STRUCTURED ROBUST STABILITY AND BOUNDEDNESS OF NONLINEAR HYBRID DELAY SYSTEMS

Taking different structures in different modes into account, the paper has developed a new theory on the structured robust stability and boundedness for nonlinear hybrid stochastic differential delay equations (SDDEs) without the linear growth condition. A new Lyapunov function is designed in order to deal with the effects of different structures as well as those of different parameters within the same modes. Moreover, a lot of effort is put into showing the almost sure asymptotic stability in the absence of the linear growth condition.

problems where the underlying systems were either linear or nonlinear with the linear growth condition (i.e., the coefficients are bounded by a linear function).
Hu, Mao, and Zhang [9] were the first to investigate robust stability and boundedness for nonlinear hybrid SDDEs without the linear growth condition (i.e., the coefficients are not bounded by a linear function, and we will refer to these coefficients as highly nonlinear functions).The significant contribution of [9] lies in that it shows that a given stable hybrid SDDE can tolerate not only the linear-type perturbation but also the highly nonlinear perturbation without loss of the stability, while the papers up to 2013 could only cope with the linear-type perturbation.In other words, Hu, Mao, and Zhang [9] opened a new chapter in the study of robust stability for highly nonlinear hybrid SDDEs.However, the progress in this direction is due somewhat to the difficulty of high nonlinearity, and [9] is the only paper so far, to the best of our knowledge.The aim of this paper is to make some further progress in this area.
Let us explain our key motivation briefly here, though further details will be given in section 3.As we know, hybrid SDDEs have been used to model practical systems that may experience abrupt changes in their structures and parameters (see, e.g., [3,5,13,23]).The theory in [9] is good at dealing with hybrid SDDEs that may experience abrupt changes in their parameters.To explain this, assume that a population system operates in two modes, dry and rain, and it switches from one mode to the other according to a two-state Markov chain with state 1 for dry and state 2 for rain.In the dry mode, the system is described by a stochastic delay Lotka--Volterra equation dx(t) = x(t)([a 1 -b 1 x 2 (t)]dt + \sigma 1 x(t -\tau )dB(t)), while in the rain mode by another equation dx(t) = x(t)([a 2 -b 2 x 2 (t)]dt + \sigma 2 x(t -\tau )dB(t)), where \tau > 0 stands for the time delay, a 1 , b 1 , a 2 , b 2 are all positive numbers, B(t) is a scalar Brownian motion, and \sigma 1 , \sigma 2 represent the intensities of the nonlinear stochastic perturbation.In other words, the population system is described by the hybrid SDDE dx(t) = x(t)([a r(t) - b r(t) x 2 (t)]dt+\sigma r(t) x(t - \tau )dB(t)).This can be regarded as a stochastically perturbed system of the hybrid delay system dx(t)/dt = x(t)[a r(t) - b r(t) x 2 (t)] with the highly nonlinear stochastic perturbation \sigma r(t) x(t)x(t -\tau )dB(t).Given the asymptotic boundedness of the delay system dx(t)/dt = x(t)[a r(t) -b r(t) x 2 (t)], the theory in [9] shows the upper bounds on \sigma 1 and \sigma 2 for the SDDE to remain asymptotically bounded.We observe that in this example, when the system switches from one mode to the other, only the system parameters change, but the structure of the system remains the same type of Lotka--Volterra.On the other hand, many practical systems may experience abrupt changes in their structures.For example, a population system may change from a delay geometric Brownian motion dx(t) = - 2x(t)dt + \sigma 1 x(t -\tau )dB(t) in the dry mode to a delay Lotka-Volterra equation dx(t) = x(t)[1 - 2x 2 (t)]dt+ \sigma 2 x 2 (t -\tau )dB(t) in the rain mode (see, e.g., [2]); a financial system may switch from a geometric Brownian motion dx(t) = a 1 x(t)dt + \sigma 1 x(t)dB(t) to a constant elasticity of volatility (CEV) process dx(t) = a 2 (\mu -x(t))dt + \sigma 2 x 1.5 (t)dB(t) (see, e.g., [15]).Is the theory in [9] applicable to such hybrid SDDEs?We will show a negative answer in section 3.This motivates us to develop a new theory on the robust stability and boundedness for highly nonlinear hybrid SDDEs which may experience abrupt changes in their structures.
