arXiv:1603.01751v4 [math.OC] 1 Aug 2016

Stability analysis and stabilization of stochastic linear
impulsive, switched and sampled-data systems under
dwell-time constraints
Corentin Briat
Department of Biosystems Science and Engineering, ETH–Zürich, Switzerland.

Abstract
Impulsive systems are a very flexible class of systems that can be used to represent switched and sampled-data systems.
We propose to extend here the previously obtained results on deterministic impulsive systems to the stochastic setting. The
concepts of mean-square stability and dwell-times are utilized in order to formulate relevant stability conditions for such
systems. These conditions are formulated as convex clock-dependent linear matrix inequality conditions that are applicable to
robust analysis and control design, and are verifiable using discretization or sum of squares techniques. Stability conditions
under various dwell-time conditions are obtained and non-conservatively turned into state-feedback stabilization conditions.
The results are finally applied to the analysis and control of stochastic sampled-data systems. Several comparative examples
demonstrate the accuracy and the tractability of the approach.
Key words: Stochastic impulsive systems; stochastic sampled-data systems; dwell-times; clock-dependent conditions

1

Introduction

Impulsive systems [1, 2] arise in many applications such
as such as ecology [3, 4], epidemiology [5] and sampleddata systems/control [6–8]. Lyapunov-based methods
and dwell-time notions [9] can be used to establish
various stability concepts and conditions. Various concepts of dwell-time have been considered over the past
decades. Important examples are the minimum dwelltime [9–12], the average dwell-time [9,13], the persistent
dwell-time [2, 9, 14], the maximum dwell-time [8, 12, 15],
the ranged-dwell-time [8,12,15] and the mode-dependent
dwell-time [14,16,17]. Notably, minimum dwell-time stability conditions for linear impulsive systems have been
obtained in [12] following the ideas developed in [11]
in the context of switched systems. It is shown in [12]
that despite a sufficient stability condition can be expressed as a tractable linear matrix inequality problem,
this type of conditions are only applicable to linear timeinvariant systems without uncertainties and are impossible to convert into design conditions, even in the simple case of state-feedback design [8, 12, 15]. These drawEmail address:
briatc@bsse.ethz.ch,corentin@briat.info (Corentin
Briat).
URL: http://www.briat.info (Corentin Briat).

Preprint submitted to Automatica

backs motivated the consideration of looped-functionals
[12, 15, 18, 19], a particular class of indefinite functionals
satisfying a certain boundary condition, referred to as
the looping-condition. These functionals have the merit
of leading to conditions that are convex in the matrices
of the system, hence easily applicable to linear uncertain
systems with time-varying uncertainties and to nonlinear systems [20]. The price to pay for these interesting
properties is that the resulting conditions are infinitedimensional semidefinite programs, that may then be
solved using discretization techniques [21, 22] or sum of
squares programming [8,23]. Due to the presence of additional infinite-dimensional decision functions, the complexity of the conditions may not scale very well with
the dimension of the system and/or the number of basis functions used to express infinite-dimensional variables. Moreover, looped-functionals are limited to stability analysis and are difficult to consider for design purposes. To circumvent this problem, clock-dependent conditions have been considered in [8, 17, 19, 24] where it is
shown that these conditions possess the same advantages
than looped-functional conditions (i.e. the possibility of
considering nonlinear and uncertain linear systems) together with the additional possibility of using them for
design purposes. Despite these frameworks have been recently shown to be theoretically equivalent in [19], because the same accuracy is attained with a lower compu-

2 August 2016

been thoroughly considered in the literature. This paper
therefore fills this gap by providing tractable conditions
for both the analysis and the control of such systems.
Various comparative examples demonstrate the accuracy and tractability of the approach.

tational cost, it is then preferable to use clock-dependent
stability conditions rather than looped-functional-based
ones.
We propose to address here the case of stochastic impulsive systems where the continuous-time part consists
of a linear stochastic differential equation with state
multiplicative noise [25] and the discrete-time part is a
stochastic difference equation [26]. The impulses arrival
times are considered here as purely time-dependent and
deterministic. Stochastic hybrid systems have been extensively studied in the literature; see e.g. [27, 28] and
references therein. However, very few address the case
where both parts of the impulsive system are affected
by noise [29, 30] and, when this is the case, average
dwell-time conditions, state-dependent impulse times or
stochastically arriving impulses [31, 32] are most of the
time only considered.

Outline: The structure of the paper is as follows: in
Section 2 preliminary definitions and results are given.
Section 3 is devoted to dwell-time stability analysis while
Section 4 addresses dwell-time stabilization. Sampleddata systems are finally treated in Section 5. Examples
are considered in the related sections.
Notations: The cone of symmetric (positive definite)
matrices of dimension n is denoted by Sn (Sn0 ). The sets
of integers and whole numbers are denoted by N and
N0 , respectively. Given a vector v, its 2-norm is defined
as ||v||2 = (v T v)1/2 . For a square matrix A, we define
He[A] := A + AT . The symbols ⊕ and ⊗ are used for
denoting the Kronecker sum and product, respectively.

The main goal of the paper is therefore to develop stability conditions in the same spirit as in [8, 12, 17, 19]
where the deterministic case was considered. To this
aim, we consider the notion of dwell-times and obtain
a necessary and sufficient condition characterizing the
mean-square stability under constant dwell-time, and
sufficient conditions establishing the mean-square stability under ranged and minimum dwell-time conditions.
These conditions, albeit stated in a more implicit way,
naturally generalize those obtained in the deterministic
setting as they reduce to the deterministic conditions
when the noise-related terms are set to zero. Due to
the implicit structure of the conditions, they cannot be
checked per se. In order to overcome this difficulty, and
in the same spirit as in [8, 17, 19], lifted versions of the
conditions expressed as clock-dependent linear matrix
inequalities are considered. These conditions have the
benefits of being convex in the matrices of the system,
a property that enables their use in the contexts of uncertain systems and control design. Because these conditions are infinite-dimensional, they cannot be checked directly and need to be relaxed. Possible relaxation methods include the approximation of infinite-dimensional
variables using a piecewise-linear approximation [21, 22]
or the use of sum of squares programming [8, 23]. It
is emphasized that these relaxed conditions are asymptotically exact in the sense that they can approximate
arbitrarily well the original conditions they have been
derived from. These conditions are shown to include
those of [33], which characterize the mean-square stability stochastic linear switched systems, by exploiting the
possibility of representing switched systems as impulsive systems. The approach is then non-conservatively
extended to state-feedback design, for which convex conditions are also obtained. Finally, the analysis and control of aperiodic stochastic sampled-data systems is performed using the proposed method by reformulating the
considered sampled-data system into an impulsive system. It is worth mentioning here that sampled-data systems driven by multiplicative noise do not seem to have

2

Preliminaries

From now on, the following class of linear stochastic impulsive system
dx(t) = [Ax(t) + Bc1 uc (t)]dt + Ec x(t)dW1 (t)
+Bc2 uc (t)dW2 (t), t 6= tk
1
x(t+
k ) = Jx(tk ) + Bd ud (k) + Ed x(tk )ν1 (k)

