1484 J. LU ET AL.
system states also suers a time delay. This moti-
vated (Qiu et al., 2016) to design a discrete-time state
feedback controller with a time delay to ensure the
controlled system is exponential stable in mean square.
On the other hand, the feedback controls men-
tioned above are based on discrete-time observations
of system state but they still depend on continuous-
time observations of system mode. Of course this is
perfectly ne if the mode of the system is fully observ-
able at no cost. However, the mode is not obvious
in many real-world situations and it costs to identify
the current mode of a hybrid stochastic system. Also,
Geromel and Gabriel (2015) emphasised the necessity
to design the feedback control based on discrete-time
observations of both state and mode from the numer-
ical point of view when studying the state feedback
sampled-data control design for Markov jump sys-
tems.Soitisimportantandreasonabletoobservethe
systemmodeatdiscretetimesaswellwhendesign-
ing a discrete-time feedback control (Li et al., 2017).
Lu et al. (2020) designed a feedback control not only
based on discrete-time state observations but also
mode observations to make the controlled delay sys-
tem become exponential stable in mean square.
It can be observed from above that in the case
of discrete-time feedback control, stability in mean-
square sense has been widely studied. However, in
practical problems, for instance, some digital image
processes require the stability moment order up to
50 (Fonseca, 2016; Singh & Upneja, 2014), while in
some other problems only the lower moment stability
is required. Therefore, it is necessary and meaningful
to analyse the stability of stochastic systems in dier-
ent moments (Dong & Mao, 2017;Huangetal.,2008;
Wu et al., 2013). Zhu and Zhang (2017)showed
that unstable hybrid SDEs can be stabilised in pth
moment exponential sense by linear feedback control
based on discrete-time state observations with a time
delay when p ≥ 2. Recently, by comparison theorem
between an SDEs and an SDDEs, Hu et al. (2020)
proved that a given unstable SDE can be stabilised by
delay feedback control in the more general sense, i.e.
exponential stability in pth moment (p > 0) as well as
exponential stability in almost sure sense.
Giventheabovebackground,thecoremissionof
this paper is to study on stabilisation of a given nonlin-
ear hybrid stochastic system with time-varying delay
in the sense of pth moment by feedback control based
on discrete-time state and mode observations with a
time delay (p > 1), which still remains as an impor-
tant and meaningful problem. Meanwhile, the upper
bound on the duration will be obtained. The signi-
cant contributions of this work are listed as follows:
(1) Compared to the existing feedback controls (see
e.g. Mao, 2013;Qiuetal.,2016;Lietal.,2017), our
hybrid control design technique is more practical
and reasonable, which is not only based on the
discrete-time observations of both state and the
mode, but also suering an constant time delay in
state observation.
(2) We establish a new Lyapunov functional to sta-
bilise a given hybrid SDDE in the sense of H
∞
stability, asymptotic stability, exponential stabil-
ity, and asymptotic exponential stability in pth
moment (p > 1).
Therestofthispaperisorganisedasfollows.We
state some necessary preliminaries and present the
problem formulation in Section 2.Sections3– 5 state
our main results. Section 6 gives numerical examples
to illustrate the theoretical ndings. Finally, Section 7
concludes this paper.
2. Notation and stabilisation problem
Throughout this paper, unless otherwise specied, we
let (,
F , {F
t
}
t≥0
, P) be a complete probability space
with a ltration {
F
t
}
t≥0
satisfying the usual conditions
(i.e. it is increasing and right continuous while
F
0
con-
tains all
P-null sets). Let w(t) = (w
1
(t), ..., w
m
(t))
T
be an m-dimensional Brownian motion dened on
the probability space. If A isavectorormatrix,its
transpose is denoted by A
T
.Ifx ∈ R
n
,then|x| is its
Euclidean norm. If A is a matrix, we let |A|=
(A
T
A)
be its trace norm and A=max{|Ax| : |x|=1} be
theoperatornorm.IfA is a symmetric matrix (A =
A
T
), denote by λ
min
(A) and λ
max
(A) its smallest and
largest eigenvalue, respectively. By A ≤ 0andA < 0, we
mean A is non-positive and negative denite, respec-
tively. If both a, b are real numbers, then a ∨ b =
max{a, b} and a ∧ b = min{a, b}.IfA isasubsetof,
denote by I
A
its indicator function; that is I
A
(ω) = 1
when ω ∈ A and 0 otherwise.
Let r(t), t ≥ 0, be a right-continuous Markov chain
ontheprobabilityspacetakingvaluesinanitestate
space S ={1, 2, ..., N} with generator = (γ
ij
)
N×N