To make our theory more general, we consider the case where the space of modes, S, of a given hybrid system can be divided into two proper subspaces, S 1 and S 2 , such that the system is described by the same type of SDDEs for modes in S 1 (though different parameters for different modes of course) but by a different type of SDDEs for modes in S 2 .For example, for the population system in the second half of the last paragraph, we have S = \{ dry, rain\} , S 1 = \{ dry\} , S 2 = \{ rain\} , and the system c \bigcirc 2018 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license Downloaded 07/19/18 to 130.159.82.56.Redistribution subject to CCBY license is described by a delay geometric Brownian motion for a mode in S 1 but by a delay Lotka--Volterra equation for a mode in S 2 .Of course, in our setting, both S 1 and S 2 could contain 2 or more modes (see Example 6.2).We should point out that it is possible to develop our theory to cope with the even more general case where S can be divided into more than two subspaces, and the structures of the underlying hybrid SDDE are significantly different among these subspaces.However, to avoid our notation becoming too complicated, we will only concentrate on the case of two subspaces in this paper.
The key contributions of our paper are highlighted below: \bullet This is the first paper that takes the different structures in different modes into account to develop a new theory on structured robust stability and boundedness for highly nonlinear hybrid SDDEs.\bullet The new theory established in this paper is applicable to hybrid SDDEs which may experience abrupt changes in both structures and parameters.\bullet The stabilities discussed in this paper include not only the pth moment and almost sure exponential stability but also the pth moment and almost sure asymptotic stability as well as H \infty stability.(For the definitions of these stabilities we refer the reader to [9,23].)\bullet A significant amount of new mathematics has been developed to deal with the difficulties due to the structured difference and those without the linear growth condition.For example, a new Lyapunov function will be designed in order to deal with the effects of different structures for S 1 -modes and S 2modes as well as the effects of different parameters within S 1 and S 2 .A lot of effort has also been put into showing the almost sure asymptotic stability without the linear growth condition.To develop our new theory, we will introduce some necessary notation in section 2. We will show in section 3 that the theory in [9] is not applicable to hybrid SDDEs which may experience abrupt changes in structures, and this motivates us to establish a new theory in this paper.Our main results on robust boundedness and stability will be discussed in sections 4 and 5.We will present some case studies and examples in section 6 to illustrate our theory.We will finally conclude our paper in section 7.
2. Notation.Throughout this paper, unless otherwise specified, we use the following notation.Let (\Omega , \scrF , \{ \scrF t \} t\geq 0 , P ) be a complete probability space with a filtration \{ \scrF t \} t\geq 0 satisfying the usual conditions (i.e., it is increasing and right continuous while \scrF 0 contains all P -null sets).Let B(t) = (B 1 (t), . . ., B m (t)) T be an m-dimensional Brownian motion defined on the probability space.Let r(t), t \geq 0, be a right-continuous-left-limit Markov chain on the probability space taking values in a finite state space S = \{ 1, 2, . . ., N \} with generator \Gamma = (\gamma ij ) N \times N given by where \Delta > 0.Here \gamma ij \geq 0 is the transition rate from i to j if i \not = j while \gamma ii = -\sum j\not =i \gamma ij .We assume that the Markov chain r(\cdot ) is independent of the Brownian motion B(\cdot ).We also denote by | x| the Euclidean norm for x \in R n .If A is a vector or matrix, its transpose is denoted by A T .If A is a matrix, its trace norm is denoted by We also need some notation on M-matrices.For a vector or matrix A, by A > 0 we mean all elements of A are positive.A Z-matrix is a square matrix A = (a ij ) N \times N which has nonpositive off-diagonal entries (namely a ij \leq 0 for all i \not = j).The following lemma provides us with two useful criteria to verify if a given Z-matrix is a nonsingular M-matrix (see, e.g., [4,9,23]).