+Bd2 ud (k)ν2 (k),

(1)

k∈N

x(0) = x0
is considered where x, x0 ∈ Rn , uc ∈ Rmc and ud ∈ Rmd
are the state of the system, the initial condition, the
continuous control input and the discrete control input, respectively. The notation x(t+ ) is a shorthand for
lims↓t x(s), i.e. the trajectories are assumed to be leftcontinuous. The sequence of impulse instants {tk }k∈N
is defined such that Tk := tk+1 − tk ≥ Tmin for some
Tmin > 0. This then implies that {tk }k∈N0 , t0 = 0,
is increasing without bound. The processes W1 (t) and
W2 (t) are independent scalar Wiener processes; i.e. for
i = 1, 2, we have that Wi (0) = 0, Wi (t) is almost
surely everywhere continuous and has independent increments Wi (t) − Wi (s), 0 ≤ s < t, that are normally
distributed with zero mean and variance t − s. The sequence {νi (k)}k∈N0 , i = 1, 2, is sequence of independent
identically distributed random variables with zero mean
and unit variance that are independent of x(tk ) for all
k ∈ N. Let (Ω, F, (Ft,k )t≥0,k∈N0 , P) be a complete probability space with σ-algebra F and natural (hybrid) filtration Ft,k (for more details on hybrid filtrations see [34]).
Let E[·] be the expectation operator with respect to P.
Definition 1 We say that the system (1) with ud , uc ≡ 0

2

is mean-square asymptotically stable if E[||x(t)||22 ] → 0
as t → ∞.

retrieved by setting Ec = 0. Associated with Φ(t), we
define for any Z ∈ Sn the following quadratic expression

Lemma 2 Let us consider a sequence {tk } for which
0 < Tmin ≤ Tk ≤ Tmax < ∞ for all k ∈ N0 . Then, the
system (1) with ud , uc ≡ 0 is mean-square asymptotically
2
stable if and only if E[||x(t+
k )||2 ] → 0 as k → ∞.

ΞZ (s) := Φ(s)T ZΦ(s), s ≥ 0.

(3)

With these definitions in mind, we can now move forward
to the main results of the section.

Proof : It is clear that if the system is mean-square
2
asymptotically stable then E[||x(t+
k )||2 ] → 0 as k → ∞.
To prove the converse, first note that, when ud , uc ≡ 0,
dE[||x(t)||22 ]
= E[x(t)T (He[A]+EcT Ec )x(t)] ≤ βE[||x(t)||22 ]
dt
for all t ∈ (tk , tk+1 ) and for any large enough β > 0. This
2
then implies that E[||x(tk + τ )||22 ] ≤ eβTmax E[||x(t+
k )||2 ],
+ 2
which implies in turn that if E[||x(tk )||2 ] → 0 as k → ∞,
then supτ ∈(0,Tk ] E[||x(tk + τ )||22 ] → 0 as k → ∞. The
proof is complete.
♦

3.1

Constant dwell-time

Let us first consider the case of constant dwell-time, that
is, the case where, for some T̄ > 0, we have that Tk = T̄
for all k ∈ N0 . We then have the following result:
Theorem 5 The following statements are equivalent:
(i) The system (1) with uc ≡ 0, ud ≡ 0 is meansquare asymptotically stable under constant dwelltime, that is, for the sequence of impulse times verifying Tk = T̄ , k ∈ N,
(ii) The matrix M(T̄ ) := exp(AT̄ )J defined with

The above result remains valid as long as the impulses
are persistent, i.e. Tk < ∞. In the case where the impulses are not persistent (i.e. there exists a k ∗ ∈ N0 such
that Tk < ∞ for k = 0, . . . , k ∗ − 1 and Tk∗ = ∞), then
the asymptotic mean-square stability of (1) eventually
becomes equivalent to the asymptotic mean-square stability of the continuous-time part of (1). Note also that
this result remains valid when state-dependent control
inputs are considered (e.g. state-, output- and dynamicoutput feedback).

A := A ⊕ A + Ec ⊗ Ec and J := J ⊗ J + Ed ⊗ Ed
is Schur stable.
(iii) There exists a matrix P ∈ Sn0 such that
E[J T ΞP (T̄ )J + EdT ΞP (T̄ )Ed ] − P ≺ 0

(4)

holds where ΞP (·) is defined in (3).
(iv) There exist a matrix-valued function S : [0, T̄ ] →
Sn , S(0)  0, and a scalar ε > 0 such that the
conditions

Remark 3 By considering the change of variables
z(t) = eαt x(t) and applying the above result to z(t), we
immediately get that E[||x(t)||22 ] exponentially converges
2
to 0 with rate 2α as t → ∞ if and only if E[||x(t+
k )||2 ] ge−2αTmin
ometrically converges to 0 with rate e
as k → ∞.

− Ṡ(τ ) + AT S(τ ) + S(τ )A + EcT S(τ )Ec  0 (5)
and

3

Mean-square stability of stochastic linear impulsive systems

J T S(T̄ )J − S(0) + EdT S(T̄ )Ed + ε I  0

hold for all τ ∈ [0, T̄ ].
(v) There exist a matrix-valued function S : [0, T̄ ] →
Sn , S(T̄ )  0, and a scalar ε > 0 such that the
conditions

This section is devoted to the mean-square stability analysis of the system (1) with no input (i.e. uc ≡ 0 and
ud ≡ 0) under constant, ranged and minimum dwelltime. Before stating the main results of this section, it
is convenient to introduce here the following definition:

Ṡ(τ ) + AT S(τ ) + S(τ )A + EcT S(τ )Ec  0
Definition 4 The fundamental solution of the
continuous-time part of the system (1) with uc ≡ 0 and
ud ≡ 0 is denoted by Φ : R≥0 → Rn×n where
dΦ(t) = AΦ(t)dt + Ec Φ(t)dW1 (t), t ≥ 0

(6)

(7)

and
J T S(0)J − S(T̄ ) + EdT S(0)Ed + ε I  0

(8)

(2)
hold for all τ ∈ [0, T̄ ].

where Φ(0) = I and W1 (t) is defined as in (1).
Proof : Proof that (iii) ⇒ (i). The proof of this
implication follows from the fact that by pre- and
T
post-multiplying (4) by x(t+
and x(t+
k)
k ), respec2
tively, we immediately get that E[||x(t+
)||
k+1 2,P ] − (1 −

Note that for every t ≥ 0, Φ(t) is an Ft,k -measurable random variable. This fundamental solution therefore naturally generalizes the deterministic one, which is readily

3

2
ε)E[||x(t+
k )||2,P ] ≤ 0 for some ε ∈ (0, 1), all k ∈
N0 and where ||v||22,P := v T P v. This implies that
2
E[||x(t+
k )||2,P ] → 0 as k → ∞, proving then the meansquare asymptotic (exponential) stability of the system
(invoking Lemma 2 and Remark 3).

