Stabilization in distribution of hybrid stochastic differential delay equations with Lévy noise by discrete-time state feedback controls

1.   Introduction

Stochastic differential equations (SDEs) with Markovian switching (also known as hybrid SDEs) has been widely used to model many systems in biological systems, financial systems, and other fields. A field of common interest in the study of hybrid SDEs is automatic control, taking subsequent emphasis on the stability analysis ^[1,2]. Most of the literature, such as ^[3,4,5,6,7], only consider Brown motions. However, Brown motions are continuous and cannot describe discontinuous noises like jump-type noises. Compared with Brown motions, Lévy noise, which contains both continuous Brown motions and discontinuous Poisson jumps, can model the extreme sudden events, such as earthquakes, storms, floods, wars, and so on. For example, in ^[8] some sufficient conditions were put forward to achieve almost surely exponential stability of neural networks with Markovian switching and Lévy noise. Therefore, with the development of stochastic analysis, stochastic differential equations with Markovian switching and Lévy noise are considered by many researchers, see ^[9,10,11,12].

It is well known that time delays are often and inevitably encountered for various reasons in many fields such as population systems, manufacturing, chemistry and chemical engineering, finance, etc. Meanwhile, a time delay is often one of the main causes of poor performance in systems, see ^[13,14]. Hence, taking time delays into account is reasonable and necessary when studying the stability of SDDEs-MS-LN. Nowadays, stability and stabilization of such SDDEs have been studied, see ^[15,16,17]. For example, Yuan et al. in ^[16] investigated sufficient conditions for stability of delay jump diffusion processes. Li in ^[17] focused on the mean square stability of stochastic differential equations with Lévy noise.

A common feature in these papers is that most of the research is focused on the stability of the trivial solutions. However, many hybrid systems do not have an equilibrium state or their solutions do not converge to zero, see ^[18,19]. It is not sufficient to only study the stability of trivial solutions in the real world. For example, for many population systems under realistic backdrops, stochastic permanence is a more suitable control goal than extinction (see ^[20,21,22]). In this case, it is of great significance to know whether the solution will converge to some distribution or not (but not necessarily to zero). This property is known as asymptotic stability in distribution. Stability in distribution of SDEs-MS with Brownian motion has attracted some attention of scholars recently, for example, Yuan et al. ^[19] and You et al. ^[23]. In 2010, Yuan et al. ^[16] studied stability in distribution of hybrid delay systems with jumps. As a classical area of stability of hybrid systems, Li et al. ^[24] recently considered to employ delay feedback controls to stabilize a given SDEs-MS-LN in distribution. But for the stabilization in distribution of SDDEs-MS-LN, the discussion is still open. In addition, to reduce the practical cost of control design, feedback controls based on discrite-time state observations^[1,14] are considered in this paper.

Mathematically speaking, let us consider an unstable SDDE-MS-LN

where $X(t) \in \mathbb{R}^{d}$, $h$ is a time delay of the system, $r(t)$ is a Markov chain, $B(t)$ is a Brownian motion, $\tilde{N}(\mathrm{\; d} t, \mathrm{\; d} z)$ is a compensated Poisson random measure, and $\mathbb{R}_{0}^{n} = \mathbb{R}^{n}-\{0\}$ (For formal definitions, see Section $2$.) Such a regular feedback control requires the continuous observations of the state $X(t)$ for all $t\geq 0$. This is of course expensive and sometimes not possible as the observations are often of discrete time. Now we can design a feedback control $u(X([t/\tau]\tau), r(t))$ based on the discrete-time observations of the state $X(t)$ at times $0$, $\tau$, $2\tau$, ..., so that the controlled system

becomes stable in distribution.

The main aim of this paper is to explore how to use feedback control $u(X([t/\tau]\tau), r(t))$ to stabilize a given unstable SDDE-MS-LN in distribution. The key points of this paper are as follows.

● We introduce Lévy noise to remodel hybrid stochastic delay systems and study the stability in distribution for controlled SDDEs-MS-LN.

● Due to the discontinuity of Lévy noise, we need to study the stability in distribution for SDDEs-MS-LN in functional space $\mathcal{D}_{h}$ (for formal definitions, see Section $2$) rather than $\mathcal{C}_{h}$ in ^[23].

● Making use of the generalized It$\hat o$ formula for Lévy-type stochastic integrals ^[25], we construct a special Lyapunov functional based on Lévy noise, the property of stability and discrete-time feedback control to achieve the asymptotic stability in distribution for controlled SDDEs-MS-LN.

