {\cal Q}(t) = \{({\mathfrak{F}}_{2}z)(t): z \in \overline{\cal B}_{l}\} $ is relatively compact in $ Z. $ By using step 3–5 along with Arzela-Ascoli theorem, we obtain that the $ {\mathfrak{F}}_{2} $ is a completely continuous operator. Hence, by Krasnoselskii's theorem [43], we realize that the operator $ \mathfrak{F}_{1}+\mathfrak{F}_{2} $ has a fixed point, which is a mild solution of the stochastic system (1.1).
In this section, we prove the existence of optimal controls for the stochastic system. Let $ z^{v} $ be the mild solution of the stochastic system (1.1) corresponding to the control $ v \in {\cal U}_{ad} $. We consider the Lagrange problem $ ({\cal LP}): $ Find an optimal state-control pair $ (z^{*}, v^{*}) \in {\cal BPC}\times {\cal U}_{ad} $ such that
$ \mathfrak{J}(z^{*},v^{*}) \leq \mathfrak{J}(z^{v},v)\;\;\mbox{for all}\;v \in {\cal U}_{ad}, $ |
where
$ \begin{eqnarray*} \mathfrak{J}(z^{v},v) = \mathbb{E}\int_{0}^{b}\mathfrak{M}(t,z_{t}^{v},z^{v}(t),v(t))dt. \end{eqnarray*} $ |
For the existence of optimal controls, we shall introduce the following hypotheses
[H7]: The function $ \mathfrak{M} : {\cal J}_{1}\times {\mathfrak{B}} \times Z \times {\cal T} \rightarrow \mathbb{R}\cup\{\infty\} $ satisfies:
(a) $ \mathfrak{M} $ is Borel measurable
(b) $ \mathfrak{M}(t, z_{1}, z_{2}, \cdot) $ is convex function on $ {\cal T} $ for each $ z_{1} \in {\mathfrak{B}}, $ $ z_{2} \in Z $ and almost all $ t \in {\cal J}_{1} $.
(c) $ \mathfrak{M}(t, \, \cdot\, ,\cdot\, ,\cdot\, ) $ is sequentially lower semi-continuous on $ {\mathfrak{B}}\times Z \times {\cal T} $ for almost all $ t \in {\cal J}_{1}. $
(d) There exist constants $ \omega_{1}, \omega_{2} \geq 0, $ $ \omega_{3} > 0 $ and $ \Phi $ is non-negative function in $ L^{1}({\cal J}_{1}, \mathbb{R}) $ such that
$ \mathfrak{M}(t,z_{1},z_{2},v) \geq \Phi(t) + \omega_{1}\|z_{1}\|_{\mathfrak{B}}+ \omega_{2}\|z_{2}\| + \omega_{3}\|v\|^{2}_{\cal T}. $ |
[H8]: The operator $ {\cal C} $ is strongly continuous.
Theorem 4.1. Assume that the presumptions [H1]–[H8] are fulfilled. Then the problem $ ({\cal LP}) $ admits at least one optimal control pair on $ {\cal BPC} \times {\cal U}_{ad}. $
Proof. If $ \inf\{\mathfrak{J}(z^{v}, v) : v \in {\cal U}_{ad}\} = +\infty $, there is nothing to prove. Next, we choose $ \inf\{\mathfrak{J}(z^{v}, v) : v \in {\cal U}_{ad}\} = \epsilon < +\infty $ and using the hypotheses [H7], we obtain
$ \begin{eqnarray*} \mathfrak{J}(z^{v},v) &\geq & \int_{0}^{b} \Phi(t)dt+ \omega_{1}\int_{0}^{b} \|z^{v}_{t}(t)\|_{\mathfrak{B}}dt+\omega_{2}\int_{0}^{b} \|z^{v}(t)\|dt+ \omega_{3}\int_{0}^{b} \|v(t)\|_{\cal T}^{2}\;\geq\;\epsilon\; \gt \;-\infty. \end{eqnarray*} $ |
By definition of infimum, there exists a minimizing sequence $ \{(z^{k}, v^{k})\} \subset \mathscr{R}_{ad} $, where $ \mathscr{R}_{ad} $ = $ \{\, (z, v)\, :\, z $ be the mild solution of the stochastic system (1.1) corresponding to the control $ v \in {\cal U}_{ad}\} $ such that
$ \mathfrak{J}(z^{k},v^{k}) \rightarrow \epsilon\;\mbox{as}\;k\,\rightarrow\,+\infty . $ |
Since $ \{v^{k}\} $ $ \subseteq {\cal U}_{ad} $, $ \{v^{k}\} $ is bounded in the space $ L^{2}_{\cal F}({\cal J}_{1}, {\cal T}) $, then exists a subsequence, relabeled as $ \{v^{k}\}, $ and $ v^{*} \in L^{2}_{\cal F}({\cal J}_{1}, {\cal T}) $ such that $ v^{k} $ converges weakly to $ v^{*} $ in $ L^{2}_{\cal F}({\cal J}_{1}, {\cal T}) $ as $ k \rightarrow \infty $. Since $ {\cal U}_{ad} $ is convex and closed, then by Marzur Lemma, we have $ v^{*} \in {\cal U}_{ad} $.