The conditions stated in the two last statements are referred to as clock-dependent conditions as they explicitly
depend on the time τ (the clock value) elapsed since the
last impulse. At each event, the clock is reset to 0 (i.e.
τ (t+
k ) = 0) and then grows continuously with slope 1
until the next impulse time. Clocks are extensively considered in the analysis of hybrid systems and timed automata; see e.g. [2, 35]. Clocks here are used to measure
the current dwell-time value and explicitly consider it in
the conditions.

Proof that (i) ⇒ (iii). The proof of this statement is
based on the explicit construction of a matrix P ∈ Sn0
that verifies the condition (4) whenever the system (1) is
mean-square asymptotically (exponentially) stable. To
do this, let us define first the following expression
Q̇(t) = AT Q(t) + Q(t)A + EcT Q(t)Ec , t 6= tk
Q(t+
k)

T

= J Q(tk )J +

EdT Q(tk )Ed ,

k∈N

It is interesting to note that when the system without any control input is deterministic, i.e. Ec = 0 and
Ed = 0, then the conditions stated in the previous result reduce to those of [8], emphasizing then their higher
degree of generality. However, unlike in the deterministic setting, the condition (4) is not directly tractable in
this form due to the presence of the expectation operator
and random matrices that are difficult to compute. The
conditions of the other statements, although stated as
infinite-dimensional LMI problems, can be turned into
finite-dimensional tractable conditions using several relaxation techniques. This will be discussed in more details in Section 3.4.

(9)

with Q0 = Q(0) = Y for any Y ∈ Sn0 . Note
that the mean-square exponential stability of the system is equivalent to the exponential stability of the
above matrix-valued linear differential equations since
E[x(t)T Y x(t)] = xT0 Q(t)x0 . Let us define P ∗ =
P
∞
+
k=0 Qk  0, Qk := Q(tk )  0, Q0 = Y  0. Note
that the latter sum is well-defined because (9) is exponentially stable and also observe that


E J T ΞQk (T̄ )J + EdT ΞQk (T̄ )Ed = Qk+1 .
(10)

3.2

Substituting then P ∗ in place of P in (4) and using (10)
yield
∞
∞
X
X
Qk −
Qk = −Q0 = −Y ≺ 0,
(11)
k=1

Ranged dwell-time

Let us consider now the ranged dwell-time case, that is,
the case where Tk ∈ [Tmin , Tmax ], 0 < Tmin ≤ Tmax <
∞, for all k ∈ N0 . We then have the following result:

k=0

which proves the result.
Theorem 6 The following statements are equivalent:
Proof that (i) ⇔ (ii). The mean-square asymptotic stability of (1) is equivalent to the asymptotic stability of
(9). Vectorizing this matrix differential equation yields
the system ż(t) = Az(t), t 6= tk , and z(t+
k ) = J z(tk ),
k ∈ N. Since the impulses arrive periodically, then the
stability of this system is equivalent to the stability of
+
the discretized system z(t+
k+1 ) = exp(AT̄ )J z(tk ) and
the result follows.

(i) There exists a matrix P ∈ Sn0 such that
E[J T ΞP (θ)J + EdT ΞP (θ)Ed ] − P ≺ 0

(12)

holds for all θ ∈ [Tmin , Tmax ] where ΞP (·) is defined
in (3).
(ii) There exist a matrix-valued function S :
[0, Tmax ] → Sn , S(0)  0, and a scalar ε > 0 such
that the conditions

Proof that (iv) ⇒ (iii). Integrating (5) with S(0) =
P , for simplicity, and using the definition of Φ, we get
that S(τ )  E[Φ(τ )T S(0)Φ(τ )]. Substituting then this
expression in (6) yields the condition (4). The proof is
complete.

− Ṡ(τ ) + AT S(τ ) + S(τ )A + EcT S(τ )Ec  0 (13)
and
J T S(θ)J − S(0) + EdT S(θ)Ed + ε I  0

Proof that (iii) ⇒ (iv). Assume that (4) holds and
define S ∗ (τ ) = E[Φ(τ )T S(0)Φ(τ )], S(0) = P . This gives
that −Ṡ ∗ (τ ) + AT S ∗ (τ ) + S(τ )A∗ + EcT S(τ )∗ Ec = 0,
hence (5) holds. Substituting then the value S ∗ (T̄ ) in the
place of S(T̄ ) in (6) yields an expression that is identical
to (4). The condition (6) then readily follows.

(14)

hold for all τ ∈ [0, Tmax ] and all θ ∈ [Tmin , Tmax ].
Moreover, when one of the above equivalent statements
holds, then the system (1) with uc ≡ 0 and ud ≡ 0 is
mean-square asymptotically stable under ranged dwelltime (Tmin , Tmax ), that is, for any sequence of impulse
times verifying Tk ∈ [Tmin , Tmax ] for all k ∈ N.

Proof that (iv) ⇔ (v). The equivalence follows from
the change of variables S(τ ) ← S(T̄ − τ ).
♦

4

Proof : The proof of this result follows from the same
lines as the proof of Theorem 5.
♦

Using now (18), we can conclude that
θ ≥ 0. This, in turn, implies that

dfT̄
dθ

 0 for all

E[J T ΞP (T̄ + θ)J + EdT ΞP (T̄ + θ)Ed ]
3.3

Minimum dwell-time

 E[J T ΞP (T̄ )J + EdT ΞP (T̄ )Ed ]

for any sufficiently small θ ≥ 0. Noting finally that the
above inequality also holds when substituting T̄ by T̄ +µ
for any µ ≥ 0 we get that

We finally consider the minimum dwell-time case, that
is, Tk ≥ T̄ for all k ∈ N0 . We then have the following
result:

E[J T ΞP (T̄ + θ0 )J + EdT ΞP (T̄ + θ0 )Ed ]

Theorem 7 The following statements are equivalent:
(i) There exists a P ∈

Sn0

 E[J T ΞP (T̄ )J + EdT ΞP (T̄ )Ed ]

such that the LMIs

E[J T ΞP (θ)J + EdT ΞP (θ)Ed ] − P ≺ 0

(23)

(24)

(15)

for all θ0 := θ + µ ≥ 0. Using finally (17) yields the
result. The proof is complete.
♦

(16)

3.4

and
AT P + P A + EcT P Ec ≺ 0

hold for all θ ≥ T̄ where ΞP (·) is defined in (3).
(ii) There exists a P ∈ Sn0 such that the LMIs
E[J T ΞP (T̄ )J + EdT ΞP (T̄ )Ed ] − P ≺ 0

The conditions of the theorems stated in the previous
sections are infinite-dimensional LMI feasibility problems which cannot be verified directly. In what follows,
we describe two relaxation methods turning the original untractable conditions into tractable ones. Note that
even though we only provide these relaxations for Theorem 5, (iv), similar ones can be obtained for the conditions of Theorem 6 and Theorem 7.