● In order to reduce the cost of the continuous working time of the controller, feedback control $u(X([t/\tau]\tau), r(t))$ based on the discrete-time observations is an efficient strategy to stabilize the unstable systems. Moreover, we show that there is a positive number $\tau^{*}$ such that the feedback control $u(X([t/\tau]\tau), r(t))$ will make the controlled system $(1.2)$ asymptotically stable in distribution provided $\tau\leq \tau^{*}$. We will also give a lower bound on $\tau^{*}$ which is computable numerically.

The structure of this paper is organized as follows. In Section $2$, we present some notations, definitions, and assumptions related to Eq $(1.1)$. In Section $3$, we study the stability in distribution of the solution to Eq $(1.2$) based on the Lyapunov functionl and It$\hat o$ formula. In Section $4$, the method for designing the control function is discussed. In Section $5$, we provide a numerical example to verify the effectiveness of the new results. Finally, this article is concluded in Section $6$.

2.   Notations and assumptions

Throughout this paper, unless otherwise specified, we use the following notations. Let $\mathbb{R}^{d}$ be the $d$-dimensional Euclidean space and $\mathcal{B}\left(\mathbb{R}^{d}\right)$ denote the family of all Borel measurable sets in $\mathbb{R}^{d}$. Let $|\cdot|$ denote the Euclidean norm or the matric trace norm, respectively. For a matrix $A$, $|A| = \sqrt{\operatorname{trace}\left(A^{T} A\right)}$ denotes its trace norm and $\|A\| = max\{|Ax|:|x| = 1\}$ denotes its operator norm. If $A$ is a symmetric matrix, the largest and smallest eigenvalue are denoted by $\lambda_{\max }(A)$ and $\lambda_{\min }(A)$, respectively. In general, $\left(\Omega, \mathcal{F}, \left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$ signifies a complete probability space whose filtration $\left\{\mathcal{F}_{t}\right\}_{t \geq 0}$ satisfies the usual conditions. Denote by $\mathcal{D}_{h}$ (or $D\left([-h, 0];\mathbb{R}^{d}\right))$ the family of all càdlàg (i.e., right continuous with left limits) functions $\xi$ : $[-h, 0] \rightarrow \mathbb{R}^{d}$ in the Skorokhod topology. For any $\xi_1, \xi_2 \in \mathcal{D}_{h}$, define the Skorohod metric $d_{S}\left(\xi_{1}, \xi_{2}\right) = \inf _{\lambda \in \Lambda}\left\{\|\lambda\|^{\circ} \vee\left\|\xi_{1}-\xi_{2} \circ \lambda\right\|_{h}\right\}$, where $\Lambda$ denotes the class of strictly increasing, continuous mappings of $[-h, 0]$ onto itself, $\xi_{2} \circ \lambda$ denotes the composition of two functions $\xi_{2}$ and $\lambda$, $\|\lambda\|^{\circ} = \sup _{-h \leq s < t \leq 0}\left|\log \frac{\lambda(t)-\lambda(s)}{t-s}\right|$, and $\|\xi\|_{h} = \sup _{-h \leq s \leq 0}|\xi(s)|$. Under the Skorohod metric $d_{S}, D\left([-h, 0]; \mathbb{R}^{d}\right)$ is complete and separable (^[26], Theorem 12.2, p. 128). In addition, $\mathcal{B}\left(\mathcal{D}_{h}\right)$ denotes the family of all Borel measurable sets in $\mathcal{D}_{h}$. Let $B(t) = \left(B_{1}, \ldots, B_{m}\right)$ be an $m$-dimensional Brownian motion. Denote by $N(t, z)$ an $n$-dimensional Poisson process, and denote the compensated Poisson random measure by

where $\left\{N_{k}, k = 1, \ldots, n\right\}$ are independent 1-dimensional Poisson random measures with characteristic measure $\left\{\nu_{k}, k = \right. \; 1, \ldots, n\}$ coming from $n$ independent 1-dimensional Poisson point processes.

Let $r(t)$, $t \geq 0$, be a right-continuous irreducible Markov chain on the probability space taking values in a finite state space $\mathbb{S} = \{1, 2, \ldots, N\}$ with the generator $\Gamma = \left(\gamma_{i j}\right)_{N \times N}$ given by

where $\Delta > 0$ satisfies $\lim\limits_{\Delta\rightarrow 0} \frac{o(\Delta)}{\Delta} = 0$ and $\gamma_{i j}$ is the transition rate from $i$ to $j$ satisfying $\gamma_{i j} > 0$ if $i \neq j$ while $\gamma_{i i} = -\sum_{i \neq j} \gamma_{i j}$. We assume that $r(t), B(t)$, and $N(t, z)$ are independent of each other.