Let $ z^{k} $ be the sequence of mild solutions of the stochastic system (1.1) corresponding to $ v^{k} $ and $ z^{k} $ fulfills the consecutive integral equations
$ \begin{equation*} z^{k}(t) = \begin{cases} \Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,\overline{z^{k}}_{t})\\+\int_{0}^{t}\Re(t-s){\cal C}(s)v^{k}(s)ds+\int_{0}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{k}}_{\rho(s,\overline{z^{k}}_{s})})dB^{\cal H}(s),& \text{}\; t \in [0,t_{1}],\,j = 0,\\ {\cal E}_{j}(t,\overline{z^{k}}_{t}), & \text{}\; t \in (t_{j},p_{j}],\,j \geq 1,\\ \Re(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z^{k}}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z^{k}}_{p_{j}})]+{\cal F}_{1}(t,\overline{z^{k}}_{t})\\+\int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v^{k}(s)ds+\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{k}}_{\rho(s,\overline{z^{k}}_{s})})dB^{\cal H}(s),& \text{}\; t \in (p_{j},t_{j+1}],\,j \geq 1. \end{cases} \end{equation*} $ |
Let $ {\cal F}_{2}^{k}(s)\equiv {\cal F}_{2}(s, \overline{z^{k}}_{\rho(s, \overline{z^{k}}_{s})}) $. Then, for each $ z^{k} \in \overline{\cal B}_{l}\subset {\cal BPC}, $ by hypotheses [H5], we obtain that $ {\cal F}_{2}^{k}:{\cal J}_{1} \rightarrow L_{2}^{0}(Y, Z) $ is bounded operator. Hence, $ {\cal F}_{2}^{k}(\cdot)\in L^{2}({\cal J}_{1}, L_{2}^{0}(Y, Z)) $. Furthermore, $ \{{\cal F}_{2}^{k}(\cdot)\} $ is bounded in $ L^{2}({\cal J}_{1}, L_{2}^{0}(Y, Z)) $, there are subsequence, relabeled as $ \{{\cal F}_{2}^{k}(\cdot)\} $ and $ {\cal F}_{2}^{*}(\cdot) \in L^{2}({\cal J}_{1}, L_{2}^{0}(Y, Z)) $ such that $ {\cal F}_{2}^{k}(\cdot) \xrightarrow[]{w} {\cal F}_{2}^{*}(\cdot)\; \mbox{in}\; L^{2}({\cal J}_{1}, L_{2}^{0}(Y, Z))\; \mbox{as}\; k\; \rightarrow\; \infty. $
Next, we consider the following stochastic system
$ \begin{equation} \begin{cases} d\mathfrak{D}(t,z_{t}) = {\cal A}\big[\mathfrak{D}(t,z_{t})+\int_{0}^{t}{\cal G}(t-s)\mathfrak{D}(t,z_{s})ds\big]dt+{\cal C}(t)v^{*}(t)dt+{\cal F}_{2}^{*}(t)dB^{\cal H}(t)\\ \qquad\qquad\qquad\quad\;t\in (p_{j},t_{j+1}],\;j = 0,1,\ldots,{\cal M}, & \text{}\\ z(t) = {\cal E}_{j}(t,z_{t}),\,\,\,t\in (t_{j},p_{j}],\;j = 1,2,\ldots,{\cal M}, & \text{}\\ z_{0} = \Omega\in \mathfrak{B}.& \text{} \end{cases} \end{equation} $ | (4.1) |
By Theorem 3.1, we know that the stochastic system (4.1) has a mild solution
$ \begin{equation*} z^{*}(t) = \begin{cases} \Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,\overline{z^{*}}_{t})\\+\int_{0}^{t}\Re(t-s){\cal C}(s)v^{*}(s)ds+\int_{0}^{t}\Re(t-s){\cal F}_{2}^{*}(s)dB^{\cal H}(s),& \text{}\qquad\; t \in [0,t_{1}],\,j = 0,\\ {\cal E}_{j}(t,\overline{z^{*}}_{t}), & \text{}\qquad\; t \in (t_{j},p_{j}],\,j \geq 1,\\ \Re(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z^{*}}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z^{*}}_{p_{j}})]+{\cal F}_{1}(t,\overline{z^{*}}_{t})\\+\int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v^{*}(s)ds+\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}^{*}(s)dB^{\cal H}(s),& \text{}\qquad\; t \in (p_{j},t_{j+1}],\,j \geq 1. \end{cases} \end{equation*} $ |
For any $ t \in [0, t_{1}] $, we have
$ \begin{eqnarray*} \mathbb{E}\|z^{k}(t) - z^{*}(t)\|^{2} \leq 3[\Upsilon_{1}^{k}(t)+\Upsilon_{2}^{k}(t)+\Upsilon_{3}^{k}(t)],\qquad\qquad \end{eqnarray*} $ |
where
$ \begin{eqnarray*} \Upsilon_{1}^{k}(t) & = & \mathbb{E}\|{\cal F}_{1}(t,\overline{z^{k}}_{t})-{\cal F}_{1}(t,\overline{z^{*}}_{t})\|^{2}\\ & \leq & L_{{\cal F}_{1}}\|\overline{z^{k}}_{t}-\overline{z^{*}}_{t}\|^{2}_{\mathfrak{B}}\\ &\leq & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}} \sup\{\mathbb{E}\|\overline{z^{k}}(s)-\overline{z^{*}}(s)\|^{2}: 0 \lt s \lt t\}\\ &\leq & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\sup\limits_{s \in [0,b]}\mathbb{E}\|\overline{z^{k}}(s)-\overline{z^{*}}(s)\|^{2}\\ & = & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\sup\limits_{s \in [0,b]}\mathbb{E}\|z^{k}(s)-z^{k}(s)\|^{2},\;\;(\mbox{since}\;\overline{z} = z\;\mbox{in}\;[0,b])\\ & = & 2[{\cal K}_{2}^{*}]^{2}L_{{\cal F}_{1}}\|z^{k}-z^{*}\|^{2}_{\cal PC},\\ \Upsilon_{2}^{k}(t)& = &\mathbb{E}\bigg\|\int_{0}^{t} {\cal R}(t-s){\cal C}(s) [v^{k}(s)-v^{*}(s)]ds\bigg\|^{2}\\ & \leq & N^{2}t_{1}\mathbb{E}\int_{0}^{t}\|{\cal C}(s)v^{k}(s)-{\cal C}(s)v^{*}(s)\|^{2}ds,\\ \Upsilon_{3}^{k}(t)& = &\mathbb{E}\bigg\|\int_{0}^{t}{\cal R}(t-s)[{\cal F}_{2}^{k}(s)-{\cal F}_{2}^{*}(s)]dB^{H}(s)\bigg\|^{2}\\ & \leq & 2{\cal H}t_{1}^{2{\cal H}-1}\int_{0}^{t}\mathbb{E}\|\Re(t-s)[{\cal F}_{2}^{k}(s)-{\cal F}_{2}^{*}(s)]\|^{2}_{L^{0}_{2}}ds. \end{eqnarray*} $ |