Lemma 2.1.Let A = (a ij ) N \times N be a Z-matrix.Then A is a nonsingular M-matrix if and only if one of the following statements holds: (1) A - 1 exists and its elements are all nonnegative.
(2) There exists x > 0 in R N such that Ax > 0.
By this lemma, we see, for example, that for any positive numbers \varepsi i (i \in S), is a nonsingular M-matrix as A(1, . . ., 1) T = (\varepsi 1 , . . ., \varepsi N ) > 0. This useful technique will be used quite often when we discuss some special cases in section 6 below.

Motivation.
To motivate our new study in this paper, let us recall a key result on robust stability from [9].Consider an n-dimensional hybrid differential equation on t \geq 0 and assume that this hybrid system is subject to a stochastic delay perturbation and the perturbed system is described by a hybrid SDDE Here r(t), B(t), and \tau have been defined in section 2; both F : R n \times R + \times S \rightar R n and G : R n \times R + \times S \rightar R n\times m are Borel measurable and locally Lipschitz continuous in the first variable.In [9], the following assumption was imposed.
Assumption 3.1.Let q > p \geq 2 and assume that for each i \in S, there are a real number \= \beta i2 and a nonnegative number \= \beta i4 such that for all (x, t) \in R n \times R + , and It is shown in [9] that this assumption along with the local Lipschtiz condition guarantees the pth moment exponential stability of the given equation (3.1).The study of the robust stability is to investigate how much of the stochastic delay perturbation G(x(t -\tau ), t, r(t))dB(t) the given stable equation (3.1) can tolerate so that its perturbed system (3.2) remains stable.To measure the stochastic delay perturbation more precisely, the following assumption was then imposed in [9].Assumption 3.2.Let q > p \geq 2 be the same as in Assumption 3.1 and assume that for each i \in S, there are nonnegative numbers \= \beta i3 and \= \beta i5 such that for all (y, t) \in R n \times R + .The study of the robust stability is then to give the bounds on the parameters \= \beta i3 and \= \beta i5 in order for the perturbed system (3.2) to remain stable.The following theorem describes this situation.Theorem 3.3 (see [9,Theorem 3.4]).Let Assumptions 3.1 and 3.2 hold.Assume that F (0, t, i) = G(0, t, i) = 0 for all t \geq 0 and i \in S. Define  for all i \in S, then the perturbed system (3.2) is exponentially stable in the pth moment.
The significant contribution of this theorem lies in that it shows not only how much of the linear perturbation (controlled by \sqrt{} \= \beta i3 | y| ) but also how much of the nonlinear perturbation (controlled by \sqrt{} \= \beta i5 | y| q - p+2 ) the given stable equation (3.1) can tolerate without loss of the stability, while the existing papers up to 2013 could only cope with the linear perturbation as pointed out in section 1.
However, we shall now point out its limitation.Recall the population system stated in section 1: It operates in two modes: dry and rain.Assume that the switching between the two modes is modeled by a Markov chain r(t) on the state space S = \{ 1, 2\} (1 for dry and 2 for rain) with the generator .
The system is modeled by the hybrid SDDE (3.9) dx(t) = F (x(t), r(t))dt + G(x(t -\tau ), r(t))dB(t), where B(t) is a scalar Brownian motion and for x, y \in R, in which both \sigma 1 and \sigma 2 are positive constants.That is, the system satisfies a delay geometric Brownian motion dx(t) = - 2x(t)dt+\sigma 1 x(t - \tau )dB(t) in the dry mode but a delay Lotka--Volterra equation dx(t) = x(t)[1 -2x 2 (t)]dt + \sigma 2 x 2 (t -\tau )dB(t) in the rain mode.In other words, the system experiences abrupt changes in their structures when it switches from one mode to the other.If both \sigma 1 = 0 and \sigma 2 = 0, (3.9) becomes In other words, (3.9) is a stochastically perturbed system of (3.10).Noting that xF (x, 1) = - 2x So \= \scrA is a nonsingular M-matrix.In other words, Assumption 3.1 is satisfied.This implies that (3.10) is exponentially stable in mean square.We expect that (3.10) can tolerate a liner perturbation \sigma 1 x(t -\tau )dB(t) in mode 1 and a nonlinear perturbation \sigma 2 x 2 (t -\tau )dB(t) in mode 2 given its linear and nonlinear structure in modes 1 and 2, respectively.The aim here is to obtain upper bounds on \sigma 1 and \sigma 2 so that the perturbed system (3.9)remains stable.Noting that Unfortunately, we never have \sigma 2 2 < 0 so Theorem 3.3 is not applicable to the hybrid SDDE (3.9).This indicates that the theory in [9] may not be applicable to the hybrid SDDEs that may experience abrupt changes in their structures.
