Post on 20-Oct-2020
transcript
Model Periodic releases Feedback control Mixed strategies Sparse measures
Control Strategies for Sterile InsectTechniques
Pierre-Alexandre Bliman* (Inria & LJLL, Paris)Daiver Cardona-Salgado (UAO, Cali, Colombia)
Yves Dumont (CIRAD, Montpellier & U. Pretoria, South Africa)Olga Vasilieva (Univalle, Cali, Colombia)
June 14, 2019
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Dengue epidemicsI ' 400 millions infected per year
(96 millions symptomatic) – 30times more than 1960!
I ' 3.9 billion humans at risk in128 countries
I ' 500 000 people with severedengue, fatality ' 2.5% (≤1%with proper medical care)
I 4 different strains, complexcross-immunity: no treatment(except symptoms), no vaccine
I Aedes aegypti and Ae.albopictus main vectors (and forzika, chikungunya. . . )
[World Health Organization, 2016]
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Vector repartition (1/2)
Ae. Aegypti
Presence probability(from 0 to 1)
Spatial resolution:5 km × 5 km
Ae. Albopictus
[MUG Kraemer et al., eLife, June 2015]
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Vector repartition (2/2)
Year of classification ofAe. albopictus “implantedand active” by department
[Direction générale de la santé, May 2019]
4 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
NEW Control methods. . .
[NL Achee et al., PLoS Negl Trop Dis, March 2019]5 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
. . . and NEW Control theory problems
I Issues: Modeling, Observation, Control in PopulationDynamics
I Spatial aspects, but not only: age, sex and genotypicstructure may be important
I Methods: Control theory for (Sub-)Monotone Systems
Today’s talk:
Control Strategies for Sterile Insect Techniques
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Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
7 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
8 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
A sex-structured model
M,F : number of males, females
{Ṁ = rρe−β(M+F )F − µMM,Ḟ = (1− r)ρe−β(M+F )F − µFF .
(1)
r Primary sex ratioρ Mean number of eggs deposited per female per day (day−1)
µM , µF Mean death rate for male, female per day (day−1)
β Characteristic of the competition effect per individual
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Equilibria of the modelI All trajectories are ultimately uniformly bounded.I Denote E ∗0 = (0, 0) the mosquito-free equilibrium of (1).
Define the basic offspring numbers for males and females:
NF :=(1− r)ρµF
, NM :=rρ
µM(2)
Theorem 1 (Equilibria of the entomological model)
• If NF < 1, then (1) has E ∗0 as unique equilibrium.• If NF > 1, then (1) also has a unique positive equilibrium E ∗:
F ∗ =NF
NF +NM1
βlogNF , M∗ =
NMNF +NM
1
βlogNF
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Stability of the equilibria
Theorem 2 (Stability properties of the entomological model)
I If NF < 1, then E ∗0 is Globally Asymptotically Stable(GAS).
I If NF > 1, then E ∗0 is unstable, and E ∗ is GAS inR2+ \ {(M, 0),M ∈ R+}
0 M*0
F* Phase portrait for NF > 1.
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Entomological model controlled by sterile males
MS : number of sterile males.Ṁ = rρ MM+γMS (t)e
−β(M+F )F − µMM,Ḟ = (1− r)ρ MM+γMS (t)e
−β(M+F )F − µFFṀS = Λ− µSMS
(3)
Λ Number of sterile males released per day (day−1)µS Mean death rate for sterile male per day (day
−1)γ Relative reproductive efficiency
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Model Periodic releases Feedback control Mixed strategies Sparse measures
General assumptions
