In this section we give some examples to illustrate our results with a constant delayτ >0.
Example 5.7.1. Let us consider the case when
a(n, j) = 2n+ 1
(n+j+ 1)(n+j+ 2) for all 0≤j≤n.
We have a(0,0) = 12,a(n,0) = (n+1)(n+2)2n+1 , supn≥1a(n,0) = 12,
n
X
j=1
2n+ 1
(n+j+ 1)(n+j+ 2) = n(2n+ 1) 2(n+ 1)(n+ 2), and
sup
n≥0 n
X
j=0
2n+ 1
(n+j+ 1)(n+j+ 2) = 1. (5.7.1)
One can easily see that conditions (5.2.4) and (5.2.5) with µ := kψkτ, ψ ∈ S([−τ,0],R+) are equivalent to the inequalities
1
2φ(kψkτ) +kh(0)k ≤v, (5.7.2)
and
2n+ 1
(n+ 1)(n+ 2)φ(kψkτ) + n(2n+ 1)
2(n+ 1)(n+ 2)φ(v) +kh(n)k ≤v, n≥1. (5.7.3) Consider the scalar equation
x(n+ 1) =
n
X
j=0
(2n+ 1)
(n+j+ 1)(n+j+ 2)xp(j−τ) +h(n), n≥0, (5.7.4) with the initial condition
x(n) =ψ(n), −τ ≤n≤0, (5.7.5)
whereh(n)∈R+, ψ∈S([−τ,0],R+),n≥0 andp >0. In fact there are three cases with respect to the value ofp.
(a1) p ∈ (0,1) and φ(t) = tp, t > 0. If supn≥0h(n) < ∞ and (5.7.1) are satisfied, then for any ψ≥0, the solution of (5.7.4) and (5.7.5) is bounded by Theorem 5.4.1.
Chapter5. Boundedness of Solutions of Nonlinear VDEs with Constant Delay 59 (a2) p = 1 and φ(t) = t, t > 0. It is not difficult to see that for any v > 0 large enough the
inequalities (5.7.2) and (5.7.3) are equivalent to the inequalities 1
Letk= supn≥1nh(n)<∞, it is easily to see that the inequality (5.7.6) is satisfied if 4
5kψkτ+4 3k≤v.
Hence (5.2.4) and (5.2.5) are satisfied, by Lemma 5.4.6, ψ has property (P0). It is worth to note that in this case our Theorem5.4.5is applicable for anyψ(n)∈S([−τ,0],R+), but the results in [5,53,55,56,71,72,92] are not applicable in this case.
(a3) p >1 and φ(t) =tp, t >0. Assume k= supn≥1h(n), and forψ(n) ∈S([−τ,0],R+), there
hold. Hence ψ has property (P0), and by Theorem 5.4.7, for small ψ, the solution of
(5.7.4)-(5.7.5) is bounded.
Summarizing the observations and applying Theorem 5.4.5 and Theorem 5.4.7, we get the next results respectively.
Proposition 5.7.1. If Equation (5.7.4) is linear, that is p = 1, and supn≥1n h(n) < ∞, then every positive solution of(5.7.4) with initial condition (5.7.5) is bounded.
Chapter5. Boundedness of Solutions of Nonlinear VDEs with Constant Delay 60 Proposition 5.7.2. Assume equation (5.7.4) is super-linear, that is p > 1. If for some ψ ∈ S([−τ,0],R+) there exists v∈
hold, then the positive solution of (5.7.4) with initial condition (5.7.5) is bounded.
The following example shows the applicability of our result Corollary5.5.7in the critical case.
Example 5.7.2. Consider the equation
x(n+ 1) =
If q+c <1 and supn≥0h(n)< ∞,then our condition (a) in Corollary 5.5.7holds. Relation q+c= 1 implies
∞
X
n=0
cqn= 1.