Let us consider a $d$-dimension SDDE-MS-LN ($1.1$), where $f: \mathbb{R}^{d} \times\mathbb{R}^{d} \times\mathbb{S} \rightarrow \mathbb{R}^{d}$, $g: \mathbb{R}^{d} \times\mathbb{R}^{d}\times \mathbb{S} \rightarrow \mathbb{R}^{d \times m}$, and $H: \mathbb{R}^{d} \times\mathbb{R}^{d}\times \mathbb{S} \times \mathbb{R}_{0}^{n} \rightarrow \mathbb{R}^{d \times n}$ are Borel measurable functions, $X(t^{-}) = \lim_{s \uparrow t}X(s)$. We note that each column $H^{(k)}$ of the $d \times n$ matrix $H = \left[H_{l j}\right]$ depends on $z$ only through the $k$th coordinate $z_{k}$, that is

We refer to ^[16,27] where this type of dependence is discussed and investigated for SDDEs-MS-LN. We can rewrite out in detail component $X_{l}(t)$, $1 \leq l \leq d$, in (1.1), that is

Next we will state an assumption about the coefficients of SDDE-MS-LN $(1.1)$.

Assumption 2.1. There exist positive constants $a_{1}$, $a_{2}$, and $a_{3}$ such that

and

for all $x, \bar{x}, y, \bar{y} \in \mathbb{R}^d$ and $i\in \mathbb{S}$.

It is easy to see from Assumption ($2.1)$ that

and

for all $(x, \bar{x}, i) \in \mathbb{R}^{d} \times\mathbb{R}^{d}\times \mathbb{S}$ and $i\in \mathbb{S}$, where $a_{0} = 2 \max _{i \in \mathbb{S}}\left(|f(0, 0, i)|^{2} \vee\right. \; \left.|g(0, 0, i)|^{2}\right) \vee \sum_{k = 1}^{n} \int_{\mathbb{R} \backslash\{0\}}\left|H^{(k)}\left(0, 0, i, z_{k}\right)\right|^{2} \nu_{k}\left(d z_{k}\right).$

By Assumption 2.1, it is known (see ^[16]) that the SDDE-MS-LN (1.1) has a unique global solution $X(t)$ for all $t \geq 0$. Assume that the original SDDE-MS-LN (1.1) does not have the desired property of stability in distribution. Therefore we need to design a feedback control to stabilize the system (1.1). To make the design more concise and simple, we use the linear form of feedback control $u(X(\delta(t)), r(t)) = A(r(t)) X(\delta(t))$, where $A(i) \equiv A_{i} \in \mathbb{R}^{d \times d}(1 \leq i \leq N)$, $\delta(t) = [t/\tau]\tau$. In addition, throughout this paper, we will set $a_{4} = {\max}_{i\in \mathbb{S}}\|A_i\|^2$. The controlled system (1.2) therefore becomes

with the initial data as

It is well known to all (see ^[28]), under Assumption 2.1, SDDE-MS-LN $(2.3)$ has a unique global solution for any initial data $(2.4)$. Define $X_{t} = \{X(t+s):-h \leq s \leq 0\}$ for $t \geq 0$, which is a $\mathcal{D}_{h}$-valued process. $X^{\xi, i}(t)$ denotes the solution of SDDE-MS-LN $(2.3)$ with initial data $(2.4)$. $r^{i}(t)$ denotes the Markov chain starting from $i$. It is also known that (see ^[29])

where $C_{t}$ is a positive constant that depends on $t$ but is independent of the initial data $(\xi, i)$.

We notice that the joint process $\left(X_{t}, r(t)\right)$ is not a time-homogeneous Markov process. But when $h$ can be divisible by $\tau$, for $k\geq0$, we can easily get that the joint process $\left(X_{k\tau}, r(k\tau)\right)$ is a time-homogeneous Markov process with transition probability $p(k, \xi, i; d \zeta \times\{j\})$, where $p(k, \xi, i; d \zeta \times\{j\})$ denotes the transition probability measure on $\mathcal{D}_{h} \times \mathbb{S}$, that is

for any $E \in \mathcal{B}\left(\mathcal{D}_{h}\right)$ and $J \in \mathbb{S}$.