For any $ t \in (t_{j}, p_{j}] $, $ j = 1, 2, \ldots, {\cal M} $, we have
$ \begin{eqnarray*} \mathbb{E}\|z^{k}(t) - z^{*}(t)\|^{2} \leq 2[{\cal K}_{2}^{*}]^{2} L_{{\cal E}_{j}}\|z^{k}-z^{*}\|^{2}_{\cal PC}. \end{eqnarray*} $ |
For any $ t \in (p_{j}, t_{j+1}] $, $ j = 1, 2, \ldots, {\cal M}, $ we have
$ \begin{eqnarray*} \mathbb{E}\|z^{k}(t) - z^{*}(t)\|^{2} \leq 4[\Psi_{1}^{k}(t)+\Psi_{2}^{k}(t)+\Psi_{3}^{k}(t)+\Psi_{4}^{k}(t)], \end{eqnarray*} $ |
where
$ \begin{eqnarray*} \Psi_{1}^{k}(t)& = & 2N^{2}\mathbb{E}\|{\cal E}_{j}(p_{j},\overline{z^{k}}_{p_{j}})-{\cal E}_{j}(p_{j},\overline{z^{*}}_{p_{j}})\|^{2}\qquad\qquad\quad\\ & \leq & 4[{\cal K}_{2}^{*}]^{2}N^{2} L_{{\cal E}_{j}}\|z^{k}-z^{*}\|^{2}_{\cal PC},\\ \Psi_{2}^{k}(t)& = &2N^{2}\mathbb{E}\|{\cal F}_{1}(p_{j},\overline{z^{k}}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z^{*}}_{p_{j}})\|^{2}\\&&+\mathbb{E}\|{\cal F}_{1}(t,\overline{z^{k}}_{t})-{\cal F}_{1}(t,\overline{z^{*}}_{t})\|^{2}\\ &\leq & (2N^{2}+1)L_{{\cal F}_{1}}\|\overline{z^{k}}_{t}-\overline{z^{*}}_{t}\|^{2}_{\mathfrak{B}}\\&\leq & 2[{\cal K}_{2}^{*}]^{2}(2N^{2}+1)L_{{\cal F}_{1}}\|z^{k}-z^{*}\|^{2}_{\cal PC}, \end{eqnarray*} $ |
$ \begin{eqnarray*} \Psi_{3}^{k}(t)& = &\mathbb{E}\bigg\|\int_{p_{j}}^{t} {\cal R}(t-s){\cal C}(s) [v^{k}(s)-v^{*}(s)]ds\bigg\|^{2}\\ & \leq & N^{2}t_{j+1}\mathbb{E}\int_{p_{j}}^{t}\|{\cal C}(s)v^{k}(s)-{\cal C}(s)v^{*}(s)\|^{2}ds,\qquad\,\, \end{eqnarray*} $ |
$ \begin{eqnarray*} \Psi_{4}^{k}(t)& = &\mathbb{E}\bigg\|\int_{p_{j}}^{t}{\cal R}(t-s)[{\cal F}_{2}^{k}(s)-{\cal F}_{2}^{*}(s)]dB^{H}(s)\bigg\|^{2}\\ & \leq & 2{\cal H}t_{j+1}^{2{\cal H}-1}\int_{p_{j}}^{t}\mathbb{E}\|\Re(t-s)[{\cal F}_{2}^{k}(s)-{\cal F}_{2}^{*}(s)]\|^{2}_{L^{0}_{2}}ds. \end{eqnarray*} $ |
For $ t \in [0, b] $, we have
$ \begin{eqnarray*} \mathbb{E}\|z^{k}(t) - z^{*}(t)\|^{2} &\leq & L_{0}\|z^{k}-z^{*}\|^{2}_{\cal PC}+ \Phi_{1}^{k}(t)+\Phi_{2}^{k}(t), \end{eqnarray*} $ |
where
$ \begin{eqnarray*} L_{0}& = & \max\limits_{1 \leq j \leq {\cal M}}\Big[16[{\cal K}_{2}^{*}]^{2}N^{2}L_{{\cal E}_{j}}+8[{\cal K}_{2}^{*}]^{2}(2N^{2}+1)L_{{\cal F}_{1}}+2[{\cal K}_{2}^{*}]^{2} L_{{\cal E}_{j}}\Big] \lt 1,\\ \Phi_{1}^{k}(t)& = & 4N^{2}b\,\mathbb{E}\int_{0}^{t}\|{\cal C}(s)v^{k}(s)-{\cal C}(s)v^{*}(s)\|^{2}ds,\\ \Phi_{2}^{k}(t)& = &8{\cal H}b^{2{\cal H}-1}\int_{0}^{t}\mathbb{E}\|\Re(t-s)[{\cal F}_{2}^{k}(s)-{\cal F}_{2}^{*}(s)]\|^{2}_{L^{0}_{2}}ds. \end{eqnarray*} $ |
Thus, we have
$ \begin{eqnarray*} \|z^{k} - z^{*}\|^{2}_{\cal PC} &\leq & \dfrac{\Phi_{1}^{k}(t)+\Phi_{2}^{k}(t)}{1-L_{0}}. \end{eqnarray*} $ |
By [H8], we have
$ \begin{eqnarray} \|{\cal C}v^{k}-{\cal C}v^{*}\|^{2}_{L^{2}({\cal J}_{1},Z)}\rightarrow 0\;\mbox{as}\;k \rightarrow \infty. \end{eqnarray} $ | (4.2) |
By the Lebesgue dominated convergence theorem and Eq. (4.2), we have
$ \begin{eqnarray*} \Phi_{1}^{k}(t),\;\Phi_{2}^{k}(t) \rightarrow 0\;\mbox{as}\;k \rightarrow \infty. \end{eqnarray*} $ |
Hence,
$ \begin{eqnarray*} z^{k}\;\rightarrow\;z^{*}\;\;\mbox{in}\;\;{\cal BPC}\;\;\mbox{as}\;\;k\;\;\rightarrow\;\;\infty. \end{eqnarray*} $ |
By [H5], we obtain
$ {\cal F}_{2}^{k}(\,\cdot\,) \rightarrow {\cal F}_{2}(\,\cdot\,,\overline{z^{*}}_{\rho(\,\cdot\,,\overline{z^{*}}_{\,\cdot\,})})\;\;\mbox{in}\;\;{\cal BPC}\;\;\mbox{as}\;\;k\;\;\rightarrow\;\;\infty. $ |
Limit is unique, so we obtain
$ {\cal F}_{2}^{*}(t) = {\cal F}_{2}(t,\overline{z^{*}}_{\rho(t,\overline{z^{*}}_{t})}). $ |
Thus, $ z^{*} $ can be given