Note that the results in [5, 53, 55, 56, 71, 72] are not applicable. But our Corollary 5.5.7 is applicable under condition
sup
n≥0
h(n) qn <∞, since it implies condition (b) in Corollary5.5.7:
sup Remark 5.7.3. By mathematical induction, it is easy to see that the solution of (5.7.7) and (5.7.8) is given in the form
x(n) =c(q+c)n−1ψ+
Chapter5. Boundedness of Solutions of Nonlinear VDEs with Constant Delay 61 ψ ∈R+. At the same time condition (b) in Corollary 5.5.7 does not hold, and hence condition (β) in Proposition 1.1 is not necessary.
Therefore by statements (a) and (b) of Corollary 5.5.7, we get the following proposition.
Proposition 5.7.4.The solution of (5.7.7)-(5.7.8)is bounded if eitherq+c <1andsupn≥0h(n)<
∞, or q+c= 1, and
sup
n≥0
h(n) qn <∞.
The next example shows the sharpness of our Corollary 5.5.8.
Example 5.7.3. Consider the equation x(n+ 1) = mathematical induction it is easy to prove that the sequence (x(n))n≥0 is strictly increasing.
Now forx0 >1, we prove thatx(n)→ ∞ asn→ ∞. Assume, for the sake of contradiction, that the sequence (x(n))n≥0 is bounded. Since it is strictly increasing,x∗ = limn→∞x(n) is finite and x∗ > x0. On the other hand x(n+ 1) ≥ xp(n), and hence we get x∗ ≥ (x∗)p > x∗, which is a contradiction. Hence for allx0>1 the solution of (5.7.9) is unbounded.
Now applying Corollary 5.5.8to (5.7.9), there exists
v∈
Then the solution of (5.7.9) with initial value x0 ≤(1−q)p(p−1)2 is bounded, but the solution of (5.7.9) with initial value x0 >1 is unbounded. Our results do not give any information about the boundedness of the solutions, whenever x0 ∈
(1−q)
2 p(p−1),1
i
, but this gap tends to zero if eitherpis large enough orq is very close to zero. Hence, our results for the super-linear case are
Chapter5. Boundedness of Solutions of Nonlinear VDEs with Constant Delay 62 sharp in some sense. As a special case letp = 2. In this case, the solution of (5.7.9) with initial conditionx(0) =x0 is bounded wheneverx0 ∈[0,1−q] and it is unbounded whenever x0 >1.
Chapter 6
BIBO Stability of Difference Equations
In this chapter we investigate the BIBO stability of controlled nonlinear discrete time systems described by difference equations with time-dependent delay. This work is related to our work on continuous case in Chapter4.
6.1 Problem statement
In this section we consider the nonlinear discrete time system
x(n+ 1)−x(n) =g(n, x(n−σ(n))) +u(n), n≥0 y(n) =Cx(n), n≥0.
(6.1.1)
Here x(n) is state vector, u(n) is input vector and y(n) is output vector of the system (6.1.1), C∈Rd1×d is constant matrix andg(n,·) :Rd→Rd,n≥0 satisfies
kg(n,x)k ≤˜ b(n)ϕ(k˜xk), x˜∈Rd, (6.1.2) whereb(n)>0,ϕ:R+→R+ is a monotone nondecreasing mapping.
We assume that the uncontrolled system, i.e., (6.1.1) with u ≡0 may have unbounded solu-tions. Our goal is to find a control law of the form
u(n) =−Dx(n−k) +r(n) (6.1.3)
63
Chapter6. BIBO Stability of Difference Equations 64 which guarantees that the closed-loop delayed system
x(n+ 1)−x(n) =g(n, x(n−σ(n)))−Dx(n−k) +r(n), n >0;
y(n) =Cx(n);
x(n) =ψ(n), −τ ≤n≤0,
(6.1.4)
is BIBO stable. D = diag(λ1, . . . , λd), λi > 0, i = 1, . . . , d. r(n) is the reference input, 0≤σ(n)≤`,τ := max{`, k},ψ∈S([−τ,0],Rd) andkψkτ := max
−τ≤n≤0kψ(n)k.
In Theorem 6.2.1 and 6.2.2 we give sufficient conditions on how to select the feedback gain and the time delaykto guarantee the boundedness of the solutions.