Denote by $\mathcal{P}\left(\mathcal{D}_{h}\right)$ the family of probability measures on the measurable space $\left(\mathcal{D}_{h}, \mathcal{B}\left(\mathcal{D}_{h}\right)\right)$. For $\mathbb{P}_{1}, \mathbb{P}_{2} \in \mathcal{P}\left(\mathcal{D}_{h}\right)$, metric $d_{\mathbb{L}}$ is given by

where $\mathbb{L} = \left\{\phi: \mathcal{D}_{h} \rightarrow \mathbb{R}\right.$ satisfying $|\phi(\xi)-\phi(\zeta)| \leq d_{\mathrm{S}}(\xi, \zeta)$ and $|\phi(\xi)| \leq 1$ for $\left.\xi, \zeta \in D_{h}\right\}$. In addition, let $\mathcal{L}\left(X_{t}\right)$ denote the probability measure generated by $X_{t}$ on $\left(\mathcal{D}_{h}, \mathcal{B}\left(\mathcal{D}_{h}\right)\right)$.

Definition 2.1. The SDDE-MS-LN $(2.3)$ is said to be asymptotically stable in distribution if there exists a probability measure $\mu_{h} \in \mathcal{P}\left(\mathcal{D}_{h}\right)$ such that

for all $(\xi, i) \in \mathcal D_{h} \times \mathbb{S}$.

3.   Stabilization in distribution

Let $C^{2}\left(\mathbb{R}^{d} \times \mathbb{S}; \mathbb{R}_{+}\right)$ denote the family of all non-negative continuous functions $\Psi(x, i)$ defined on $\mathbb{R}^{d} \times \mathbb{S}$ which are twice continuously differentiable in $x$ for all $i \in \mathbb{S}$. Assume that there exists one $\Psi \in C^{2}\left(\mathbb{R}^{d} \times \mathbb{S}; \mathbb{R}_{+}\right)$, and define an operator $L \Psi$ from $\mathbb{R}^{d} \times\mathbb{R}^{d} \times \mathbb{S}$ to $\mathbb{R}$ by:

where $\quad \Psi_{x}(x, i) = \left(\frac{\partial \Psi(x, i)}{\partial x_{1}}, \frac{\partial \Psi(x, i)}{\partial x_{2}}, \ldots, \frac{\partial \Psi(x, i)}{\partial x_{d}}\right), \quad \Psi_{x x}(x, i) = \; \left(\frac{\partial^{2} \Psi(x, i)}{\partial x_{i} \partial x_{j}}\right)_{d \times d}$.

The difference between two solutions of the system ($2.3$) with different initial values is as follows:

Let $\Phi \in C^{2}\left(\mathbb{R}^{d} \times \mathbb{S}; \mathbb{R}_{+}\right)$, and define an operator $\mathbb{L} \Phi: \mathbb{R}^{d} \times \; \mathbb{R}^{d} \times\mathbb{R}^{d} \times\mathbb{R}^{d} \times \mathbb{S} \rightarrow \mathbb{R}$ concerning Eq (3.2) by

To study stabilization in distribution of system ($2.3$), we need the following assumptions.

Assumption 3.1. There exist positive constants $c_{1}$, $\theta_{1}$, $b_{2}$, and $b_{0} > b_{1}\ge 0$, function $\Psi(x, i) \in C^{2}\left(\mathbb{R}^{d} \times \mathbb{S}; \mathbb{R}_{+}\right)$, and $Q_{1}(x) \in C\left(\mathbb{R}^{d}; \mathbb{R}_{+}\right)$ such that

for all $(x, \bar{x}, i) \in \mathbb{R}^{d} \times\mathbb{R}^{d} \times \mathbb{S}$.

Assumption 3.2. There exist positive constants $c_{2}$, $\theta_{2}$, and $b_{3} > b_{4}\ge 0$, function $\Phi(x, i) \in C^{2}\left(\mathbb{R}^{d} \times \mathbb{S}; \mathbb{R}_{+}\right)$, and $Q_{2}(x) \in C\left(\mathbb{R}^{d}; \mathbb{R}_{+}\right)$ such that

for all $(x, y, \bar {x}, \bar {y}, i) \in \mathbb{R}^{d} \times \mathbb{R}^{d} \times \mathbb{R}^{d} \times\mathbb{R}^{d} \times\mathbb{S}$.

3.1.   Lyapunov functionals

To obtain our results, we need to establish the Lyapunov functional on the segments $\hat{X}_{t}: = \; \{X(t+s):- \tau-h \leq s \leq 0\}$ and $\hat{r}_{t} = \{r(t+s): -\tau-h \leq s \leq 0\}$ for $t \geq \tau$. Let $r(s) = r(0)$ for $-\tau-h \leq s \leq 0$. Evidently, $\hat{X}_{t}$ is $D\left([- \tau-h, 0]; \mathbb{R}^{d}\right)$-valued which is different with $X_{t}$. The Lyapunov functional will be of the form

where

and $\alpha$ is a positive constant selected later.