$ \begin{equation*} z^{*}(t) = \begin{cases} \Re(t)[\Omega(0)-{\cal F}_{1}(0,\Omega)]+{\cal F}_{1}(t,\overline{z^{*}}_{t})\\+\int_{0}^{t}\Re(t-s){\cal C}(s)v^{*}(s)ds+\int_{0}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{*}}_{\rho(s,\overline{z^{*}}_{s})})dB^{\cal H}(s),& \text{}\; t \in [0,t_{1}],\,j = 0,\\ {\cal E}_{j}(t,\overline{z^{*}}_{t}), & \text{}\; t \in (t_{j},p_{j}],\,j \geq 1,\\ \Re(t-p_{j})[{\cal E}_{j}(p_{j},\overline{z^{*}}_{p_{j}})-{\cal F}_{1}(p_{j},\overline{z^{*}}_{p_{j}})]+{\cal F}_{1}(t,\overline{z^{*}}_{t})\\+\int_{p_{j}}^{t}\Re(t-s){\cal C}(s)v^{*}(s)ds+\int_{p_{j}}^{t}\Re(t-s){\cal F}_{2}(s,\overline{z^{*}}_{\rho(s,\overline{z^{*}}_{s})})dB^{\cal H}(s),& \text{}\; t \in (p_{j},t_{j+1}],\,j \geq 1. \end{cases} \end{equation*} $ |
Since $ {\cal BPC} \hookrightarrow L^{1}({\cal J}_{1}, Z) $, by using the [H7] and Balder's theorem [44], we have
$ \begin{eqnarray*} &&\epsilon \leq \mathfrak{J}(z^{*},v^{*}) = \mathbb{E}\int_{0}^{b} \mathfrak{M}(t,z_{t}^{*},z^{*},v^{*})dt\leq\lim\limits_{k\rightarrow \infty}\mathbb{E}\int_{0}^{b} \mathfrak{M}(t,z_{t}^{k},z^{k},v^{k})dt = \epsilon, \end{eqnarray*} $ |
which shows that $ \mathfrak{J} $ attains its infimum at $ (z^{*}, v^{*}) \in {\cal BPC}\times{\cal U}_{ad}. $
Consider the following non-instantaneous impulsive stochastic partial neutral integro-differential control system driven by fBm with state-dependent delay:
$ \begin{eqnarray} && d\mathfrak{D}(t,\mu_{t})(\varepsilon) = \dfrac{\partial^{2}}{\partial \varepsilon^{2}}\bigg[\mathfrak{D}(t,\mu_{t})(\varepsilon)+\int_{0}^{t}\mathfrak{S}(t-s)\mathfrak{D}(s,\mu_{s})(\varepsilon)ds\bigg]dt\\&&\qquad\qquad\quad\;\;+\int_{0}^{1}\mathfrak{K}(\varepsilon,s)v(s,t)ds\,dt+\int_{-\infty}^{t}\omega_{3}(t,s-t,\varepsilon,\mu(s-\rho_{1}(t)\rho_{2}(\|\mu(t)\|),\varepsilon))ds\,dB^{\cal H}(t),\\&&\qquad\qquad\quad\;\; \;v\,\in\,{\cal U}_{ad},\;(t,\varepsilon )\in \cup_{j = 0}^{\cal M}(p_{j},t_{j+1}] \times [0,\pi],\\ &&\quad\mu(t,\varepsilon) = \int_{-\infty}^{t} \overline{\omega_{j}}(s-t,\varepsilon)\mu(s,\varepsilon)ds,\;\;(t,\varepsilon )\in \cup_{j = 1}^{\cal M}(t_{j},p_{j}] \times [0,\pi]\\ &&\quad\mu(t,0) = 0\;\; = \mu(t,\pi),\\ &&\quad\mu(s,\varepsilon) = \Omega(s,\varepsilon),\;(s,\varepsilon) \in (-\infty,0] \times [0,\pi], \end{eqnarray} $ | (5.1) |
with cost functional as
$ \begin{eqnarray*} \mathfrak{J}(\mu,v) = \mathbb{E}\int_{0}^{1}\int_{0}^{\pi}\int_{-\infty}^{0}\|\mu(t+s,\varepsilon)\|^{2}dsd\varepsilon dt+\mathbb{E}\int_{0}^{1}\int_{0}^{\pi}\|\mu(t,\varepsilon)\|^{2}d\varepsilon dt+\mathbb{E}\int_{0}^{1}\int_{0}^{\pi}\|v(t,\varepsilon)\|^{2}_{\cal T}d\varepsilon dt, \end{eqnarray*} $ |
where $ 0 = t_{0} = p_0 < t_1 < p_1 < \cdots < t_{\cal M} < p_{\cal M} < t_{{\cal M}+1} = b = 1, $ $ \mathfrak{K}:[0, \pi]\times[0, 1] $ is continuous and $ B^{\cal H} $ is a fBm with the Hurst index $ 1/2 < {\cal H} < 1. $ In this system
$ \begin{eqnarray*} \mathfrak{D}(t,\mu_{t})(\varepsilon) = \mu(t,\varepsilon)-\int_{-\infty}^{t}\omega_{1}(s-t)\mu(s,\varepsilon)ds. \end{eqnarray*} $ |
Consider the space $ Z = {\cal T} = L^{2}[0, \pi] $ and $ {\cal A}:D({\cal A}) \subset Z \rightarrow Z $ by $ {\cal A}\theta = \theta'' $ and domain of $ {\cal A} $ is defined as
$ \begin{eqnarray} D({\cal A}) = \{\theta\,\in\,Z :\;\theta,\;\theta'\;\mbox{are absolutely continuous, } \theta''\,\in\,Z,\;\theta(0) = \theta(\pi) = 0\}. \end{eqnarray} $ | (5.2) |
Then $ {\cal A} $ generates a $ C_{0} $-semigroup $ \Re(t) $ which is compact, self-adjoint. And there exist normalized set $ \theta_{n}(v) = \sqrt{2/\pi}\, sin\, (nv), \; n \in \mathbb{N} $ of eigenvectors of $ {\cal A} $ corresponding to eigenvalues $ n^{2}, \; n\in\mathbb{N}. $ Since the resolvent operator $ \Re(t) $ is compact, there exists a constant $ N > 0 $ such that $ \|\Re(t)\| \leq N $, then the hypotheses [H1] is fulfilled. Next, we define the admissible control set $ {\cal U}_{ad} = \{v(\, \cdot\, ,\varepsilon)|[0, 1] \rightarrow {\cal T} $ is measurable, $ {\cal F}_{t} $-adapted stochastic processes, and $ \|v\|_{L^{2}_{\cal F}} \leq \alpha, \alpha > 0\} $.