Similar to the notion of BIBO stability defined for the continuous case (Definition2.1.4), we introduce the next definition
Definition 6.1.1. The controlled system (6.1.4) is said to be BIBO stable by the control law (6.1.3) if there exist positive constants θ1 and θ2 = θ2(kψkτ) such that every solution of the system (6.1.4) satisfies
ky(m)k ≤θ1krk∞+θ2, m∈N for every reference inputr ∈Ld∞.
The following definition is related to our Definition 4.1.1of local BIBO stability on the con-tinuous case.
Definition 6.1.2. The controlled system (6.1.4) is said to be locally BIBO stable by the control law (6.1.3) if there exist positive constants δ1,δ2 and θsatisfying
ky(m)k ≤θ, m∈N provided that kψkτ < δ1 andkrk∞< δ2.
Our approach is the following. We associate the linear system
z(n+ 1)−z(n) =−Dz(t−k), n≥0 (6.1.5)
with the constant delaykand the initial condition
z(n) =ψ(n), −k≤n≤0 (6.1.6)
to (6.1.4). Then the state equation (6.1.4) can be considered as the nonlinear perturbation of (6.1.5), and by the variation of constants formula (see Lemma2.3.3) we get
x(n) =z(n) +
n−1
X
j=0
W(n−j−1) [g(j, x(j−σ(j))) +r(j)], n≥1, (6.1.7)
Chapter6. BIBO Stability of Difference Equations 65 wherez(n) is the solution of (6.1.5)-(6.1.6) andW is the fundamental solution of (6.1.5), i.e., the solution of the IVP
W(n+ 1)−W(n) =−DW(n−k), n≥0, W(n) =
( 0, −k≤n≤ −1;
I, n= 0,
where I is the identity matrix and 0 is the zero matrix. Since D is a diagonal matrix, it is easy to see that W is diagonal matrix too.
We can rewrite the equation (6.1.7) as a VDE x(n+ 1) =z(n+ 1) +
n
X
j=0
W(n−j)g(j, x(j−σ(j))) +
n
X
j=0
W(n−j)r(j), n≥0,
and so it is equivalent to
x(n+ 1) =
n
X
j=0
f(n, j, x(j−σ(j))) +h(n), n≥0, (6.1.8)
where 0≤σ(n)≤`,τ := max(k, `),
h(n) :=z(n+ 1) +
n
X
j=0
W(n−j)r(j), (6.1.9)
and
f(n, j,x) :=˜ W(n−j)g(j,x),˜ 0≤j≤n, x˜∈Rd. (6.1.10) In Section 5.1 we studied the boundedness of the solutions of nonlinear VDEs, and next we apply our results for the boundedness of (6.1.4).
6.2 Main results
Our main goal in this section is to formulate sufficient conditions which guarantee BIBO stability of the controlled system (6.1.4).
Our first result is given for the case when g in (6.1.2) has a sub-linear estimate, i.e., when φ(t) =tp,with 0< p <1 in (6.1.2).
Theorem 6.2.1. Let g:Rd→Rd be a function which satisfies inequality (6.1.2) with ϕ(t) =tp, 0< p <1, t≥0. The feedback control system(6.1.4) is BIBO stable if
kbk∞:= sup
n≥0
b(n)<∞, 0< λi <2 cos kπ
2k+ 1, (6.2.1)
Chapter6. BIBO Stability of Difference Equations 66 hold for all i= 1, . . . , d, and D=diag(λ1, . . . , λd).
Proof. By [65] and under our conditions, fori= 1, . . . , d,the solutionzi of the IVP
zi(n+ 1)−zi(n) =−λizi(n−k), n≥0 (6.2.2) with initial condition
zi(n) =ψi(n), −k≤n≤0 (6.2.3)
satisfies the inequality
|zi(n)| ≤Mkψkτρn, n≥0, (6.2.4) whereM is positive constant, ρ∈(0,1) and kψkτ := max−τ≤j≤0kψ(j)k.