It is very easy to verify this local Lipschitz assumption.For example, the assumption is satisfied if f and g are continuously differentiable in x and y or they are differentiable in x and y with locally bounded derivatives.It is known that this classical assumption covers many hybrid SDDEs in the real world (see, e.g., the books [23,24] and the references therein).Of course, this assumption is not enough to guarantee the global solution (i.e., no explosion at a finite time).A standard additional condition for the existence and uniqueness of the global solution of the SDDE (4.1) would be the linear growth condition (see, e.g., [18,23]).However, our aim here is to study the structured robust stability and boundedness of highly nonlinear SDDEs that do not satisfy the linear growth condition.We hence need to propose alternative assumptions.
Assumption 4.2.Assume that the state space S of the Markov chain is divided into two proper subspaces S 1 and S 2 , and we may, without loss of any generality, let S 1 = \{ 1, . . ., N 1 \} and S 2 = \{ N 1 + 1, . . ., N \} , where 1 \leq N 1 < N .Assume also that there are two constants q > p \geq 2. Assume furthermore that for each i \in S 1 , there are constants \alpha i2 \in R and \alpha i1 , \alpha i3 \in R + such that, for all (x, y, t) \in R n \times R n \times R + , (4.2) while for each i \in S 2 , there are constants \alpha i2 \in R, \alpha i4 > 0, and \alpha i1 , \alpha i3 , \alpha i5 \in R + such that The reason why S is divided into two proper subspaces S 1 and S 2 is because the structure of the underlying hybrid SDDE in S 1 -modes differs from that in S 2modes, as explained in section 1.In terms of mathematics, conditions (4.2) and (4.3) describe the difference in structure.More understandably, condition (4.2) means that the hybrid SDDE in S 1 -modes satisfies the classical Khasminskii-type condition (see, e.g., [14,23]) while condition (4.2) means that the hybrid SDDE in S 2 -modes satisfies the generalized Khasminskii-type condition (see, e.g., [10]).In layman's terms, the coefficients of the SDDE in S 1 -modes may grow linearly in the delay component x(t -\tau ) while in S 2 -modes it may grow polynomially.It is easy to show whether a function grows linearly or polynomially, and hence it is not difficult to verify our Assumption 4.2, as demonstrated in our examples in section 6.
Noting that in Assumption 4.2, we only require \alpha i2 \in R for all i \in S. According to the Khasminskii-type theorems (see, e.g., [14,10,23]), the solution of the hybrid SDDE may grow exponentially.But our aim in this paper is to study the asymptotic boundedness and stability.We therefore need to impose some additional conditions on \alpha i2 's.This assumption means that some \alpha i2 must be negative; otherwise A and D could not be nonsingular M-matrices.Hence, the SDDE in mode i with \alpha i2 < 0 should be asymptotically bounded or stable.Of course, the SDDE in mode i with \alpha i2 \geq 0 could still grow.However, conditions (4.4) and (4.5) mean that the switchings from those modes with \alpha i2 \geq 0 to those with \alpha i2 < 0 are sufficiently fast so that, overall, the underlying hybrid SDDE is still asymptotically bounded or stable.We should also point out that Assumption 4.3 can be verified easily.In fact, compute A - 1 and D - 1 easily using MATLAB or R and then check if their elements are all nonnegative.If so, by Lemma 2.1, they are nonsingular M-matrices.