Remark 3.1. Since our feedback control is based on discrete-time state observations, the Lyapunov functional in ^[23,24] is not appropriate which is employed for the stabilization in distribution problem by delay feedback control. Therefore, we consider to employ a new Lyapunov functional motivated by ^[14,24] to prove the stability in distribution of controlled system (2.3).

We can observe that

For convenience, $X(t)$ denotes $X^{\xi, i}(t)$ and we fix the initial data $(\xi, i)$ arbitrarily. Applying the generalized functional It$\hat o$ formula to the Lyapunov functional defined by $(3.6)$ yields

for $t \geq \tau$, where $M(t)$ is a martingale with $M(0) = 0$, and

By Assumption 2.1, we can derive

Under Assumption 3.1, we get from Eqs (3.8) and (3.9) that

for $t \geq \tau$, where $b = b_0-2 \alpha \tau\left(2 a_{1}\tau+a_{2}+a_{3}+2 a_{4}\tau\right)/c_1.$

3.2.   Lemmas

Before proving the key theorem, we need to prove two lemmas, where Lemma $3.1$ will prove the uniform boundedness and Lemma $3.2$ will prove the exponential convergence.

Lemma 3.1. Let Assumptions 2.1 and 3.1 hold. If $\tau > 0$ is sufficiently small for

then the solution of Eq $(2.3)$ with initial data $(2.4)$ satisfies

for all $t \geq 0$, where $C$ is a positive constant.

Proof. Applying the functional It$\hat o$ formula to $e^{\beta_{0} t}(V(\hat{X}_{t}, \hat{r}_{t}, t))$, we can show that

for $t\ge \tau$, where $\beta_0$ is a positive number to be chosen later. Using Eqs ($2.5$) and ($3.7$), one can see that

where $\beta_{1}$ is a positive number. Moreover, we note that

Substituting Eqs ($3.10$) and ($3.14$) into Eq ($3.13$), we can obtain

It follows from Eq ($2.3$) that

By Itô isometry,

Set $\alpha = \frac{15a_{4}}{ 4\theta_{1}}$ and $\tau < \frac{1}{ \sqrt{15{a_{4}}}}$, and we have that $3(\frac{a_{4}}{4 \theta_{1}}+4a_{4}\alpha\tau^2)-\alpha < 0$. Then we can find a sufficiently small $\beta_{0}$ which satisfies the following condition:

Using Eqs $(3.17)$ and $(3.18)$, we derive that

It follows from Eq ($3.15$) that one gains

Note that

By condition ($3.18$), we derive that

where $\beta_{2}$ is a positive number. Hence

After that, we can make an estimate of the segment process $X_{t}$. Let $t \geq \tau+h$ and $\theta \in[0, \tau]$. According to the It$\hat o$ formula and Eq ($2.3$), we obtain that

According to Kunita's inequality (^[30], Corollary 2.12, p. 332),

where $c_{3}$ is a positive constant. It follows from Eqs ($2.1$) and ($2.2$) that

where $c_{4}$ and $c_{5}$ are positive numbers. By Eqs ($3.20$) and ($3.21$), it is easy to show

where $\beta_{4}$ is a positive number. Together with Eq (2.5), the assertion (3.12) holds. The proof is hence complete. □

Lemma 3.2. Let Assumptions 2.1 and 3.2 hold. If $\tau > 0$ is sufficiently small enough for

then for any $(\xi, \zeta, i) \in \mathcal{D}_{h} \times \mathcal{D}_{h} \times \mathbb{S}$,

for all $t \geq \tau+h$, where $\alpha_{1}$ and $\alpha_{2}$ are positive constants.

Proof. Denote by $O(t) = X^{\xi, i}(t)-X^{\zeta, i}(t)$ for any $(\xi, \zeta, i) \in \mathcal{D}_{h} \times \mathcal{D}_{h} \times \mathbb{S}$. Moreover, $O_{t} = \{O(t+s):-\tau \leq s \leq 0\}$ for $t \geq 0$ and $\hat{O}_{t} = \{O(t+s):- \tau-h \leq s \leq 0\}$ for $t \geq \tau+h$. Design a new Lyapunov functional $\tilde{V}\left(\hat{O}_{t}, \hat{r}_{t}, t\right)$:

for $t \geq \tau$. Applying the functional It$\hat o$ formula, we have

for $t \geq \tau$, where $\tilde{M}(t)$ is a martingale with $\tilde{M}(0) = 0$, and

By Assumptions ($2.1$) and ($3.2$), we have

for $t\ge \tau$, where $\bar{b} = b_3- \alpha \tau\left(2 a_{1}\tau+a_{2}+a_{3}+2a_{4}\tau\right)/c_2$.