Hence every solution of (6.2.2) tends to zero as n→ ∞, and kzk∞:= sup
n≥0
kz(n)k ≤Mkψkτ <∞. (6.2.5) From (6.2.4) it follows , limn→∞wi(n) = 0 for i= 1, . . . , d, and
$:= max
0≤i≤d
∞
X
n=0
|wi(n)|<∞. (6.2.6)
From (6.1.7), for alli= 1, . . . , d, we have xi(n+ 1) =zi(n+ 1) +
n
X
j=0
wi(n−j)[gi(j, x(j−σ(j))) +ri(j)], t≥0,
where x(n) = (x1(n), . . . , xd(n))T, g(n, x) = (g1(n, x), . . . , gd(t, x))T, r(n) = (r1(n), . . . , rd(n))T and z(n) = (z1(n), . . . , zd(n))T. Therefore (6.1.9) and (6.1.10) imply
fi(n, j, x(j−σ(j))) =wi(n−j)gi(j, x(j−σ(j))) and hi(n) =zi(n+ 1) +
n
X
j=0
wi(n−j)ri(j).
Hence by (6.1.2)
|fi(n, j, x(j−σ(j)))| ≤ |wi(n−j)| |gi(j, x(j−σ(j)))| ≤ |wi(n−j)|b(j)φ
−τ≤¯maxn≤jkx(¯n)k
, so condition (B2) in Section5.1holds withai(n, j) :=|wi(n−j)|b(j), 0≤j≤n.
Chapter6. BIBO Stability of Difference Equations 67 By (6.2.1), (6.2.6), (6.2.5) and the definition of the supremum norm, we obtain
γ := max
wherekzk∞is defined by (6.2.5) and$is defined by (6.2.6), therefore the condition (5.2.8) holds.
By conditions (6.2.1) and (6.2.6) we get α: = max
i.e., (5.2.9) holds. Then the conditions of Theorem 5.4.1for 0< p <1 with N = 0 are satisfied, hence the solutionx of (6.1.4) is bounded.
Now we show that the inequality (A.3.3) is satisfied with v:= max
Chapter6. BIBO Stability of Difference Equations 68 whereα0,γ0 andα are defined by (A.3.1), (A.3.2) and (6.2.9). Therefore (A.3.3) in the proof of Lemma5.4.2is satisfied, hence by Theorem 5.4.1we get forn≥ −τ
kx(n)k< v = max
3($kbk∞kψkpτ+kzk∞+$krk∞),(3$kbk∞)1−p1
<¯λkrk∞+ max
3($kbk∞kψkpτ+kzk∞),(3$kbk∞)1−p1
. Then
ky(n)k ≤ kCkkx(n)k ≤θ1krk∞+θ2, n≥0, where θ1 := 3$kCk, θ2 := kCkmax
3($kbk∞kψkpτ +kzk∞),(3$kbk∞)1−p1
and 0 < p < 1.
Hence by Definition6.1.1the feedback control system (6.1.4) is BIBO stable.
The following theorem gives a sufficient condition for the BIBO stability in the case of a linear estimate.
Theorem 6.2.2. Let g:Rd→Rd be a continuous function which satisfies inequality(6.1.3) with φ(t) =t, t≥0. The feedback control system (6.1.4) is BIBO stable if
kbk∞< 1
$ and 0< λi<2 cos kπ
2k+ 1, t≥0, (6.2.10) hold for alli= 1, . . . , d, D=diag(λ1, . . . , λd) and $ is defined by (6.2.6).
Lemma 6.2.3. Assume the conditions of Theorem6.2.2 are satisfied. The inequalities
ai(0,0)kψkτ +|hi(0)|< v, i= 1, . . . , d (6.2.11) and
ai(n,0) kψkτ +
n
X
j=1
ai(n, j) v+|hi(n)| ≤v, n≥1, i= 1, . . . , d (6.2.12) are satisfied with
v:= max
$krk∞+Mkψkτ 1−$kbk∞ ,kψkτ
, (6.2.13)
where λ0 := min(λ1, . . . , λd), W =diag(w1, . . . , wd) is the fundamental solution of (6.1.5), z is the solution of the IVP (6.1.5)-(6.1.6), M is defined in (6.2.4), kψkτ := sup−τ≤n≤0|x(n)| and ai(n, j) :=wi(n−j)b(n),0≤j ≤n.