Let $ l \geq 0 $, $ 1 \leq q < \infty $, $ \Lambda:(-\infty, -l]\rightarrow \mathbb{R} $, be a measurable and non-negative function. We denote by $ {\cal PC}_{l}\times L^{q}(\Lambda, Z) $ the set consists of all classes of functions $ \Omega:(-\infty, 0] \rightarrow Z $ such that $ \Omega|_{[-l, 0]}\in {\cal PC}([-l, 0], Z) $, $ \Lambda\|\Omega\|^{q} $ is Lebesgue integrable on $ (-\infty, -l) $ and $ \Omega(\cdot) $ is Lebesgue measurable on $ (-\infty, -l) $ with norm
$ \|\Omega\|_{\mathfrak{B}} = \sup\{\|\Omega(\kappa)\|\,:\,-l \leq \kappa \leq 0\}+\Big(\int_{-\infty}^{-l} \Lambda(\kappa)\;\|\Omega(\kappa)\|^{q}d\kappa\Big)^{1/q}. $ |
The space $ {\cal PC}_{0}\times L^{2}(\Lambda, Z) $ satisfies the axioms [A1]–[A3] with choice $ {\cal K}_{1} = 1 $, $ {\cal K}_{3}(t) = \gamma(-t)^{1/2} $, $ {\cal K}_{2}(t) = 1+\Big(\int_{-t}^{0} \Lambda(\kappa)d\kappa\Big)^{1/2} $, for $ t \geq 0 $. To get points of interest about the phase space, see [21,38].
Let $ \eta(\kappa)(\varepsilon) = \eta(\kappa, \varepsilon) $, $ (\kappa, \varepsilon) \in (-\infty, 0]\times [0, \pi]. $ Set
$ \mu(t)(\varepsilon) = \mu(t,\varepsilon),\;\;\rho(t,\eta) = \rho_{1}(t)\rho_{2}(\|\eta(0)\|), $ |
we have
$ \begin{eqnarray*} &&{\cal F}_{1}(t,\eta)(\varepsilon) = \int_{-\infty}^{0}\omega_{1}(\kappa)\eta(\kappa)(\varepsilon)d\kappa,\\&&{\cal F}_{2}(t,\eta)(\varepsilon) = \int_{-\infty}^{0}\omega_{2}(t,\kappa,\varepsilon,\eta(\kappa)(\varepsilon))d\kappa,\\&& {\cal C}(t)v(t)(\varepsilon) = \int_{0}^{1}\mathfrak{K}(\varepsilon,s)v(s,t)ds,\\&&{\cal E}_{j}(t,\eta)(\varepsilon) = \int_{-\infty}^{0}\overline{\omega_{j}}(\kappa,\varepsilon)\eta(\kappa)(\varepsilon)d\kappa,\;j = 1,2,\ldots,{\cal M}. \end{eqnarray*} $ |
Moreover, we assume that
1. $ \rho_{i}:[0, \infty)\rightarrow [0, \infty), \; i = 1, 2, $ are continuous functions.
2. $ \omega_{1} : \mathbb{R} \rightarrow \mathbb{R} $ is continuous function, and
$ l_{{\cal F}_{1}} = \bigg(\int_{-\infty}^{0}\dfrac{(\omega_{1}(s))^{2}}{\Lambda(s)}ds\bigg)^{1/2} \lt \infty. $ |
3. There exist continuous functions $ a_{31}, a_{32}:\mathbb{R}\rightarrow \mathbb{R} $ such that continuous function $ \omega_{3}:\mathbb{R}^{4} \rightarrow \mathbb{R} $ satisfies the conditions
$ |\omega_{3}(t,s,\varepsilon,y)| \leq a_{31}(t)a_{32}(s)|y|,\;\;(t,s,\varepsilon,y) \in \mathbb{R}^{4},\;\;\mbox{with}\;\;l_{{\cal F}_{2}} = \bigg(\int_{-\infty}^{0}\dfrac{(a_{32}(s))^{2}}{\Lambda(s)}ds\bigg)^{1/2} \lt \infty. $ |
4. There exist continuous functions $ c_{j}:\mathbb{R}\rightarrow \mathbb{R} $ such that continuous functions $ \overline{\omega_{j}} : \mathbb{R}^{2} \rightarrow \mathbb{R}, \; j = 1, 2, \ldots, {\cal M} $ satisfies the conditions
$ |\overline{\omega_{j}}(s,\varepsilon)| \leq c_{j}(s),\;(s,\varepsilon)\in \mathbb{R}^{2},\;\;\mbox{with}\;\;l_{{\cal E}_{j}} = \bigg(\int_{-\infty}^{0}\dfrac{(c_{j}(s))^{2}}{\Lambda(s)}ds\bigg)^{1/2} \lt \infty. $ |
From the above facts, we obtain