In all the sequel, we make the following hypotheses:
I NF > 1I µS ≥ µM ≥ µFI γ < 1
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Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Impulse periodic releases of constant amplitudeFor Λ > 0 constant, consider the release policy
Ṁ = rρFM
M + γMSe−β(M+F ) − µMM, (4a)
Ḟ = (1− r)ρ FMM + γMS
e−β(M+F ) − µFF , (4b)
ṀS = τΛ∑n∈N
δnτ − µSMS (4c)
When t → +∞, MS converges towards the periodic function
MperS (t) =τΛe−µS(t−b
tτcτ)
1− e−µSτ. (5)
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Behaviour of the asymptotic periodic system
We thus consider the periodic system:
Ṁ = rρFM
M + γMperS (t)e−β(M+F ) − µMM, (6a)
Ḟ = (1− r)ρ FMM + γMperS (t)
e−β(M+F ) − µFF . (6b)
Theorem 3 (Stability for constant periodic impulses)
Assume
Λ ≥ Λcritper :=cosh (µSτ)− 1
µSτ21
eβγ
×min{
2NM , 2NF ,max{r , 1− r}max{NMr,NF
1− r
}}. (7)
Then E ∗0 is GAS for system (6).16 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Proof of Theorem 3 (extracts)
Ṁ =
(rρ
F
M + γMperSe−β(M+F ) − µM
)M (8a)
Ḟ =
((1− r)ρ M
M + γMperSe−β(M+F ) − µF
)F (8b)
• MM+γMperS
e−β(M+F ) ≤ MM+γMperS
e−βM ≤ αM+γMperS
≤ αγMperS
, where
α := max{xe−βx : x ≥ 0
}= 1eβ . Thus
F((n + 1)τ
)≤ e
((1−r)ραγ
〈1
MperS
〉−µF
)τF (nτ)
• Let V(M,F ) := 12 (M2 + F 2), then
V̇ ≤(
max{r , 1− r}α 1γMperS
− 2 min{µM , µF})V
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Constant impulsive releases
0 100 200 300 400 500 600 700
time (days)
-3
-2
-1
0
1
2
3
4
5
Log
10(N
um
ber
of In
div
iduals
)
Male
Female
Sterile Male
0 100 200 300 400 500 600 700 800
time (days)
-3
-2
-1
0
1
2
3
4
5
Log
10(N
um
ber
of In
div
iduals
)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) Total # released males # weeks
τ = 7 924 627 84
τ = 14 942 869 84
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Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Framework
Instead of (4), we consider, for Λn ≥ 0, n ∈ N,
Ṁ = rρFM
M + γMSe−β(M+F ) − µMM, (9a)
Ḟ = (1− r)ρ FMM + γMS
e−β(M+F ) − µFF , (9b)
ṀS = τ∑n∈N
Λnδnτ − µSMS (9c)
Assume available periodic measurements M(nτ),F (nτ), n ∈ N.
How to choose Λn, n ∈ N,to have M(t),F (t)→ 0 for t → +∞?
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Step 1 - Acting directly on MSProposition 4 (Stabilization by direct control of MS)
Let k ∈ (0, µF(1−r)ρ). If M(t)M(t) + γMS(t)
< k , t ≥ 0 (10)
i.e. γMS(t) ≥(
1k − 1
)M(t) for any t ≥ 0, then E ∗0 is GAS for
system (4).
Proof of Proposition 4.When (10) holds, then(
Ṁ
Ḟ
)≤ A
(MF
), A :=
(−µM rρk
0 −µF + (1− r)ρk
)(11)
A is Metzler and Hurwitz: for
(Ṁ ′
Ḟ ′
)= A
(M ′
F ′
),
(M ′(0)F ′(0)
)=
(M(0)F (0)
),
0 ≤(M(t)F (t)
)≤(M ′(t)F ′(t)
)→ 0 for t → +∞
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Step 2 - Shaping control compliant with Step 1
To obtain (10), that is: γMS(t) ≥(
1k − 1
)M(t) on (nτ, (n + 1)τ ],
choose Λn such that
∀t ∈ (nτ, (n + 1)τ ], γMS(t) ≥(
1
k− 1)M ′(t)
that is: ∀ s ∈ (0, τ ],
γ(Λnτ + MS(nτ))e−µS s ≥
(1
k− 1)(
1 0)eAs
(M(nτ)F (nτ)
)(12)
It turns out that (12) has to be verified only for s = τ .
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Stabilization by impulsive feedback control
Theorem 5 (Stabilization by feedback control)For a given k ∈
(0, µF(1−r)ρ
), assume that for any n ∈ N:
τΛn ≥∣∣∣∣K (M(nτ)F (nτ)
)−MS(nτ)
∣∣∣∣+
(13a)
K :=1
γ
(1−kk e
(µS−µM )τ
rρ(1−k)µM−µF +(1−r)ρk
(e(µS−µF +(1−r)ρk)τ − e(µS−µM )τ
))T (13b)Then E∗0 is GAS for (9). If moreover
τΛn ≤ K(M(nτ)F (nτ)
)(13c)
then+∞∑n=0
Λn < +∞.
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Proof of Theorem 5
I One first verifies that if Λn is chosen as in (13a), then∀t ∈ (nτ, (n + 1)τ ], γMS(t) ≥
(1k − 1
)M(t).