Applying the functional It$\hat o$ formula to $e^{\alpha_{2} t} \mathbb{E}\left(\tilde V\left(\hat{O}_{t}, \hat{r}_{t}, t\right)\right)$, we have

where $\alpha_{4}$ is a positive number and $\alpha_{2}$ is a positive number to be determined. Substituting Eq ($3.25$) into Eq (3.26) yields

Moreover, by Assumption $3.2$, one can see that

Substituting Eq ($3.28$) into Eq ($3.27$), we can get

Moreover, we can obtain that

Set $\alpha = \frac{15a_{4}}{ 4\theta_{2}}$ and $\tau < \sqrt{\frac{2}{ 15 a_{4}}}$, and we have that $3(\frac{a_{4}}{4 \theta_{2}}+2a_{4}\alpha\tau^2)-\alpha < 0$. Then we can find a sufficiently small $\alpha_{2}$ which satisfies the following condition:

By Eq ($3.30$), we can show that

Following the condition ($3.30$), we derive that

This implies that

for $t \geq \tau$, where $\alpha_{5}$, $\alpha_{3}$, and $\alpha_{2}$ are positive constants. However, for $t \geq \tau+h$, we have that

From Assumption 2.1, one can see that

where $c_{6}$ and $c_{7}$ are all positive numbers. By Eq (3.32), we have

where $\alpha_{1}$ is a positive number. Therefore, the required assertion (3.23) holds. This completes the proof. □

3.3.   Key theorem

Next, let us prove that the SDDE-MS-LN ($2.3$) is stable in distribution by Lemmas $3.1$ and $3.2$.

Theorem 3.1. Let Assumptions $2.1$ and $3.2$ hold. Let $\tau_{1}^{*}, \tau_{2}^{*}, \tau_{3}^{*}$, and $\tau_{4}^{*}$ be the unique positive roots to the following equations

respectively, and set $\tau^{*} = \tau_{1}^{*} \wedge \tau_{2}^{*} \wedge \tau_{3}^{*} \wedge \tau_{4}^{*}$. Then for each $\tau < \tau^{*}$ and where $h$ can be divisible by $\tau$, there exists a unique probability measure $\mu_h \in \mathcal{P}\left(\mathcal{D}_{h}\right)$ such that

for all $(\xi, i) \in \mathcal{D}_{h} \times \mathbb{S}$.

Proof. Step 1: We first claim that for any compact set $\mathcal{K} \subset \mathcal{D}_{h}$,

uniformly in $(\xi, \zeta, i, j) \in \mathcal{K} \times \mathcal{K} \times \mathbb{S} \times \mathbb{S}$. Define the sequence of the stopping time $\kappa_{i j} = \inf \left\{k\tau: r^{i}(k\tau) = r^{j}(k\tau), k \geq 0\right\}$. Using the ergodicity of the Markov chain, we can obtain that $\kappa_{i j} < \infty$ a.s. Consequently, for any $\varepsilon \in(0, 1)$, there exists a number $T_{1} > 0$ such that

Recalling a known result that

we can find enough large $T_{2} > 0$ such that

where $\Omega_{\xi, i} = \left\{\omega \in \Omega: \sup _{-\tau \leq t \leq T_{1}}\left|X^{\xi, i}(t, \omega)\right| \leq q\right\}$. For any $\phi \in \mathbb{L}$ and $k\tau \geq T_{1}$, we obtain

where $\rho(t): = \mathbb{E}\left(I_{\left\{\kappa_{i j} \leq T_{1}\right\}}\left|\phi\left({X}_{k\tau}^{\xi, i}\right)-\phi\left({X}_{k\tau}^{\zeta, j}\right)\right|\right)$. Set $\Omega_{1} = \; \Omega_{\xi, i} \cap \Omega_{\zeta, i} \cap\left\{\kappa_{i j} \leq T_{1}\right\}$. By the Markov property of joint process $\left({X}_{k\tau}^{\xi, i}, r^{i}(k\tau)\right)$ and the property of conditional expectation, we derive

It is known (see ^[31]) that $d_{S}\left(\xi_{1}, \xi_{2}\right) \leq \left\|\xi_{1}-\xi_{2}\right\|,$ for any $\xi_{1}, \xi_{2} \in \mathcal{D}_{h}.$ Using this and the Proposition 1.17 in ^[32], we derive that

It is easy to observe that $\|\tilde{\xi}\| \vee\|\tilde{\zeta}\| \leq q$, for any $\omega \in \Omega_{1}$. By using Lemma 3.2, we are able to find positive number $T_{2}$ such that

for any given $\omega \in \Omega_{1}$. Then we have that

Due to the arbitrariness of $\phi$, for all $(\xi, \zeta, i, j) \in \mathcal{K} \times \mathcal{K} \times \; \mathbb{S} \times \mathbb{S}$, we get

Our claim is proved.