$ \begin{eqnarray*} \mathbb{E}\|{\cal F}_{1}(t,\eta)\|^{2} & = &\mathbb{E}\bigg[\bigg(\int_{0}^{\pi}\bigg(\int_{-\infty}^{0} \omega_{1}(\kappa)\eta(\kappa)(\varepsilon)d\kappa \bigg)^{2}d\varepsilon\bigg)^{1/2}\bigg]^{2}\qquad\qquad\qquad\qquad\\ &\leq & \mathbb{E}\bigg[\bigg(\int_{-\infty}^{0}\dfrac{(\omega_{1}(\kappa))^{2}}{\Lambda(\kappa)}d\kappa\bigg)^{1/2}\bigg(\int_{-\infty}^{0} \Lambda(\kappa)\|\eta(\kappa)\|^{2}d\kappa \bigg)^{1/2}\bigg]^{2}\\ &\leq &\bigg[l_{{\cal F}_{1}}\bigg(\|\eta(0)\|+\bigg(\int_{-\infty}^{0} \Lambda(\kappa)\|\eta(\kappa)\|^{2}d\kappa\bigg)^{1/2}\bigg)\bigg]^{2}\\ & = & L_{{\cal F}_{1}}\|\eta\|_{\mathfrak{B}}^{2}, \end{eqnarray*} $ |
where $ L_{{\cal F}_{1}} = [l_{{\cal F}_{1}}]^{2}. $
$ \begin{eqnarray*} &&\mathbb{E}\|{\cal F}_{1}(t,\eta_{1})-{\cal F}_{1}(t,\eta_{2})\|^{2}\\&&\qquad\qquad\qquad\qquad\quad = \mathbb{E}\bigg[\bigg(\int_{0}^{\pi}\bigg(\int_{-\infty}^{0} \omega_{1}(\kappa)[\eta_{1}(\kappa)(\varepsilon)-\eta_{2}(\kappa)(\varepsilon)]d\kappa \bigg)^{2}d\varepsilon\bigg)^{1/2}\bigg]^{2}\\ &&\qquad\qquad\qquad\qquad\quad\leq\mathbb{E}\bigg[\bigg(\int_{-\infty}^{0}\dfrac{(\omega_{1}(\kappa))^{2}}{\Lambda(\kappa)}d\kappa\bigg)^{1/2}\bigg(\int_{-\infty}^{0} \Lambda(\kappa)\|\eta_{1}(\kappa)-\eta_{2}(\kappa)\|^{2}d\kappa \bigg)^{1/2}\bigg]^{2}\\ &&\qquad\qquad\qquad\qquad\quad\leq\bigg[l_{{\cal F}_{1}}\bigg(\|\eta_{1}(0)-\eta_{2}(0)\|+\bigg(\int_{-\infty}^{0} \Lambda(\kappa)\|\eta_{1}(\kappa)-\eta_{2}(\kappa)\|^{2}d\kappa\bigg)^{1/2}\bigg)\bigg]^{2}\\ &&\qquad\qquad\qquad\qquad\quad = L_{{\cal F}_{1}}\|\eta_{1}-\eta_{2}\|_{\mathfrak{B}}^{2}, \end{eqnarray*} $ |
where $ L_{{\cal F}_{1}} = [l_{{\cal F}_{1}}]^{2}. $ Similarly, we have $ \mathbb{E}\|{\cal F}_{2}(t, \eta)\|^{2}\leq L_{{\cal F}_{2}}\|\eta\|_{\mathfrak{B}}^{2}, $ $ \mathbb{E}\|{\cal E}_{j}(t, \eta_{1})-{\cal E}_{j}(t, \eta_{2})\|^{2}\leq L_{{\cal E}_{j}}\|\eta_{1}-\eta_{2}\|_{\mathfrak{B}}^{2}, $ and $ \mathbb{E}\|{\cal E}_{j}(t, \eta)\|^{2} \leq L_{{\cal E}_{j}}\|\eta\|_{\mathfrak{B}}^{2}, $ where $ L_{{\cal E}_{j}} = [l_{{\cal E}_{j}}]^{2}, \; L_{{\cal F}_{2}} = [\|a_{31}\|_{\infty}l_{{\cal F}_{2}}]^{2}. $ Further, we can impose some suitable conditions on the above-defined functions to verify the hypotheses of the Theorems 3.1 and 4.1. Therefore, the problem $ ({\cal LP}) $ corresponding to the stochastic system (5.1) has at least one optimal control pair.
In this manuscript, we studied the stochastic optimal control problem for a class of non-instantaneous impulsive stochastic neutral integro-differential equation driven by fBm. We define a concept of the piecewise continuous mild solutions for the proposed system, which is used to construct a suitable operator and apply fixed point technique to derive the existence result. Also, we prove the existence of optimal controls for the proposed system, which is used to derive optimization conditions. Finally, the obtained results have been verified through an example. There are two direct issues which require further study. First, we will investigate the optimal control problems for the non-instantaneous impulsive stochastic delay differential equations driven by Lévy processes [45]. Second, we will be devoted to studying the approximate controllability for the Markov and semi-Markov switched stochastic system [46,47].
We are very thankful to the anonymous reviewers and editor for their constructive comments and suggestions which help us to improve the manuscript.
All authors declare no conflicts of interest in this paper.