I Therefore, using the linear comparison system
0 ≤(M(t)F (t)
)≤(M ′(t)F ′(t)
)→ 0 for t → +∞
and F and M vanish exponentially.
I Due to this exponential decrease, (13c) implies+∞∑n=0
Λn < +∞.
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Closed-loop periodic impulsive control, kNF = 0.2
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
6
7
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
6
7
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) Total # released males # weeks
τ = 7 2 251 052 64
τ = 14 2 390 676 64
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Closed-loop periodic impulsive control, kNF = 0.99
0 200 400 600 800 1000 1200 1400 1600 1800
time (days)
-2
-1
0
1
2
3
4
5
6
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 200 400 600 800 1000 1200
time (days)
-2
-1
0
1
2
3
4
5
6
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) Total # released males # weeks
τ = 7 794 807 240
τ = 14 909 344 130
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Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Mixed control strategies
Theorem 6 (Stabilization by saturated control)Assume that, for any n ∈ N and Λ̄ positive constant,
Λn ≥ min{
Λ̄ ;
∣∣∣∣K (M(nτ)F (nτ))−MS(nτ)
∣∣∣∣+
}(14)
and that one of the following conditions is fulfilled:
• Case 1.Λ̄ = 2
(cosh (µSτ)− 1)µSτ 2
1
eβγNF , k ∈
(0,
µF(1− r)ρ
)(15)
• Case 2.
Λ̄ =(cosh (µSτ)− 1)
µSτ 21
eβγmax{r , 1− r}max
{NMr,NF
1− r
},
k ∈
0, 2 µMρ
1− rr2
√1 + µFµM
(r
1− r
)2− 1
(16)Then E∗0 is GAS for (9). 28 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Proof of Theorem 6
Using the proof of Theorem 3, one shows that, along thetrajectories and whatever the mode,
I Case 1. F((n + 1)τ
)≤ e
((1−r)ρα
γ
〈1
MperS
〉−µF
)τF (nτ).
I Case 2. V̇ ≤ −εV, for V(M,F ) = 12 (M2 + F 2) and some
∃ε > 0.
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Mixed periodic impulsive control, kNF = 0.2
0 100 200 300 400 500 600 700
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600 700
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) Total # released males # weeks
τ = 7 450 668 72
τ = 14 465 187 72
30 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Mixed periodic impulsive control, kNF = 0.99
0 200 400 600 800 1000 1200 1400 1600 1800 2000
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 200 400 600 800 1000 1200
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) Total # released males # weeks
τ = 7 457, 489 69
τ = 14 427, 701 74
31 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
A controlled sex-structured entomological model
Impulse periodic releases
Feedback control approach
Mixed control strategies
Sparse measurements
32 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Feedback control with sparse measurementsMeasurements are long and costly. We now adapt Theorem 5 tomeasurements achieved with period pτ , for given p ∈ N∗.
Theorem 7 (Impulsive control with sparse measurements)For given k ∈
(0, µF(1−r)ρ
), assume for any n ∈ N, m = 0, 1, . . . , p − 1,
τΛnp+m ≥
∣∣∣∣∣Kp(M(nτ)F (nτ)
)−MS(npτ)e−mµSτ −
m−1∑i=0
Λnp+ie−µS (m−i)τ
∣∣∣∣∣+
(17a)
Kp :=eµSτ
γ
(1−kk e−(m+1)µMτ
rρ(1−k)µM−µF +(1−r)ρk
(e−(µF−(1−r)ρk)(m+1)τ − e−µM (m+1)τ
))T (17b)Then E∗0 is GAS for (4).
If moreover τΛnp+m ≤ Kp(M(nτ)F (nτ)
), then
+∞∑n=0
Λn < +∞.