When we design our Lyapunov function (see (4.15)), we will need two sets of numbers where \beta is a free positive parameter.Under Assumption 4.3, we see, by Lemma 2.1, that all \theta i (i \in S) and \eta i (i \in S 1 ) are positive.We will see that \beta plays a key role in balancing the effects of different structures for S 1 -modes and S 2 -modes.In particular, if we choose \beta sufficiently small, then all \eta i will be small too.This means that we can always make condition (4.9) in the following theorem possible by choosing \beta sufficiently small.In particular, let us state a remark where we show a simple method on how to determine \beta to guarantee condition (4.9 real world while they can be verified easily.Remark 4.4 shows at least one way of determining \beta to make condition (4.9) hold.The right-hand-side terms of inequalities (4.10)--(4.12)can then be computed straightaway, and these inequalities give the bounds on the nonlinear perturbation intensities \alpha i3 and \alpha i5 so that the underlying hybrid SDDE is bounded in L p asymptotically as well as in the time-average of L q .
Proof.The proof is very technical.To make it more understandable, we will divide it into several steps.
Step 1.In this step, we will define a Lyapunov function V : R n \times S \rightar R + by and show that it has some nice properties.First, it is easy to see that where .
By the generalized It\ô formula (see, e.g., [23, Theorem 1.45, page 48]), we have that on t \geq 0, where M (t) is a continuous local martingale with M (0) = 0 (the explicit form of M (t) is of no use in this paper but can be found in [23]), and the function LV : R n \times R n \times R + \times S \rightar R is defined by in which Let us first estimate LV (x, y, t, i) for i \in S 1 .In this case, we have By Assumption 4.2, we then have But, by (4.6) and (4.7), we have \gamma ij \eta j = - \beta . Hence Note that p\theta i \alpha i3 \leq 1 by condition (4.10), while by the well-known Young inequality (see [23, Similarly, for i \in S 2 , we can show that By condition (4.10) and the Young inequality, we then obtain that, for i \in S 2 , LV (x, y, t, i) Combining (4.21) and (4.25), we see that, for all i \in S, q\eta i \alpha i1 \Bigr) .
Then both \beta 1 and \beta 2 are positive numbers.Noting that \beta -\\beta (q -2) we obtain from (4.26) that, for all i \in S, Step 2. In this step, we will show the existence and uniqueness of the global solution of the SDDE (4.1) given any initial data \xi \in C([ - \tau , 0]; R n ).Under Assumption 4.1, it is known (see, e.g., [23, Theorem 7.12, page 278]) that there is a unique maximal local solution x(t) on t \in [ - \tau , \sigma \infty ), where \sigma \infty is the explosion time.To show this is a unique global solution, we need to show \sigma \infty = \infty a.s.Let k 0 > 0 be a sufficiently large integer such that \| \xi \| < k 0 .For each integer k \geq k 0 , define the stopping time where throughout this paper we set inf \emptyse = \infty (as usual \emptyse denotes the empty set).It is easy to see that \tau k is increasing as k \rightar \infty and \tau \infty := lim k\rightar\infty \tau k \leq \sigma \infty a.s.Hence the aim of this step will be done if we can show that \tau \infty = \infty a.s.
for all (x, y, t, i) \in R n \times R n \times R + \times S. Set Substituting this into (4.28)yields Applying the generalized It\ô formula, we then have ) we have This, along with (4.16) and (4.30), implies that (4.31) where Consequently c 1 k p P (\tau k \leq t) \leq c 5 + c 4 t.
Step 3. We shall show assertion (4.13In this section we will discuss the robust stability of the SDDE (4.1).For this purpose, we will assume that f (0, 0, t, i) = 0 and g(0, 0, t, i) = 0 for all (t, i) \in R + \times S. Hence the SDDE (4.1) admits a trivial solution x(t) = 0 for all t \geq 0 when the initial data \xi = 0.It is also natural to let \alpha i1 = 0 for all i \in S in Assumption 4.2.The following theorem gives a criterion on the H \infty -stability in L q .Theorem 5.1.Let all the conditions in Theorem 4.5 hold and, moreover, \alpha i1 = 0 for all i \in S. Then for any initial data \xi \in C([ - \tau , 0]; R n ), the unique global solution x(t) of the SDDE (4.1) has the property that (5.1) Proof.We use the same notation as in the proof of Theorem 4.5.Clearly, everything we showed there is correct.In particular, c 3 = 0 in (4.27) given that \alpha i1 = 0 for all i \in S. Hence, (4.27) becomes It is then easy to show by the generalized It\ô formula that 2\beta Letting t \rightar \infty yields assertion (5.1).