Step 2: Next, we claim that for any $(\xi, i) \in \mathcal{D}_{h} \times \mathbb{S}$, $\left\{\mathcal{L}\left(X_{k\tau}^{\xi, i}\right)\right\}_{k \in \mathbb{N}_{+} }$ is a Cauchy sequence in $\mathcal{P}\left(\mathcal{D}_{h}\right)$ with metric $d_{\mathbb{L}}$. That is, we need to show that for any $\varepsilon > 0$, there is a positive number $k_{0}$ such that

for all integers $s \geq k_{0}$ and $v\ge 1$. Let $\varepsilon \in(0, 1)$ be arbitrary. By Lemma 3.1, there is a $\bar{q} > 0$ such that

For any $\phi \in \mathbb{L}$ and integers $s\ge 1$, we can then derive, using $(2.6)$ and $(3.38)$, that

where $Z_{\bar{q}} = \left\{\zeta \in \mathcal{D}_{h}:\|\zeta\| \leq \bar{q}\right\}$. But, by (3.36), there is a positive integer $k_{0}$ such that

whenever $(\zeta, j) \in Z_{\bar{q}} \times \mathbb{S}$. We, therefore, obtain

for $s \geq k_{0}$ and $v\ge 1$. As this holds for any $\phi\in \mathbb{L}$, we must have ($3.37$) as claimed.

Step 3: By Eq ($3.37$), there exists a unique $\mu_{h} \in \; \mathcal{P}\left(\mathcal{D}_{h}\right)$ such that

which together with Eq ($3.36$) gains

for all $(\xi, i) \in \mathcal{D}_{h} \times \mathbb{S}$. That is assertion (3.35). The proof is, hence, complete. □

4.   Design of matrices $A_i's$

To simplify the calculation and design of matrices, we choose the forms of the function as follows:

for some $N$ symmetric positive definite matrices $W_{i}(i \in \mathbb{S})$. It follows easily from Eqs $(3.1)$ and $(3.3)$ that

and

Assumption 4.1. Let Assumption $2.1$ hold. There exist positive numbers ${j}_{0}$, ${b}_{1}$, ${j}_{3}$, ${b}_{2}$, ${b}_{4}$, ${j}_{0}\geq{b}_{1}$, ${j}_{3}\geq{b}_{4}$, and positive definite matrices $W_{i}$ such that

for all $x$, $y \in \mathbb{R}^{d}$ and $i \in \mathbb{S}$.

If we set $b_0 = j_{0} -4\check{c}^2\theta_{1}$, $b_3 = j_{3} -4\check{c}^2\theta_{2}$, it reaches the desired conditions ($3.4$) and ($3.5$). That is to say, we have shown that Assumption 4.1 implies Assumptions $3.1$ and $3.2$.

By Lemmas $3.1$, $3.2$, and Theorem $3.1$, the following corollary therefore follows.

Corollary 4.1. Let Assumptions $3.1$ and $3.2$ be replaced by Assumption 4.1. If $h$ can be divisible by $\tau$ and is small enough for

where $\bar{\theta} \in\left(0, \frac{ j_{0}-b_{1}}{4\check{c}^{2}} \wedge \frac{j_{3}-b_{4}}{4\check{c}^{2}}\right)$, $c = \min _{i \in \mathbb{S}} \lambda_{\min }W_{i}$, $\check{c} = \max _{i \in \mathbb{S}}\left\|W_{i}\right\|$, and $\alpha = \frac{15 a_{4}}{4\bar{\theta}}$. Then the SDDE-MS-LN ($2.3$) is stable in distribution.

The key to the problem of stabilization in distribution lies in the design of the matrices $A_i(i\in\mathbb{S})$. Therefore we need to find the matrices of the form $A_i = F_iG_i$ with $F_i\in R^{n\times l}$ and $G_i \in R^{n\times l}$ for some positive integer $l$. The two cases of state feedback and output injection are discussed below.

(i) State feedback

When $G_{i}$'s are given, we need to seek suitable $F_{i}$'s to make SDDE-MS-LN $(2.3)$ stable in distribution. The matrices are designed in two steps.