[1] | G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge: Cambridge University Press, 1992. |
[2] | B. ∅ksendal, Stochastic Differential Equations: An Introduction with Applications, Springer Science & Business Media, 2013. |
[3] | X. Mao, Stochastic Differential Equations and Applications, Chichester: Horwood Publishing Limited, 2008. |
[4] |
X. Yang and Q. Zhu, pTH moment exponential stability of stochastic partial differential equations with Poisson jumps, Asian J. Control, 16 (2014), 1482-1491. doi: 10.1002/asjc.918
![]() |
[5] | X. Yang and Q. Zhu, Existence, uniqueness, and stability of stochastic neutral functional differential equations of Sobolev-type, J. Math. Phys., 56 (2015), Article ID 122701: 1-16. |
[6] |
B. Boufoussi and S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Stat. Probab. Lett., 82 (2012), 1549-1558. doi: 10.1016/j.spl.2012.04.013
![]() |
[7] |
Y. Ren, X. Cheng and R. Sakthivel, On time-dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay, Math. Methods Appl. Sci., 37 (2014), 2177-2184. doi: 10.1002/mma.2967
![]() |
[8] |
N. T. Dung, Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochastics, 87 (2015), 142-159. doi: 10.1080/17442508.2014.924938
![]() |
[9] | 1.R. Dhayal, M. Malik and S. Abbas, Approximate controllability for a class of non-instantaneous impulsive stochastic fractional differential equation driven by fractional Brownian motion, Differ. Equations Dyn. Syst., (2019). Available from: http://sci-hub.tw/10.1007/s12591-019-00463-1. |
[10] |
A. Boudaoui, G. T. Caraballo and A. Ouahab, Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. B, 22 (2017), 2521-2541. doi: 10.3934/dcdsb.2017084
![]() |
[11] |
Q. Zhu and J. Cao, Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays, IEEE Trans. Neural Networks Learn. Syst., 23 (2012), 467-479. doi: 10.1109/TNNLS.2011.2182659
![]() |
[12] |
X. B. Shu, Y. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 74 (2011), 2003-2011. doi: 10.1016/j.na.2010.11.007
![]() |
[13] | X. B. Shu and Y. J. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476. |
[14] | S. Li, L. X. Shu, X. B. Shu, et al. Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, (2018), DOI: 10.1080/17442508.2018.1551400. |
[15] |
S. F. Deng, X. B. Shu and J. Z. Mao, Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point, J. Math. Anal. Appl., 467 (2018), 398-420. doi: 10.1016/j.jmaa.2018.07.002
![]() |
[16] |
Q. Zhu, pth moment exponential stability of impulsive stochastic functional differential equations with Markovian switching, J. Franklin Inst., 351 (2014), 3965-3986. doi: 10.1016/j.jfranklin.2014.04.001
![]() |
[17] | E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141 (2013), 1641-1649. |
[18] | J. Wang and M. Fečkan, A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal., 46 (2015), 915-933. |
[19] | M. Malik, R. Dhayal, S. Abbas, et al. Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, RACSAM Rev. R. Acad. A, 113 (2019), 103-118. |
[20] | M. Malik, R. Dhayal and S. Abbas, Exact controllability of a retarded fractional differential equation with non-instantaneous impulses, Dynam. Cont. Dis. Ser. B, 26 (2019), 53-69. |
[21] |
Z. Yan and X. Jia, Existence and controllability results for a new class of impulsive stochastic partial integro-differential inclusions with state-dependent delay, Asian J. Control, 19 (2017), 874-899. doi: 10.1002/asjc.1413
![]() |
[22] |
S. Liu and J. Wang, Optimal controls of systems governed by semilinear fractional differential equations with not instantaneous impulses, J. Optim. Theory Appl., 174 (2017), 455-473. doi: 10.1007/s10957-017-1122-3
![]() |
[23] | M. Malik, A. Kumar and M. Fečkan, Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, J. King Saud Univ. Sci., 30 (2018), 204-213. |
[24] |
R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. Control, 83 (2010), 387-393. doi: 10.1080/00207170903171348
![]() |
[25] |
Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay, Int. J. Control, 85 (2012), 1051-1062. doi: 10.1080/00207179.2012.675518
![]() |
[26] | X. Fu and R. Huang, Existence of solutions for neutral integro-differential equations with state-dependent delay, Appl. Math. Comput., 224 (2013), 743-759. |
[27] |
R. P. Agarwal, B. de Andrade and G. Siracusa, On fractional integro-differential equations with state-dependent delay, Comput. Math. Appl., 62 (2011), 1143-1149. doi: 10.1016/j.camwa.2011.02.033
![]() |
[28] | H. Huang, Z. Wu, L. Hu, et al. Existence and controllability of second-order neutral impulsive stochastic evolution integro-differential equations with state-dependent delay, J. Fixed Point Theory Appl., 20 (2018), Article 9: 1-27. |
[29] |
K. Ezzinbi, S. Ghnimi and M. A. Taoudi, Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst., 4 (2010), 54-64. doi: 10.1016/j.nahs.2009.07.006
![]() |
[30] |
V. Vijayakumar, Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces, Int. J. Control, 91 (2018), 204-214. doi: 10.1080/00207179.2016.1276633
![]() |
[31] |
W. Wei, X. Xiang and Y. Peng, Nonlinear impulsive integro-differential equations of mixed type and optimal controls, Optimization, 55 (2006), 141-156. doi: 10.1080/02331930500530401
![]() |
[32] |
1.Y. R. Jiang, N. J. Huang and J. C. Yao, Solvability and optimal control of semilinear nonlocal fractional evolution inclusion with Clarke subdifferential, Appl. Anal., 96 (2017), 2349-2366. doi: 10.1080/00036811.2017.1321111
![]() |
[33] |
J. Wang, Y. Zhou, W. Wei, et al. Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl., 62 (2011), 1427-1441. doi: 10.1016/j.camwa.2011.02.040
![]() |
[34] |
P. Balasubramaniam and P. Tamilalagan, The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators, J. Optim. Theory Appl., 174 (2017), 139-155. doi: 10.1007/s10957-016-0865-6
![]() |
[35] |
Z. Yan and F. Lu, Solvability and optimal controls of a fractional impulsive stochastic partial integro-differential equation with state-dependent delay, Acta Appl. Math., 155 (2018), 57-84. doi: 10.1007/s10440-017-0145-y
![]() |
[36] |
Z. Yan and X. Jia, Optimal controls of fractional impulsive partial neutral stochastic integro-differential systems with infinite delay in Hilbert spaces, Int. J. Control Autom. Syst., 15 (2017), 1051-1068. doi: 10.1007/s12555-016-0213-5
![]() |