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Closed-loop control, sparse measures, kNF = 0.2, p = 4
0 50 100 150 200 250 300 350 400 450 500
time (days)
-3
-2
-1
0
1
2
3
4
5
6
7
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600
time (days)
-3
-2
-1
0
1
2
3
4
5
6
7
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) # released males # weeks # releases
τ = 7 4 363 430 54 34
τ = 14 2 896 835 56 17
34 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Closed-loop control, sparse measures, kNF = 0.99, p = 4
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
6
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
6
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) # released males # weeks # releases
τ = 7 1 221 593 58 37
τ = 14 1 043 107 62 20
35 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Mixed control, sparse measures, kNF = 0.2, p = 4
0 100 200 300 400 500 600
time (days)
-3
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) # released males # weeks # releases
τ = 7 534 849 65 53
τ = 14 499 497 66 25
36 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Mixed control, sparse measures, kNF = 0.99, p = 4
0 100 200 300 400 500 600
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
0 100 200 300 400 500 600 700
time (days)
-2
-1
0
1
2
3
4
5
Lo
g10
(Nu
mb
er
of
Ind
ivid
ua
ls)
Male
Female
Sterile Male
τ = 7 days τ = 14 days
Period (days) # released males # weeks # releases
τ = 7 450, 077 69 53
τ = 14 449, 059 74 28
37 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Thank you for your attention
For more details:
I “Implementation of Control Strategies for Sterile InsectTechniques”, to appear in Mathematical Biosciences
I pierre-alexandre.bliman@inria.fr
38 / 42
mailto:pierre-alexandre.bliman@inria.fr
Model Periodic releases Feedback control Mixed strategies Sparse measures
Kamke’s theoremDefinition 1 (Function of type K )f : Rm → Rm is of type K in a set S if: ∀x , y ∈ S , ∀i ∈ {1, . . . ,m},
xi = yi ∧ x ≥ y ⇒ fi (x) ≥ fi (y)Proposition 8 (Differential characterisation of type K )When S convex, f C1 is of type K if ∂fi∂xj (x) ≥ 0, x ∈ S , i 6= j.
Theorem 9 (e.g. [1965, Coppel])Let f (t, ·) be of type K , ∀t ∈ [a, b], let x fulfil ẋ = f (t, x) on [a, b].
I If y C0 satisfies D−(y)(t) := lim suph→0+
y(t)−y(t−h)h > f (t, y(t)) on
(a, b] and y(a) > x(a), then y(t) > x(t), t ∈ [a, b]
I If y C0 satisfies D−(y)(t) := lim infh→0+
y(t+h)−y(t)h < f (t, y(t)) on
(a, b] and y(a) < x(a), then y(t) < x(t), t ∈ [a, b]
In other words: a differential inequality induces comparison results.39 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Metzler matrices and linear positive systems
Definition 2 (Metzler matrix)A ∈ Rm×m is called Metzler (or essentially nonnegative) matrix if
i 6= j ⇒ aij ≥ 0
Theorem 10 (e.g. [2010, Haddad et al.])A is Metzler iff componentwise: ∀t ≥ 0, eAt ≥ 0m
Definition 3 (Positive system)Let A ∈ Rm×m,B ∈ Rm×p,C ∈ Rq×m, the linear input/output system
ẋ = Ax + Bu, y = Cx , x(0) = x0 (18)
is positive if: x ≥ 0m, y ≥ 0q whenever x0 ≥ 0m and u ≥ 0p.
Corollary 11 (e.g. [2000, Farina, Rinaldi])System (18) is positive iff A Metzler and B,C ≥ 0.
40 / 42
Model Periodic releases Feedback control Mixed strategies Sparse measures
Switched systems and stability
Definition 4 (e.g. [2003, Liberzon])
Given systems ẋ = fi (x(t)), fi : Rm → Rm locally Lipschitz,i ∈ I, and σ : [0,+∞)→ I piecewise constant, one defines aswitched system as:
ẋ = fσ(t)(x(t)), x(0) = x0 (19)
Stability of all ẋ = fi (x(t)) does not imply stability for (19)!
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Model Periodic releases Feedback control Mixed strategies Sparse measures
Common Lyapunov functions
Let a common equilibrium exist, e.g. fi (0m) = 0m, i ∈ I.
Definition 5 (e.g. [2003, Liberzon])
A positive definite C1 map V : Rm → R+ is called commonLyapunov function for the family of systems ẋ = fi (x(t)),i ∈ I, if exists a positive definite C0 map W : Rm → R+ with
∀i ∈ I, ∀x , ∂V∂x
fi (x) ≤ −W (x)
Theorem 12 (e.g. [2003, Liberzon])
Let the family of systems ẋ = fi (x(t)), i ∈ I, possess aradially unbounded common Lyapunov function. Then theswitched system ẋ = fσ(t)(x(t)) is GAS, uniformly wrt σ.
42 / 42
bbleu A controlled sex-structured entomological modelbbleu Impulse periodic releasesbbleu Feedback control approachbbleu Mixed control strategiesbbleu Sparse measurements