In general it is not possible to imply lim t\rightar\infty E| x(t)| q = 0 from (5.1).On the other hand, You et al. [28] showed this is possible if both coefficients f and g of the SDDE (4.1) satisfy the linear growth condition.However, we are interested in the SDDEs which do not satisfy the linear growth condition in this paper.It is therefore useful if we can show lim t\rightar\infty E| x(t)| q = 0 from (5.1) without the linear growth condition.The following theorem describes this possibility which is one of our new contributions in this paper.
Theorem 5.2.In addition to the same conditions as in Theorem 5.1, assume that there is a positive constant K such that for all (x, y, t) \in R n \times R n \times R + .Then for any initial data \xi \in C([ - \tau , 0]; R n ), the unique global solution x(t) of the SDDE (4.1) has the property that Proof.Fix any initial data \xi \in C([ - \tau , 0]; R n ).If (5.4) were not true, there must exist a positive number \varepsi and a sequence of positive numbers \{ t k \} k\geq 1 such that t k \rightar \infty as k \rightar \infty and (5.
On the other hand, for any k \geq k 0 and t \in [t k -\tau , t k ], it is easy to show by the It\ô formula that By condition (5.3) and inequality (5.6), we derive This, together with (5.5), implies (5.9) \varepsi \leq E| x(t k )| q -\varepsi \leq E| x(t)| q \forall t \in [t k -\tau , t k ].
In general it is not possible to imply lim t\rightar\infty | x(t)| = 0 a.s.from (5.1).However, this is possible in our case and we will show this under the same conditions of Theorem 5.1 without any additional condition, unlike Theorem 5.2 which needs the additional condition (5.3).We should also point out that You et al. [28] showed lim t\rightar\infty | x(t)| = 0 a.s.from E \int \infty 0 | x(t)| 2 dt < \infty (please note it is 2 but not q) under the linear growth condition.Our new proof given below not only overcomes the difficulty without the linear growth condition but is also much simplified.Fix k from now on and define the stopped process y(t) = x(t \wedge \tau k ) for t \geq 0. Clearly, y(t) is an It\ô process of the form (5.17) By Assumption 4.1 as well as f (0, 0, t, i) = 0 and g(0, 0, t, i) = 0, we see that \= This, together with (5.20), implies Noting that \rho 2i - 1 \leq T if \tau 2j \leq T , we can derive from (5.21) and the above inequality that | y(t)| q dt \Bigr) \geq \varepsi q+1 \delta j.
But this contradicts the second inequality in (5.20).Therefore the desired assertion (5.11) must hold.
The theorems above do not show how quickly the solution will tend to the equilibrium state 0 as t \rightar \infty .It would be more desirable if we could describe the rate of this asymptotic convergence.The exponential stability meets this desire.Let us now discuss the robustness of the pth moment and almost sure exponential stability to close this section.Proof.In the same way that (4.27) was proved, we can show from (4.20) and (4.24) that where \\alpha := max i\in S p\theta i \alpha i3 < 1 by condition (5.24).This implies By the generalized It\ô formula, we have that + LV (x(s), x(s -\tau ), s, r(s)) \Bigr) ds + M (t) (5.31) on t \geq 0, where M (t) is a continuous local martingale with M (0) = 0. Making use of (4.16) and (5.28)--(5.30),we can then easily show where c 11 is a positive number dependent on the initial data only.Since M (t) is a local martingale, there is a sequence \{ \\tau k \} \infty k=1 of stopping times such that \\tau k \rightar \infty as k \rightar \infty while for each k, M (t \wedge \\tau k ) is a martingale on t \geq 0. It follows from (5.32) that, for each k \geq 1, Taking the expectations on both sides yields (5.34) Letting k \rightar \infty , we get assertion (5.25) immediately.Moreover, by the nonnegative semimartingale convergence theorem (see, e.g., [23, Theorem 1.10, page 18]), we have lim sup t\rightar\infty \Bigl( c 1 e \lambda t | x(t)| p \Bigr) < \infty a.s., which implies another assertion (5.26).