Step 1: Seek $N$ couples of positive-definite symmetric matrices $\left(W_{i}, \hat{R}_{i}, \hat{S}_{i}\right)$ such that

and then

Step 2: Seek a solution of matrices $F_{i}$ to the following linear matrix inequalities ensuring that ${j}_{3} > {b}_{4}$:

Corollary 4.2. Under Assumption $2.1$, seek matrices $F_{i}(i\in\mathbb{S})$ with the above Steps 1 and 2. Then Corollary $4.1$ holds with $A_{i} = F_{i} G_{i}$ and

(ii) Output injection

When $F_{i}$'s are given, we need to seek $G_{i}$'s. This case is similar to the case of state feedback, and therefore we can get another corollary.

Corollary 4.3. Under Assumption $2.1$, find matrices $W_{i}$, $\hat{R}_{i}$, $\hat{S}_{i}$, and $G_{i}(i\in\mathbb{S})$ with the above Steps 1 and 2. Then Corollary $4.1$ holds with $A_{i} = F_{i} G_{i}$, and moreover ${j}_{3}$ and ${b}_{4}$ are the same as in Corollary $4.2$.

5.   Example

In this section, we will give an example to illustrate our results.

Example 5.1. Let us consider the following unstable SDDE-MS-LN:

with initial value $X(0) = 1$, $r(0) = 1$, and $N(0) = 0$, where the coefficients f, g, and H are defined by

for all $x, \bar{x}\in \mathbb{R}$, $B(t)$ is a scalar Brownian motion, $N(t)$ is a scalar Poisson process with intensity $\lambda$, $\tilde{N}(t)$ denotes its compensated Poisson random measure, $r(t)$ is a Markov chain on the state space $\mathbb{S} = \{1, 2\}$ with its generator

and the time delay $h = 0.01$.

From Figure 1, we know the SDDE-MS-LN $(5.1)$ is unstable. Let us now apply our new theory to design a linear feedback control to make the SDDE-MS-LN $(5.1)$ stable in distribution. Suppose that the type of linear feedback control is $-k(i) X(\delta(t))$, where $k(i)$ will be computed later. Then the controlled system becomes

Let $W_{i}(i \in 1, 2)$ be the identity matrix. Set $\lambda = 1$. After the calculation, we can get that ${b}_{4} = 2.12$. Next, we need to choose a number $j_{3}$ which satisfies $j_{3} > {b}_{4}$ and $-j_{3} = -\max _{i \in \mathbb{S}} \left(1.08-2k(i)\right) < 0$ for $i \in \mathbb{S}$. Hence, Corollary $4.1$ holds for $j_{3} = 6$. Then we can deduce that $k(1) = 3.54$ and $k(2) = 4.26$. It is easy to verify that Assumption $2.1$ holds for $a_{1} = 0.18$, $a_{2} = 0.08$, and $a_{3} = 2$. Furthermore, by $a_{4} = \max _{i \in \mathbb{S}}\|k(i)\|^{2}$, we can get $a_{4} = 18.1476$ and $\check{c} = 1$. Thus, setting $c = 1$, $\bar{\theta} = 0.1$, we can derive

Consequently, $\tau^{*} = 0.00158$. By Corollary $4.1$, the controlled system $(5.2)$ is stable in distribution when $\tau < 0.00158$ and $h$ can be divisible by $\tau$.

In addition, we plot the marginal density function of $X(t)$ by using of the Euler-Maruyama method with step size 0.001 in Figure 2. From the figure, as t increases, the change in distribution becomes smaller and smaller. This show that the controlled system $(5.2)$ is stable in distribution.

6.   Conclusions

In this paper, stabilization in distribution for given unstable SDDEs-MS-LN whose drift and diffusion coefficients are globally Lipshitz continuous has been investigated. We successfully showed that the stability in distribution of controlled SDDE-MS-LN can be achieved by linear feedback controls based on discrete-time state observations. A lower bound on duration $\tau^{*}$ is given so that the controlled SDDEs-MS-LN is stable in distribution as long as $\tau < \tau^{*}$ and $h$ can be divisible by $\tau$. We specifically discussed how to design the linear feedback control in two structure cases. Finally, a numerical example is illustrated to support our theory. But the system coefficients are still under a linear growth condition, and the stochastic systems are driven by Markovian switching. Hence we will devote our future work to releasing the linear growth condition on $f, g$ ^[5] and investigating stabilization in distribution for nonlinear stochastic differential delay equations with semi-Markovian switching and Lévy noise (SDDEs-SMS-LN) controlled by discrete feedback controls ^[33,34].

Author contributions

Jingjing Yang: Formal analysis, methodology, investigation, resources, software, writing-original draft. Jianqiu Lu: Conceptualization, methodology, supervision, writing-review and editing, funding acquisition.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

All authors declare that there are no conflicts of interest.