(17)

and
AT P + P A + EcT P Ec ≺ 0

(18)

hold where ΞP (·) is defined in (3).
(iii) There exist a matrix-valued function S : R≥0 →
Sn , S(T̄ )  0, S(T̄ +s) = S(T̄ ), s ≥ 0, and a scalar
ε > 0 such that the conditions
AT S(T̄ ) + S(T̄ )A + EcT S(T̄ )Ec ≺ 0,

3.4.1

Piecewise linear approach

The first method, referred to as the piecewise linear approximation, proposes to impose a piecewise linear structure to the general matrix-valued functions involved in
the conditions; see e.g. [22]. The following result states
the conditions that approximate those of Theorem 5,
(iv):

(19)

Ṡ(τ ) + AT S(τ ) + S(τ )A + EcT S(τ )Ec  0 (20)
and
J T S(0)J − S(T̄ ) + EdT S(0)Ed + ε I  0

Computational aspects

Proposition 8 Let N ∈ N. The following statements
are equivalent:

(21)

(a) The conditions of Theorem 5, (iv) hold with the
piecewise-linear matrix-valued function S(τ ) given
by


Si+1 − Si
iT̄
+ Si
(25)
S(τ ) =
τ−
N
T̄ /N
for τ ∈ [iT̄ /N, (i + 1)T̄ /N ], Si ∈ Sn , i = 0, . . . , N −
1.
(b) There exist matrices Si ∈ Sn , i = 1, . . . , N , S0  0,
and a scalar ε > 0 such that the LMIs

hold for all τ ∈ [0, T̄ ].
Moreover, when one of the above equivalent statements
holds, then the system (1) with uc ≡ 0 and ud ≡ 0
is mean-square asymptotically stable under minimum
dwell-time T̄ , that is, for any sequence of impulse times
verifying Tk ≥ T̄ for all k ∈ N.
Proof : The proof that (ii) is equivalent to (iii) follows
from Theorem 5. The proof that (i) implies (ii) is also
immediate. Let us then focus on the reverse implication.
Define first the function fT̄ (θ) := E[Φ(T̄ + θ)T P Φ(T̄ +
θ)]. Then, we have that

−

−

dfT̄
= E[Φ(T̄ + θ)T [He[P A] + EcT P Ec ]Φ(T̄ + θ)]. (22)
dθ

5

Si+1 − Si
+ AT Si + Si A + EcT Si Ec  0 (26)
T̄ /N

Si+1 − Si
+ AT Si+1 + Si+1 A + EcT Si+1 Ec  0
T̄ /N
(27)

and
J T SN J − S0 + EdT SN Ed + ε I  0

Regarding the conservatism, it can be shown using the
same arguments as in [17] that if the conditions of Theorem 5, (iv) are feasible, then there exists a sufficiently
large integer d such that the above SOS program is feasible for some polynomials S, Γ of degree at least 2d. Finally, it is important to stress that SOS conditions are
in general more tractable than those obtained using the
piecewise linear approximation that often require a large
discretization order; see [17, 19].

(28)

hold for all i = 0, . . . , N − 1.
When one of the above statements holds, then the conditions of Theorem 5, (iv) hold with the computed
piecewise-linear matrix S(τ ).
Proof : The proof follows from a convexity argument.
By substituting the piecewise-linear expression of S(τ )
given by (25) into the conditions (6) and (5) gives (28)
and
−

Si+1 − Si
+ He[S(τ )A] + EcT S(τ )Ec  0,
T̄ /N

3.5

Interestingly, the results developed in this section also
applies to linear stochastic switched systems. To emphasize this, let us consider the stochastic switched system

(29)

dy(t) = Gσ(t) y(t)dt + Hσ(t) y(t)dW1 (t), y(0) = y0
(30)
where y, y0 ∈ Rn and W1 (t) are the state of the system, the initial condition and the standard Wiener process, respectively. The switching signal σ : R≥0 →
{1, . . . , N }, for some finite N ∈ N, is piecewise constant
and describes the evolution of the mode of the switched
system. This system can be reformulated in the form (1)
with the matrices

respectively. Noting then that the above LMI is affine
in τ , then for each i = 0, . . . , N − 1, it is necessary
and sufficient to check the above LMI at the vertices of
the interval [iT̄ /N, (i + 1)T̄ /N ], that is, at the values
iT̄ /N and (i + 1)T̄ /N . The equivalence follows from the
losslessness of the manipulations.
♦
As the conditions stated in the above result are finitedimensional, they can be solved using standard SDP
solvers such as SeDuMi [36]. Note, moreover, than using
similar arguments as in [37], we can prove that if the conditions of Theorem 5, (iv) are feasible, then there exists
an integer N ∗ such that the conditions of Proposition 8,
(b) are feasible for all N ≥ N ∗ .
3.4.2

N

N

k=1

k=1

A = diag[Gk ], Ec = diag(Hk ) and Jij = (ei eTj ) ⊗ In ,
(31)
for all i, j = 1, . . . , µ, i 6= j, where {ei }i=1,...,N is the
standard basis of RN . Note, however, that we have here
multiple jump maps Jij (actually reset maps here). It
is immediate to see that the derived stability conditions
can be straightforwardly extended to address this case,
and we get the following result adapted from Theorem 7:

Sum of squares programs

Another possible relaxation relies on the use of sum of
squares programming [23, 38] where we impose a polynomial structure with fixed degree to the matrix-valued
functions. The following result states conditions that approximate those of Theorem 5, (iv) and that can be easily checked using SOSTOOLS [38] and the semidefinite
programming solver SeDuMi [36]:

Corollary 10 Assume that there exists a block-diagonal
matrix-valued function R = diagN
i=1 (Ri ), Ri : [0, T̄ ] →
Sn , R(T̄ )  0, such that the conditions
GTi Ri (T̄ ) + Ri (T̄ )Gi + HiT Ri (T̄ )Hi ≺ 0,

Proposition 9 Let ε, ν, T̄ > 0 be given and assume that
the following sum of squares program

(32)

Ṙi (τ ) + GTi Ri (τ ) + Ri (τ )Gi + HiT Ri (τ )Hi  0 (33)
and
Ri (0) − Rj (T̄ ) + ε I  0
(34)

Find polynomial matrices S, Γ : R → Sn such that

hold for all τ ∈ [0, T̄ ] and for all i, j = 1, . . . , N , i 6= j.

S(0) − νIn  0
Γ(τ ) is SOS
Ṡ(τ ) − He[S(τ )A] − EcT S(τ )Ec − Γ(τ )τ (T̄
S(0) − J T S(T̄ )J − EdT S(T̄ )Ed − ε I  0

Application to switched systems

Then, the linear stochastic switched system (30) is meansquare asymptotically stable under minimum dwell-time
T̄ .