Special cases and examples.
In this section we will discuss a number of special cases of hybrid SDDEs in order to demonstrate how our new theory established in the previous two sections can be applied to show the robustness of boundedness and stability of a given hybrid system subject to various types of nonlinear stochastic perturbations.As a standing hypothesis in this section, we will assume that all coefficients of SDDEs in this section will satisfy the local Lipschitz condition and, moreover, q > p \geq 2. To make our cases a bit simpler, we assume that the given hybrid system is described by a hybrid differential equation (6.1) dx(t)/dt = F (x(t), t, r(t)).
Its structured differences and various stochastic perturbations will be discussed in the following cases.We leave the situation to the reader where the given hybrid system is described by a hybrid differential delay equation dx(t)/dt = f (x(t), x(t -\tau ), t, r(t)).x T F (x, t, i) \leq for (x, t, i) \in R n \times R + \times S.Here a i2 > 0 for i \in S but, for the structured difference, we let a i1 < 0 for i \in S 1 and a i1 \in R for i \in S 2 .This means that the differential equation in mode i \in S 1 is stable, but it may not be in mode i \in S 2 .In order for the hybrid equation (6.1) to be stable, we assume moreover that (6.3) \scrA := - diag(pa 11 , . . ., pa N 1 ) -\Gamma is a nonsingular M-matrix.It is then known (see, e.g., [9]) that equation (6.1) is exponentially stable in the pth moment.Suppose that this equation is subject to a stochastic perturbation and the perturbed system is described by (6.4) dx(t) = F (x(t), t, r(t))dt + G(x(t), x(t -\tau ), t, r(t))dB(t), and the perturbation has its structured difference: when mode i \in S 1 , the perturbation is independent of x(t -\tau ), namely but when mode i \in S 2 , the perturbation is independent of x(t), namely Assume furthermore that (6.5) | G 1 (x, t, i)| \leq a i3 | x| q - p+2 , i \in S 1 , and where a i3 > 0. Our aim here is to give a bound on a i3 so that the perturbed system (6.4) remains stable.Note that for i \in S 1 Hence, if we impose the bounds (6.7) a i3 \leq 2a i2 q -1 , i \in S 1 , then Assumption 4.2 is satisfied with \alpha i1 = 0, \alpha i2 = a i1 , \alpha i3 = 0 for i \in S, \alpha i4 = a i2 , \alpha i5 = 0.
then condition (4.12) is satisfied as well.We can therefore conclude by Theorem 5.4 that the perturbed system (6.4) is both pth moment and almost surely exponentially stable provided the perturbation parameters a i3 satisfy conditions (6.7) and (6.9).

Case 2.
Assume that for each i \in S 1 , there is a number a i1 < 0 such that (6.10) while for each i \in S 2 , there is a pair of numbers a i1 \in R and a i2 > 0 such that (6.11) for (x, t) \in R n \times R + .We also assume that the matrix \scrA defined by (6.3) is a nonsingular M-matrix.Suppose that (6.1) is subject to a stochastic perturbation dependent on the delay state x(t -\tau ) and the perturbed system is described by (6.12) dx(t) = F (x(t), t, r(t))dt + G(x(t -\tau ), t, r(t))dB(t), and the perturbation has its structured difference in the sense that (6.13) | G(y, t, i)| \leq a i3 | y| 2 , i \in S 1 , and for (y, t) \in R n \times R + , where a i3 > 0 for all i \in S. Once again, we wish to obtain upper bounds on a i3 's for the perturbed system (6.12) to remain stable.Noting that for i \in S 1 x T F (x, t, i) + 0.5(q -1)| G(y, t, i)| 2 \leq a i1 | x| 2 + 0.5(q -1)a i3 | y| 2 while for i \in S 2 x T F (x, t, i) + 0. = 0, \alpha i2 = a i1 for i \in S, \alpha i3 = 0.5(q -1)a i3 for i \in S 1 , \alpha i3 = 0, \alpha i4 = a i2 , \alpha i5 = 0.5(p -1)a i3 for i \in S 2 .