[37] | D. Nualart, The Malliavin Calculus and Related Topics, New York: Springer-Verlag, 1995. |
[38] | J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial Ekvac., 21 (1978), 11-41. |
[39] |
R. Dhayal, M. Malik and S. Abbas, Approximate and trajectory controllability of fractional neutral differential equation, Adv. Oper. Theory, 4 (2019), 802-820. doi: 10.15352/aot.1812-1444
![]() |
[40] |
R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space, J. Differ. Equations, 50 (1983), 234-259. doi: 10.1016/0022-0396(83)90076-1
![]() |
[41] |
R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc., 273 (1982), 333-349. doi: 10.1090/S0002-9947-1982-0664046-4
![]() |
[42] |
E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. Real World Appl., 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014
![]() |
[43] |
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. doi: 10.1016/j.camwa.2009.06.026
![]() |
[44] |
E. J. Balder, Necessary and sufficient conditions for L1-strong-weak lower semicontinuity of integral functionals, Nonlinear Anal., 11 (1987), 1399-1404. doi: 10.1016/0362-546X(87)90092-7
![]() |
[45] |
Q. Zhu, Stability analysis of stochastic delay differential equations with L\'evy noise, Syst. Control Lett., 118 (2018), 62-68. doi: 10.1016/j.sysconle.2018.05.015
![]() |
[46] |
B. Wang and Q. Zhu, Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems, Syst. Control Lett., 105 (2017), 55-61. doi: 10.1016/j.sysconle.2017.05.002
![]() |
[47] |
B. Wang and Q. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica, 94 (2018), 72-80. doi: 10.1016/j.automatica.2018.04.016
![]() |
1. | Rajesh Dhayal, Muslim Malik, Syed Abbas, Solvability and optimal controls of non-instantaneous impulsive stochastic fractional differential equation of order q ∈ (1,2), 2020, 1744-2508, 1, 10.1080/17442508.2020.1801685 | |
2. | J. Priyadharsini, P. Balasubramaniam, Optimal control for fractional higher order damped stochastic impulsive systems, 2021, 0170-4214, 10.1002/mma.7159 | |
3. | Manuel De la Sen, Xiaodi Li, About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays, 2021, 2021, 1687-9651, 1, 10.1155/2021/5593813 | |
4. | Rajesh Dhayal, Muslim Malik, Syed Abbas, Existence, stability and controllability results of stochastic differential equations with non-instantaneous impulses, 2021, 0020-7179, 1, 10.1080/00207179.2020.1870049 | |
5. | Rajesh Dhayal, Muslim Malik, Syed Abbas, Approximate and trajectory controllability of fractional stochastic differential equation with non‐instantaneous impulses and Poisson jumps, 2020, 1561-8625, 10.1002/asjc.2389 | |
6. | Bo Li, Ranran Zhang, Ting Jin, Yadong Shu, Parametric approximate optimal control of uncertain differential game with application to counter terror, 2021, 146, 09600779, 110940, 10.1016/j.chaos.2021.110940 | |
7. | Rajesh Dhayal, José Francisco Gómez‐Aguilar, Guillermo Fernández‐Anaya, Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses, 2022, 43, 0143-2087, 386, 10.1002/oca.2805 | |
8. | Xin Wang, Guangming Zhuang, Guoliang Chen, Qian Ma, Junwei Lu, Asynchronous mixed H∞ and passive control for fuzzy singular delayed Markovian jump system via hidden Markovian model mechanism, 2022, 429, 00963003, 127253, 10.1016/j.amc.2022.127253 | |
9. | Amadou Diop, Moustapha Dieye, Bipan Hazarika, Random integrodifferential equations of Volterra type with delay : attractiveness and stability, 2022, 430, 00963003, 127301, 10.1016/j.amc.2022.127301 | |
10. | Rajesh Dhayal, Muslim Malik, Existence and controllability of impulsive fractional stochastic differential equations driven by Rosenblatt process with Poisson jumps, 2021, 130, 0022-0833, 10.1007/s10665-021-10167-7 | |
11. | Yazid Alhojilan, Hamdy M. Ahmed, New Results Concerning Approximate Controllability of Conformable Fractional Noninstantaneous Impulsive Stochastic Evolution Equations via Poisson Jumps, 2023, 11, 2227-7390, 1093, 10.3390/math11051093 | |
12. | S. Varshini, K. Banupriya, K. Ramkumar, K. Ravikumar, Existence and Stability Results of Stochastic Differential Equations with Non-instantaneous Impulse and Poisson jumps, 2022, 9, 2353-0626, 256, 10.1515/msds-2022-0159 | |
13. | Surendra Kumar, Anjali Upadhyay, Optimal control problem for fractional stochastic delayed systems with noninstantaneous impulses, 2021, 38, 0265-0754, 855, 10.1093/imamci/dnab014 | |
14. | Gunasekaran Gokul, Barakah Almarri, Sivajiganesan Sivasankar, Subramanian Velmurugan, Ramalingam Udhayakumar, New Study on the Controllability of Non-Instantaneous Impulsive Hilfer Fractional Neutral Stochastic Evolution Equations with Non-Dense Domain, 2024, 8, 2504-3110, 265, 10.3390/fractalfract8050265 | |
15. | Yihong Yuan, Danfeng Luo, Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays, 2024, 47, 0170-4214, 6471, 10.1002/mma.9932 | |
16. | K. Ramkumar, K. Ravikumar, Dimplekumar Chalishajar, Existence trajectory and optimal control of Clarke subdifferential stochastic integrodifferential inclusions suffered by non-instantaneous impulses and deviated arguments, 2023, 13, 26667207, 100295, 10.1016/j.rico.2023.100295 | |
17. | M. Johnson, V. Vijayakumar, An analysis on the optimal control results for second-order Sobolev-type delay differential inclusions of Clarke’s subdifferential type, 2024, 128, 10075704, 107649, 10.1016/j.cnsns.2023.107649 | |
18. | Syed Mohammad Abdal, Existence, Stability and Optimality of a Semilinear Control System Involving Instantaneous and Non-instantaneous Impulses, 2024, 30, 1079-2724, 10.1007/s10883-024-09713-0 | |
19. | M. Johnson, V. Vijayakumar, Kiwoon Kwon, Existence and Optimal Controls for Generalized Riemann–Liouville Fractional Sobolev‐Type Stochastic Integrodifferential Equations of Order ϑ∈(1,2)$$ \vartheta \in \left(1,2\right) $$, 2025, 0170-4214, 10.1002/mma.10662 | |
20. | S. Varshini, K. Ravikumar, K. Ramkumar, Hamdy M. Ahmed, Conformable Fractional Stochastic Differential Inclusions Driven by Poisson Jumps with Optimal Control and Clarke Subdifferential, 2025, 38, 0894-9840, 10.1007/s10959-025-01414-z | |
21. | M. Johnson, V. Vijayakumar, Kiwoon Kwon, Investigation of optimal control results for generalized Riemann-Liouville fractional integrodifferential systems of Volterra-Fredholm type with nonlocal initial conditions, 2025, 13, 2195-268X, 10.1007/s40435-025-01733-3 |