− τ ) is SOS

Proof : The proof is based on the reformulation (1)(31). Noting then the system is block-diagonal, then it
is enough to choose a matrix-valued function R(τ ) that
is also block-diagonal. Substituting the model in the
conditions of Theorem 7 and expanding them yield the

is feasible. Then the conditions of Theorem 5, (iv) hold
with the computed polynomial matrix S(τ ) and the system
(1) is mean-square asymptotically stable under constant
dwell-time T̄ .

6

Table 1
Estimated smallest constant dwell-time T̄ for the system (1)(35) for various values for κ and δ using Proposition 9 and
matrix polynomials S(τ ) and Γ(τ ) of degree 6.

♦

result.

It is interesting to note that the above condition, although formulated in a more compact form, include
those obtained in [33]. If we indeed apply now the piecewise linear approximation, then we get exactly the conditions of [33, Theorem 1].
3.6

Examples

κ/δ

0

0.6

1.2

1.8

2.4

0

1.1406

1.1568

1.2031

1.2734

1.3595

0.3

1.1918

1.2089

1.2578

1.3319

1.4225

0.6

1.3787

1.3992

1.4577

1.5458

1.6531

0.9

1.8774

1.9073

1.9920

2.1184

2.2702

1.2

3.9306

4.0011

4.1938

4.4765

4.8305

We now apply some of the previously developed results
to some academic examples.
Table 2
Smallest constant dwell-time T̄ for the system (1)-(35) for
various values for κ and δ using Theorem 5, (ii).

Example 11 Let us consider the system (1) with the
matrices [8, 12, 15]
"
A=

−1 0

#

1 −2

"
, J=

21
13

, Ec = κI2 , Ed = δI2 , (35)

Bc1 = Bc2 = Bd1 = Bd2 = 0 (i.e. no control input) for
some scalars κ, δ ≥ 0. We then choose several values for
the parameters δ and κ. The case (κ, δ) = (0, 0) corresponds to the deterministic case of [8]. For each of these
values for the parameters, we solve the sum of squares
program of Proposition 9 for the constant dwell-time and
the minimum dwell-time cases with matrix polynomials
of degree 6. Note that to adapt Proposition 9 to the minimum dwell-time case, we simply have to add the constraint He[AT S(T̄ )] + EcT S(T̄ )Ec ≺ 0 to the program.

A=

−1 2

"
, J=

0.6

1.2

1.8

2.4

0

1.1406

1.1568

1.2030

1.2732

1.3593

0.3

1.1918

1.2089

1.2577

1.3317

1.4223

0.6

1.3787

1.3992

1.4576

1.5456

1.6528

0.9

1.8773

1.9072

1.9918

2.1181

2.2700

1.2

3.9315

4.0005

4.1932

4.4752

4.8083

0.5 0

#

0 0.5

κ/δ

0

0.6

1.2

1.8

2.4

0

1.1406

1.1568

1.2031

1.2734

1.3595

0.3

1.1918

1.2089

1.2578

1.3319

1.4225

0.6

1.3787

1.3992

1.4577

1.5458

1.6531

0.9

1.8774

1.9073

1.9920

2.1184

2.2703

1.2

3.9307

4.0012

4.1941

4.4776

4.8565

the maximal value for Tmax using a variation of the SOS
program in Proposition 9 and a bisection approach. In
order to evaluate the conservatism, we compare these results with the maximum value for θ for which the LMIs
P  0 and M(θ)T P M(θ) − P ≺ 0 are feasible for all
θ ∈ [0.01, Tmax ] (quadratic stability condition). Since
this problem is not directly solvable (note that sum of
squares methods do not apply because of the presence of
exponential terms), the interval [0.01, Tmax ] is gridded
with 201 points and the LMIs are checked on these points
only. This leads to the results of Table 5 where we can
observe that slightly larger values for Tmax are found.
However, this is at the price of a much higher computational cost (see [17, 19]) and gridding imprecision. Note
also that the exponential conditions are only valid in the
time-invariant case, while the conditions of Theorem 6
are more flexible and apply also to systems affected by
time-varying uncertainties/parameters.

Example 12 We now consider the system (1) with the
matrices [8, 12, 15]
#
1 3

0

Table 3
Estimated minimum dwell-time T̄ for the system (1)-(35) for
various values for κ and δ using Proposition 9 and matrix
polynomials S(τ ) and Γ(τ ) of degree 6.

The numerical results are gathered in Table 1 and Table 3
where we can see that the deterministic results are indeed
retrieved and that, as expected, stability deteriorates as
we increase the value of the parameters κ and δ. It is also
interesting to observe that, for this example, the results
for the constant and the minimum dwell-times are quite
close. Note, however, that this is far from being a general
rule. In order to estimate the conservatism of the method,
we consider the criterion of statement (ii) of Theorem
5 and we find the results summarized in Table 2 where
we can see that the proposed method is very accurate for
this system. This then implies that the estimates of the
minimum dwell-time are also equally accurate.

"

κ/δ

#

, Ec = κI2 , Ed = δI2 ,

(36)
Bc1 = Bc2 = Bd1 = Bd2 = 0 (i.e. no control input) for
some scalars κ, δ ≥ 0. For δ = κ = 0, this system is
known (see [12]) to be stable with maximum dwell-time
Tmax ≈ 0.4620, i.e. for all Tk ≤ 0.4620. Using the ranged
dwell-time result with Tmin = 0.01, we get the results

7

Table 4
Estimated ranged dwell-time Tmax for Tmin = 0.01 for the
system (1)-(36) for various values for κ and δ using an adaptation of Proposition 9 and matrix polynomials of degree 6.

holds for all θ ∈ [Tmin , Tmax ] where Ψ1 (θ) :=
Φ1 (θ)T P Φ1 (θ) and
dΦ1 (τ ) = (A + Bc1 Kc (τ ))Φ1 (τ )dτ

κ/δ

0

0.2

0.4

0.6

0.8

0

0.4620

0.4126

0.2971

0.1647

0.0388

0.75

0.3891

0.3474

0.2502

0.1387

0.0327

1.5

0.2640

0.2357

0.1698

0.0941

0.0221

2.75

0.1312

0.1171

0.0844

0.0467

0.0110

3

0.1154

0.1031

0.0742

0.0411

0.0064

+(Ec dW1 (τ ) + Bc2 Kc (τ )dW2 (τ ))Φ1 (τ )
(39)
defined for τ ∈ [0, Tmax ] and Φ1 (0) = I.
(b) There exist matrix-valued functions S̃ : [0, Tmax ] →
Sn , S̃(0)  0, Uc : [0, Tmax ] → Rmc ×n , a matrix
Ud ∈ Rmd ×n and a scalar ε > 0 such that the conditions


Λ(τ )
?
?


 Ec S̃(τ ) −S̃(τ )
(40)
0 
0

2
Bc Uc (τ )
?
−S̃(τ )

Table 5
Estimated ranged dwell-time Tmax for Tmin = 0.01 for the
system (1)-(36) for various values for κ and δ using the gridded quadratic stability condition.