We hence conclude that under condition (6.18), the SDDE (3.9) is both mean square and almost surely exponentially stable.
To perform computer simulations, we set \sigma 1 = 0.8, \sigma 2 = 1.2, and \tau = 0.1 and let the initial data \xi (t) = 2 + sin(t) on t \in [ - 0.1, 0] and r(0) = 2.The following computer simulations (Figure 6.1) support our theoretical results clearly.Fig. 6.1.The computer simulations of the sample paths of the Markov chain and the solution of (3.9) with the parameters and initial data specified above using the Euler--Maruyama method with step size 10 - 3 .

Case 3.
In this case we will discuss the robust boundedness.Assume that (6.19) x where all a i1 and a i2 are positive numbers.Suppose that the perturbed system is described by (6. where a i3 and a i4 are all nonnegative numbers.We aim to obtain upper bounds on them so that the perturbed system (6.21)remains asymptotically bounded.It follows from these conditions that for i \in S 1 x T F (x, t, i) + 0.5(q -1)| G(y, t, i)| 2 \leq a i1 -a i2 | x| 2 + 0.5(q -1)a 13 | y| 2 , (6.24) while for i \in S 2 Consequently, the matrix A defined by (4.4) becomes A = diag(pa 12 , . . ., pa N12 , 0, . . ., 0) -\Gamma .
As a result, Assumption 4.2 is satisfied with Let S 1 = \{ 1, 2\} , S 2 = \{ 3, 4\} and p = 2, q = 4.It is straightforward to show that conditions (6.19), (6.20), (6.22), and (6.23) are satisfied with a 12 = 1.9, a 22 = 2.9, a 32 = 1.9, a 42 = 2.9, We can therefore conclude that if the perturbed parameters \sigma i satisfy (6.30), then for any initial data \xi \in C([ - \tau , 0]; R), the solution x(t) of the SDDE (6.29) has the properties that lim sup To perform a computer simulation for the second moment of the solution, we set \sigma 1 = 1, \sigma 2 = \sigma 3 = \sigma 4 = 1.3, and \tau = 0.1 and let the initial data \xi (t) = 1 + sin(t) on t \in [ - 0.1, 0] and r(0) = 1.The computer simulations in Figure 6.2 show a single sample path of the Markov chain and that of the solution, from which we can see how the Markov chain jumps from one mode to another and also the solution evolves in a bounded domain.To illustrate the boundedness of the second moment, we perform 200-sample-path simulations and then compute the average of their squares to form the approximation of E| x(t)| 2 .This is shown in Figure 6.3.

Conclusion.
To distinguish the difference in structures of the underlying hybrid system, we have considered the case where the space of modes, S, can be divided into two subspaces, S 1 and S 2 , such that the system is described by the same type of SDDEs for modes in S 1 but by a different type of SDDEs for modes in S 2 .Taking these different structures into account, we have successfully developed our new theory on the structured robust stability and boundedness for highly nonlinear hybrid SDDEs.A significant number of new techniques have been developed to deal with the difficulties due to the structured difference and those without the linear growth condition.The proofs of Theorems 4.5 and 5.3 typically represent our new techniques.We have also discussed three special cases and two examples plus some computer simulations to illustrate our theory.

Theorem 5 . 3 .
Under the same conditions of Theorem 5.1, for any initial data \xi \in C([ - \tau , 0]; R n ), the unique global solution x(t) of the SDDE (4.1) has the property that (5.11) lim t\rightar\infty | x(t)| = 0 a.s.c \bigcirc 2018 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license
.25) c \bigcirc 2018 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license Downloaded 07/19/18 to 130.159.82.56.Redistribution subject to CCBY license If we compare these with (4.2) and (4.3) in Assumption 4.2, we might attempt to have \alpha i2 = - a i2 for i \in S 1 and 0 for i \in S 2 .