Chapter6. BIBO Stability of Difference Equations 69 Proof. Under our condition (6.2.10) and from Lemma 2.3.2we have (6.2.6). Hence
α := max
Chapter6. BIBO Stability of Difference Equations 70
Hence the conditions (6.2.11) and (6.2.12) are satisfied.
Now we give the proof of Theorem 6.2.2.
Proof of Theorem 6.2.2. From conditions (6.2.7) and (6.2.10), we have γ : = max
Hence the condition (i) in Theorem 5.4.5 holds with N = 0, therefore the solution x of state equation of system (6.1.4) is bounded, more exactly
kx(n)k ≤v, n≥0.
From Lemma6.2.3 the conditions (5.2.4) and (5.2.5) hold with (6.2.13). Hence ky(n)k ≤ kCkkx(n)k ≤ kCkv
=kCkmax Then by Definition6.1.1the feedback control system (6.1.4) is BIBO stable.
Corollary 6.2.4. Letg:R+×Rd→Rdbe a continuous function which satisfies inequality(6.1.2) withφ(t) =t, t≥0. The feedback controlled system (6.1.4) is BIBO stable if
kbk∞< λi≤ kk
(k+ 1)k+1, t≥0, i= 1, . . . , d (6.2.15) hold.
Chapter6. BIBO Stability of Difference Equations 71 Proof. Under our condition (6.2.15) and from Lemma2.3.2we get that the fundamental solution wi of (6.2.2)-(6.2.3) is positive and
1≤i≤dmax
The proof is similar to the proof of Theorem6.2.2with Lemma 6.2.3and it is omitted.
Locally BIBO stability of the feedback controlled system (6.1.4) is proved under small initial condition and small reference input in the following theorem.
Theorem 6.2.5. Let g:R+×Rd→Rdbe a continuous function which satisfies inequality(6.1.2) withφ(t) =tp, p >1,t≥0. Then the solutionxof the feedback controlled system(6.1.4)is locally BIBO stable if (6.2.10) holds.
Proof. Suppose (6.2.10), kψkτ ≤ δ1 and krk∞ ≤ δ2 where δ1, δ2 will be specified later. From (6.2.5), (6.2.8) and (6.2.14) we have
kzk∞≤Mkψkτ ≤M δ1, M >0,
We can select the positive constantsδ1 and δ2 such that δ1p+M δ1+$δ2 ≤
Chapter6. BIBO Stability of Difference Equations 72 and
v >
p−1 p
1 p$kbk∞
p−11
≥δ1p+M δ1+$δ2
≥αkψkpτ+γ ≥a(0,0)kψkpτ+kh(0)k.
Therefore the conditions of Theorem 5.4.7 are satisfied, so the solution of state control system (6.1.4) is bounded, i.e.,
kx(n)k< v:=
1 p$kbk∞
p−11
, n≥0.
Hence
ky(n)k ≤ kCkkx(n)k< θ, where
θ:=kCk
1 p$kbk∞
p−11 .
By Definition 6.1.2the solution of the control system (6.1.4) is locally BIBO stable.
Chapter 7
Applications
In this chapter we give some applications of the main results of Chapter 3.
7.1 El Ni˜ no and the delayed action oscillator
Historically, the term El Ni˜no was used to refer to a warm southward flowing current that moder-ates the low sea-surface temperature that typically appears around Christmas-time and therefore named as El Ni˜no (the Little Boy or Christ Child in Spanish), which lasts several months. In certain years this current is unusually strong, bringing heavy rains and flooding inland, but also decimating fishing stocks, bird populations, and the other water based wildlife in what would normally be an abundant part of the Pacific.
Today, the term El Ni˜no is most often used when describing a far large-scale warming that can be observed across the whole of the Pacific Ocean by certain characteristic climatic conditions.
The effects of El Ni˜no can also be seen globally. Examples include spring rainfall levels in Central Europe, flooding in East Africa, and the ferocity of the hurricane season in the Gulf of Mexico [12]. El Ni˜no is just one phase of the El Ni˜no Southern Oscillation (ENSO) phenomenon, that is, an irregular cycle of coupled ocean temperature and atmospheric pressure oscillations across the whole equatorial Pacific.