4

κ/δ

0

0.2

0.4

0.6

0.8

0

0.4620

0.4126

0.2971

0.1647

0.0388

0.75

0.3891

0.3474

0.2502

0.1387

0.0327

1.5

0.2640

0.2357

0.1698

0.0941

0.0221

2.75

0.1312

0.1171

0.0844

0.0467

0.0110

3

0.1155

0.1031

0.0742

0.0411

0.0411

and




c

Moreover, when one of the above statements holds, then
the closed-loop system (1)-(37) is mean-square asymptotically stable under ranged dwell-time (Tmin , Tmax ) and
suitable controller gains can be computed from the conditions of statement (b) using the expressions
Kc (τ ) = Uc (τ )S̃(τ )−1 and Kd = Ud S̃(0)−1 .

We consider in this section the following class of statefeedback control law

(42)

Proof : A Schur complement on the inequality (40) and
a congruence transformation with respect to S(τ ) =
S̃(τ )−1 yields the inequality

(37)

ud (k) = Kd x(tk )
where Tk ∈ [Tmin , Tmax ], k ∈ N0 . The continuous
matrix-valued function Kc : [0, Tmax ] → Rmc ×n and the
matrix Kd ∈ Rmd ×n involved above are the gains of the
controller to be determined. We then have the following
result:

−S(τ ) + He[S(τ )(A + Bc1 Kc (τ ))] + EcT S(τ )Ec
+Kc (τ )T Bc2T S(τ )Bc2 Kc (τ )  0.
(43)
Similarly, (41) can be shown to be equivalent to

Theorem 13 The following statements are equivalent:

−S(0) + (J + Bd1 Ud )T S(θ)(J + Bd1 Ud )
+EdT S(θ)Ed + KdT Bd2T S(τ )Bd2 Kd ≺ 0.
(44)
Invoking now Theorem 6, Statement (b) proves the
equivalence between the statements of the result.
♦

(a) There exists a matrix P ∈ Sn0 such that the LMI

+EdT Ψ1 (θ)Ed + KdT (Bd2 )T Ψ1 (θ)Bd2 Kd ≺ 0

?

c

Stabilization under ranged dwell-time

E[(J + Bd1 Kd )T Ψ1 (θ)(J + Bd1 Kd )] − P

?

hold for all τ ∈ [0, Tmax ] and all θ ∈ [Tmin , Tmax ]
˙ ) + He[AS̃(τ ) + B 1 U (τ )].
where Λ(τ ) = S̃(τ

We extend here the results obtained in the previous
section to address the stabilization problem by statefeedback. We first consider stabilization under ranged
dwell-time and then derive stabilization conditions under minimum dwell-time.

uc (tk + τ ) = Kc (τ )x(tk + τ ), τ ∈ (0, Tk ]

?



J S̃(0) + B 1 U −S̃(θ)
?
? 


d d

  0 (41)

Ed S̃(0)
0
−S̃(θ)
? 


Bd2 Ud
0
0
−S̃(θ)

Mean-square stabilization of stochastic linear
impulsive systems

4.1

−S̃(0) + ε I

(38)

8

4.2

Moreover, when this is the case, the closed-loop system
(1)-(37) is mean-square asymptotically stable under minimum dwell-time T̄ and suitable controller gains can be
computed from the conditions of the statement (b) using
the expressions

Stabilization under minimum dwell-time

Let us consider now the minimum dwell-time case. Since
the dwell-time can be arbitrarily large in this setting, we
propose to use the following state-feedback control law
(
uc (tk + τ ) =

Kc (τ ) = Uc (τ )S̃(τ )−1

Kc (τ )x(tk + τ ), τ ∈ (0, T̄ ]

ud (k) = Kd x(tk )

Proof : The proof is based on the same manipulations
as in the proof of Theorem 13, with the difference that
Theorem 7 is invoked in place of Theorem 6.
♦

where Kc : [0, T̄ ] → Rmc ×n and Kd ∈ Rmd ×n are the
gains of the controllers that have to be determined. We
then have the following result:
Theorem 14 The following statements are equivalent:

4.3

(a) There exists a matrix P ∈ Sn0 such that the LMIs
E[(J + Bd1 Kd )T Ψ2 (T̄ )(J + Bd1 Kd )] − P
+

KdT (Bd2 )T Ψ2 (T̄ )Bd2 Kd

≺0

AT P + P A + EcT P Ec ≺ 0

"

(46)
A=

(47)

hold where Ψ2 (θ) = Φ2 (θ) P Φ2 (θ) and
dΦ2 (τ ) = (A + Bc1 Kc (τ ))Φ2 (τ )dτ
+(Ec dW1 (t) + Bc2 Kc (τ )dW2 (t))Φ2 (τ )
(48)
defined for τ ≥ 0 and Φ2 (0) = I.
(b) There exist matrix-valued functions S̃ : [0, Tmax ] →
Sn , S̃(T̄ )  0 , Uc : [0, Tmax ] → Rmc ×n , a matrix
Ud ∈ Rmd ×n and a scalar ε > 0 such that the conditions

"

−S̃(τ )

0

He[AS̃(T̄ ) + Bc1 Uc (T̄ )] Ec S̃(T̄ ) + Bc2 U (T̄ )
?

−S̃(T̄ )

1 1

#
, Bc1

1 −2
" #
1 0

=

" #
4

" #
1

, Bc2

"

#
3 1

=
,J =
,
0
0
1 2
" #
#
" #
"
0
1
1 0
2
1
.
, Bd =
, Bd =
, Ed = 0.2
Ec =
0.1
0
1 −1
1 2
(53)
Using then Theorem 14 with minimum dwell-time T̄ =
0.1 and polynomials of
h order 1 and 2, iwe obtain the
controller gains Kd = −3.9165 −2.9751 and

T

?

Example

Let us consider the system (1) with the matrices

and

#
"
Λ(τ ) Ec S̃(τ ) + Bc2 U (τ )

(52)

(45)

Kc (T̄ )x(tk + τ ), τ ∈ (T̄ , Tk ]

+EdT Ψ2 (T̄ )Ed

Kd = Ud S̃(0)−1 .

and

"
#T
−0.0299τ 2 − 0.2251τ + 0.1605
1
Kc (τ ) =
den(τ ) 0.0045τ 2 − 0.0167τ + 0.1881
where den(τ ) = 0.0291τ 2 + 0.5439τ − 0.1255. Choosing then x0 = (2, −2), we then obtain the trajectory
for (E[||x(t)||22 ])1/2 depicted in Fig. 1 where we can observe the convergence to 0, emphasizing the mean-square
asymptotic stability of the closed-loop system.

(49)

#
≺0
5
(50)

Application to aperiodic sampled-data systems

and


−S̃(0) + ε I J S̃(T̄ ) + Bd1 Ud Ed S̃(T̄ ) + Bd2 Ud



0
?
−S̃(T̄ )
0


?
?
−S̃(T̄ )
(51)
˙
hold for all τ ∈ [0, T̄ ] where Λ(τ ) = S̃(τ ) +
He[AS̃(τ ) +

As an application example, we utilize the ranged dwelltime result in order to derive a stabilization condition for
linear aperiodic stochastic sampled-data systems represented in the impulsive form:
dxsd (t) = Āxsd (t)dt +
xsd (t+
k)

Bc1 U (τ )].

9

P2

i=1

Ēi xsd (t)dWi (t)

= (J0 + B̄Kd )xsd (tk )

(54)

such that the conditions S̃(0)  0 and

3



˙ ) + He[AS̃(τ )] S̃(τ )E T S̃(τ )E T
S̃(τ
1
2



?
−S̃(τ )
0 

  0 (58)
?
?
−S̃(τ )

2.5
2
1.5

"
#
−S̃(0) + ε I (J S̃(0) + Bd Ud )T

1

−S̃(θ)

?
0.5
0

0

(59)

hold for all τ ∈ [0, Tmax ] and all θ ∈ [Tmin , Tmax ].

Fig. 1. Evolution of (E[||x(t)||22 ])1/2 along the trajectories
of the system (1)-(53) subject to two randomly generated
impulse time sequences with minimum dwell-time T̄ = 0.1.

Moreover, when one of the above statements holds, then
the (closed-loop) sampled-data system (54) is meansquare asymptotically stable under ranged dwell-time
Tk ∈ [Tmin , Tmax ] and a suitable controller gain can be
computed from the expression Kd = Ud S̃(0)−1 .

where

We now illustrate the above result by a simple example:

0

1

"
Ā =

Asd Bsd

2

#

3

"
, Ē1 =

4

Time [sec]

#
Esd 0

"
, Ē2 =

5

0 αBsd

0 0
0 0
0
" #
" #
h
i
0
I 0
, Kd = Kd1 Kd2
, B=
J0 =
I
0 0

Example 16 Let us consider the sampled-data system
(54) with α = 0.1 and the matrices

#

0

"
Asd =

(55)
and xsd := (x1sd , x2sd ) where x1sd ∈ Rn is the state of
the continuous-time system, x2sd ∈ Rm is the piecewiseconstant state modeling the zero-order hold. As before
W1 , W2 ∈ R are two independent zero mean Wiener processes. The parameter α > 0 is here to scale the amplitude of the noise on the control channel. The gain
h of thei
controller, denoted by Kd , is given by Kd = Kd1 Kd2
where Kd1 ∈ Rm×n and Kd2 ∈ Rm×m .

#

0 −1

, Bsd =

" #
0
1

and Esd =

#
"
0 0

. (60)

0 0.1

Applying then Theorem 15 with polynomials of order 2,
we get the results gathered in Table 6. Simulations results are depicted in Fig. 2 where we can see that the designed controllers effectively render the closed-loop system mean-square asymptotically stable under the considered ranged dwell-time conditions. We can also observe
that the controllers give rise to similar performance in
terms of the rate of convergence of E[||x(t)||22 ] to 0 despite
having different time-scales for the control input update.
Table 6
Various controller gains for the sampled-data system (54)(60) obtained with Theorem 15 using polynomials of order
2.
Tmin Tmax
Kd

We then have the following result:
Theorem 15 The following statements are equivalent:
(a) There exists a matrix P ∈ Sn0 such that the LMI

0.001

E[(J0 + B̄Kd )T Ψsd (θ)(J0 + B̄Kd )] − P ≺ 0
(56)
holds for all θ ∈ [Tmin , Tmax ] where Ψsd (θ) :=
Φsd (θ)T P Φsd (θ) and
dΦsd (τ ) = ĀΦsd (τ )dτ + Ē1 Φsd (θ)dW1 (t)

0 1

1

0.1

(−0.4069, −0.1734, −0.0045)

0.5

(−0.4421, −0.2137, −0.0215)

1

(−0.3410, −0.1332, 0.0036)

5

(−0.1977, −0.1931, 0.0014)

10

(−0.1053, −0.1061, 0.0011)

20

(−0.0583, −0.0559, −0.0003)

(57)

+Ē2 Φsd (θ)dW2 (t), Φsd (0) = I
6
defined for s ∈ [0, Tmax ].
(b) There exist matrix-valued functions S̃ : [0, Tmax ] →
Sn+m , a matrix Ud ∈ Rm×(n+m) and a scalar ε > 0

Conclusion

Dwell-time stability and stabilization conditions have
been obtained for stochastic impulsive systems and

10

[7] P. Naghshtabrizi, J. P. Hespanha, and A. R. Teel,
“Exponential stability of impulsive systems with application
to uncertain sampled-data systems,” Systems & Control
Letters, vol. 57, pp. 378–385, 2008.
[8] C. Briat, “Convex conditions for robust stability analysis
and stabilization of linear aperiodic impulsive and sampleddata systems under dwell-time constraints,” Automatica, vol.
49(11), pp. 3449–3457, 2013.
[9] J. P. Hespanha, “Uniform stability of switched linear
systems: extensions of LaSalle’s invariance principle,” IEEE
Transactions on Automatic Control, vol. 49(4), pp. 470–482,
2004.
[10] A. S. Morse, “Supervisory control of families of linear
set-point controllers - Part 1: Exact matching,” IEEE
Transactions on Automatic Control, vol. 41(10), pp. 1413–
1431, 1996.
[11] J. C. Geromel and P. Colaneri, “Stability and stabilization of
continuous-time switched linear systems,” SIAM Journal on
Control and Optimization, vol. 45(5), pp. 1915–1930, 2006.
[12] C. Briat and A. Seuret, “A looped-functional approach for
robust stability analysis of linear impulsive systems,” Systems
& Control Letters, vol. 61(10), pp. 980–988, 2012.
[13] J. P. Hespanha, D. Liberzon, and A. R. Teel, “Lyapunov
conditions for input-to-state stability of impulsive systems,”
Automatica, vol. 44(11), pp. 2735–2744, 2008.
[14] L. Zhang, S. Zhuang, P. Shi, and Y. Zhu, “Uniform tube based
stabilization of switched linear systems with mode-dependent
persistent dwell-time,” IEEE Transactions on Automatic
Control, vol. 60(11), pp. 2994–2999, 2015.
[15] C. Briat and A. Seuret, “Convex dwell-time characterizations
for uncertain linear impulsive systems,” IEEE Transactions
on Automatic Control, vol. 57(12), pp. 3241–3246, 2012.
[16] ——, “Affine minimal and mode-dependent dwell-time
characterization for uncertain switched linear systems,”
IEEE Transactions on Automatic Control, vol. 58, no. 5, pp.
1304–1310, 2013.
[17] C. Briat, “Convex conditions for robust stabilization of
uncertain switched systems with guaranteed minimum and
mode-dependent dwell-time,” Systems & Control Letters,
vol. 78, pp. 63–72, 2015.
[18] A. Seuret, “A novel stability analysis of linear systems under
asynchronous samplings,” Automatica, vol. 48(1), pp. 177–
182, 2012.
[19] C. Briat, “Theoretical and numerical comparisons of looped
functionals and clock-dependent Lyapunov functions - The
case of periodic and pseudo-periodic systems with impulses,”
International Journal of Robust and Nonlinear Control,
vol. 26, pp. 2232–2255, 2016.
[20] M. M. Peet and A. Seuret, “Global stability analysis of
nonlinear sampled-data systems using convex methods,” in
Delay Systems, ser. Advances in Delays and Dynamics,
T. Vyhldal, J.-F. Lafay, and R. Sipahi, Eds. Springer-Verlag,
2014, vol. 1, pp. 215–227.
[21] K. Gu, “Discretization schemes for Lyapunov-Krasovskii
Functionals in time-delay systems,” Kybernetika, vol. 37,
no. 4, pp. 479–504, 2001.
[22] L. I. Allerhand and U. Shaked, “Robust stability and
stabilization of linear switched systems with dwell time,”
IEEE Transactions on Automatic Control, vol. 56(2), pp.
381–386, 2011.
[23] P.
Parrilo,
“Structured
semidefinite
programs and semialgebraic geometry methods in robustness
and optimization,” Ph.D. dissertation, California Institute of
Technology, Pasadena, California, 2000.