The delayed-action oscillator is a nonlinear model in which the El Ni˜no oscillations in the sea-surface temperature are modeled by three terms. Let T denote the temperature anomaly, that is, the deviation from a suitably defined long term average temperature. The model is given as a first order, nonlinear differential equation
T˙(¯t) =cT(¯t)−c1T3(¯t)−c2T(¯t−σ), (7.1.1) where c, c1, c2 and σ are constants. We assume that 0 < c2 < c due to the loss of information during transit and imperfect reflection. The first term on the right-hand side of Eq. (7.1.1)
73
Chapter7. Applications 74 represents a strong positive feedback within the ocean atmosphere system and the second term is an unspecified nonlinear net damping term that is present to limit the growth of unstable perturbations. The strength of the returning emerging signals relative to that of the local non-delayed feedback is denoted byc2.
We can rewrite the Eq. (7.1.1) as the form
˙
x(t) =x(t)−x3(t)−αx(t−τ), (7.1.2)
wheret=c¯t,x=p
c1/cT,α=c2/c and τ =cσ (see [12,109]).
We can study its dynamics by analyzing its behavior close to the fixed points because we cannot solve Eq. (7.1.1) analytically. Let ˜x be an equilibrium of (7.1.2). Then
(1−α)˜x−x˜3 = 0, which yields ˜x = 0,±√
1−α for α ∈ (0,1). The trivial fixed point, corresponding to a zero temperature anomaly is unstable, therefore we shall concentrate on the other two fixed points
˜
x± = ±√
1−α. Due to symmetry, it is sufficient to consider only one of these points, and we choose ˜x+=√
1−α.
Letz=x−x˜+, then Eq. (7.1.2) becomes
˙
z(t) = (3α−2)z(t)−αz(t−τ)−(z3(t) + 3√
1−αz2(t)). (7.1.3) with initial condition
z(t) =ψ(t), −τ ≤t≤0. (7.1.4)
We associate the homogenous linear equation
˙
y(t) =−ay(t)−by(t−τ), t≥0 (7.1.5)
to (7.1.3) wherea= 2−3α and b=α. The characteristic equation of (7.1.4) is
λ=−a−be−λτ. (7.1.6)
It is known that from [58] that the equation (7.1.6) always has a leading root λo =αo+iβo. The stability region of (7.1.5) is well-known (see, e.g., [58]). To simplify notation we introduce the open set Ω(τ) ∈R2 as the points bounded below by the line b=−aand from above by the curve
a=−scos(τ s), b= s
sin(τ s), s∈h 0,π
τ i
, see Figure7.1.
Chapter7. Applications 75 From [58] we known that (a, b) ∈ Ω(τ) if and only if αo <0, moreover Theorem 3.1 in [51]
implies that the trivial solution of (7.1.5) is asymptotically stable and |βo|τ < π2 if a < b, and bτ eaτ < π
2,
hold, i.e., the trivial solution of (7.1.5) witha= 2−3α and b=α is asymptotically stable if α ∈(0,1), and ατ e(2−3α)τ < π
2, (7.1.7)
hold. We note that the second inequality of (7.1.7) is only a sufficient condition (see [51] for exact stability region).
It is known (see e.g. [58]) that if the trivial solution solution of (7.1.5) is asymptotically stable, then there existsM ≥1 such that
kyk∞:= sup
t≥0
|y(t)| ≤Mkψkτ, (7.1.8) and the fundamental solutionw of (7.1.5) satisfies
$:=
Z ∞ 0
|w(t)|dt <∞. (7.1.9) Defineϕ(v) :=v3+ 3√
1−αv2. Then the variation of constant formula gives z(t) =y(t) +
holds. Therefore Theorem 3.2.1 implies that the solution of the IVP (7.1.3)-(7.1.4) is bounded, if (7.1.7) holds,
Chapter7. Applications 76
are satisfied. Then the solution of (7.1.2) satisfies
|x(t)−x|˜ < v, t≥0.
We note (see [51]) that if
1
e < ατ e(2−3α)τ < π 2
holds, then the fundamental solutionw of (7.1.5) is oscillatory, but if ατ e(2−3α)τ ≤ 1
e, thenw(t)>0, fort≥0 and$= 2(1−α)1 .