1.5

Tmax = 0:1
Tmax = 0:5
Tmax = 1

1
0.5
0
0

5

0

5

10

15

10

15

0.3
0.2
0.1
0

Time [sec]

Fig. 2. Evolution of E[||x(t)||22 ]1/2 (top) and E[||u(t)||22 ]1/2
(bottom) along the trajectories of the system (54)-(60) with
the controllers designed for Tmin = 0.001 and several values for Tmax . The sequences of impulse times satisfying the
ranged dwell-time condition are randomly generated.

expressed as infinite-dimensional LMI problems that
can be solved using discretization or sum of squares
techniques. The approach has been shown to include
switched systems and sampled-data systems as particular cases, making then the approach quite general.
Possible future works include the design of dynamic output feedback controllers, observer and filters. The approach can also be extended to performance characterization using induced-norms such as the induced L2 -norm
and the induced L2 -L∞ -norm. Other possible extensions
would concern other concepts of dwell-times [14] and the
consideration of controller/mode mismatch due to decision delays [39].
References
[1] A. N. Michel, L. Hou, and D. Liu, Stability of dynamical
systems - Continuous, discontinuous and discrete systems.
Boston: Birkhäuser, 2008.
[2] R. Goebel, R. G. Sanfelice, and A. R. Teel, “Hybrid
dynamical systems,” IEEE Control Systems Magazine, vol.
29(2), pp. 28–93, 2009.
[3] R. Yu and P. Leung, “Optimal partial harvesting schedule
for aquaculture operations,” Marine Resource Economics,
vol. 21, pp. 301–315, 2006.
[4] E. I. Verriest and P. Pepe, “Time optimal and optimal
impulsive control for coupled differential difference point
delay systems with an application in forestry,” in Topics in
Time Delay Systems. Springer Berlin Heidelberg, 2009, pp.
255–265.
[5] C. Briat and E. I. Verriest, “A new delay-SIR model for pulse
vaccination,” Biomedical signal processing and control, vol.
4(4), pp. 272–277, 2009.
[6] H. Ye, N. Michel, and L. Hou, “Stability theory for hybrid
dynamical systems,” IEEE Transactions on Automatic
Control, vol. 43(4), pp. 461–474, 1998.

11

[24] C. Briat, “Convex lifted conditions for robust `2 -stability
analysis and `2 -stabilization of linear discrete-time switched
systems with minimum dwell-time constraint,” Automatica,
vol. 50(3), pp. 976–983, 2014.
[25] B. Øksendal, Stochastic Differential Equations - An
Introduction with Applications (6th Edition). Springer, 2003.
[26] A. Rodkina and C. Kelly, “Stochastic difference equations
and applications,” in International Encyclopedia of Statistical
Science, M. Lovric, Ed. Springer Berlin Heidelberg, 2011,
pp. 1517–1520.
[27] L. Hou and A. N. Michel, “Moment stability of discontinuous
stochastic dynamical systems,” IEEE Transactions on
Automatic Control, vol. 46(6), pp. 938–943, 2005.
[28] A. R. Teel, A. Subbaraman, and A. Sferlazza, “Stability
analysis for stochastic hybrid systems: A survey,”
Automatica, vol. 50(10), pp. 2435–2456, 2014.
[29] W.-H. Chen, J.-G. Wang, Y.-J. Tang, and X. Lu, “Robust
H∞ control of uncertain linear impulsive stochastic systems,”
International Journal of Robust and Nonlinear Control,
vol. 18, pp. 1348–1371, 2008.
[30] Z. Feng and K. Teo, “Optimal feedback control for
stochastic impulsive linear systems subject to poisson
processes,” in Optimization and Optimal Control, ser.
Springer Optimization and Its Applications, A. Chinchuluun,
P. M. Pardalos, R. Enkhbat, and I. Tseveendorj, Eds.
Springer New York, 2010, pp. 241–258.
[31] J. Hespanha and A. R. Teel, “Stochastic impulsive
systems driven by renewal processes,” in 17th International
Symposium on Mathematical Theory of Networks and
Systems, Kyoto, Japan, 2006.
[32] D. Antunes, J. P. Hespanha, and C. Silvestre, “Volterra
integral approach to impulsive renewal systems: Application
to networked control,” IEEE Transactions on Automatic
Control, vol. 57(3), pp. 607–619, 2013.
[33] U. Shaked and E. Gershon, “Robust H∞ control of stochastic
linear switched systems with dwell-time,” International
Journal of Robust and Nonlinear Control, vol. 24, pp. 1664–
1676, 2014.
[34] A. R. Teel, “Stochastic hybrid inclusions with diffusive flows,”
in 53rd IEEE Conference on Decision and Control, Los
Angeles, California, USA, 2014, pp. 3071–3076.
[35] C. Baier and J.-P. Katoen, Principles of Model Checking.
Cambridge, Massachusetts: MIT Press, 2008.
[36] J. F. Sturm, “Using SEDUMI 1.02, a Matlab Toolbox
for Optimization Over Symmetric Cones,” Optimization
Methods and Software, vol. 11, no. 12, pp. 625–653, 2001.
[37] W. Xiang, “On equivalence of two stability criteria
for continuous-time switched systems with dwell time
constraint,” Automatica, vol. 54, pp. 36–40, 2015.
[38] A. Papachristodoulou, J. Anderson, G. Valmorbida,
S. Prajna, P. Seiler, and P. A. Parrilo, SOSTOOLS: Sum
of squares optimization toolbox for MATLAB v3.00, 2013.
[Online]. Available: http://www.cds.caltech.edu/sostools/
sostools.pdf
[39] L. Zhang and W. Xiang, “Mode-identifying time estimation
and switching-delay tolerant control for switched systems:
An elementary time unit approach,” Automatica, vol. 64, pp.
174–181, 2016.

12

