http://www.BillHowell.ca/Neural nets/Paper reviews/180209 conference paper review - math only.txt
www.BillHowell.ca, Bill@BillHowell.ca 05Jan2018 initial
see also : "http://www.BillHowell.ca/Neural nets/Paper reviews/5_NN math symbols & nomenclature - time-scale systems.txt"
WARNING !! : I only just started the math review, adding most of the symbols for this paper to the cumulative Symbol listing. I ran out of time for doing the actual step-by-step deerivations for Theorem 1
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INSTRUCTIONS - text editor viewing :
View this file in a text editor with UNICODE characters (most modern text editors have this), constant width font (eg Courier 10), tabs of 3 spaces each. To follow a series of derivations that are wider than the viewing window, turn off "word wrap".
Although these will cause problems in many text editors, I have used extended Unicode characters from :
https://en.wikipedia.org/wiki/Mathematical_Alphanumeric_Symbols
Mathematical Alphanumeric Symbols upper characters (U+1D400–U+1D7FF)
I use the text editor "kwrite" in Linux, which is one of two or three that I have found to be excellent (there are many, many others!). (I should use emacs, or so they say...)
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Table of contents (without line numbers)
>>>>> "Flat-line" UNICODE nomenclature :
>>>>> Symbols
>>>>> MATH DEFINITIONS
>>>>> 1 Introduction
>>>>> 2 Preliminaries
>>>>> 3 Boundedness and stability conditions for time-varying systems on time scales
>>>>> Theorem 1 Proof.
*****************************
>>>>> "Flat-line" UNICODE nomenclature :
This style of writing expressions allows the use of any UNICODE-enabled text editor without need for subscripts or superscripts. It also facilitates faster copy-paste re-use of expression. Furthermore, although not used for this paper, the intent is to facilitae used of expressions in a symbolic processor (I've only used this in the past for very limited capablities). Admittedly, it is a nuisance to adapt and lacks the clarity of standard formatting.
Various notations may be combined...
A variable [scalar, vector, matrix] for use in this list of notations
A(i,j) (i,j)th component of A
A(i,j) (i,j)th component of A, notation just for simple indices
A_T transpose of A
|A| absolute value of matrix A (each element)
||A|| [Euclidean, spectral] norm of A ||A||2
p3h0.25 ||·|| denotes the Euclidean norm.
||A||_p = ( sum(i from 1 to n : |A(i)|^p )^(1/p) ) denotes the p-norm of vector z
A_dot time derivative of A, same as d[dt: A]
A_til A tilde over
A_ovr A overlined
A_und A underlined
A_hat A with ^ above it
A_sup(x) A superscript x - to avoid confuxion with A^x
A_sub(x) A subscript x - as distinct from A(x)
A^x A to the power of x
λ max ( λ min ) refers to maximum (minimum) eigenvalue
t_minus t approached from t < t_minus
t_plus t approached from t_plus < t
Indexing :
yi y(i), simple concatenation (might be confused with y*i)
y(i+1) array type notation
Note the more clear & easy to read indexing used in this review for mixing indexes and t or other variables
... example ai(t) -> a(i,t)
Inequalities (for faster typing, symbolic processing)
≤ <=
≥ >=
≠ !=
[super, sub]-script symbols
"_cup" denotes variable values during ??? eg p4L23 Eq (9d)
"_ovr" denotes variable values during times s ∈ [t_til(0),0]??? eg ξ_ovr(t)
"_til" denotes variable values during ??? eg p4L38
Un-numbered [equations, inequalities] in the paper are denoted by their location, enclosed by parenthesis, eg :
* y(t) ≤ ρ*sup[s∈[t−T,t]: y(s)]
means the above equation on page 5, Line 42.
Reviewer equations are denoted by some [equation, section, subsection, whatever] and index, enclosed by parenthesis, eg :
(R.10a.5) z(t**) ≤ d/c*z(t*)*e^(-θ*ln(c)/T) , ∀t ≥ t0
Calculus :
d[dt: x] total derivative of x with respect to t
dp[dt: x] partial derivative of x with respect to t
d^n[dt^n: x] nth total derivative of x with respect to t
dp^n[dt^n: x] nth partial derivative of x with respect to t
∫[ds: f] indefinite integral of f with respect to s
∫[ds, a to b: f] definite integral of f with respect to s, from a to b
Other :
sum[i=1 to n: f(n,t)] sum of a series
Π[tt_plus : (r(t,t) - r(s,t))/(t - s)) }
D_minus D_minus(t) = lim{s->t_minus: (r(t,t) - r(s,t))/(t - s)) }
******* denotes start/end of topics & sub-topics
+-----+ denotes sub-steps in a [proof, development]..
denote [start,end] of checks on specific steps by the reviewer (me), on expression (1) in the example below :
>>>>> check (?) :
>> ???? check (?) unfinished
<<<<< end check.
>> start of short reviewer comments
<< end of short reviewer comments (Often this is not used as only one or two lines of comment are involved
>>** comment for extraction to main part of review, Section "C6. MATH CHECKS - step-by-step"
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>>>>> Symbols
+-----+
first characters of line - indicates definition used in a similar paper, which makes for interesting contrasts,
plus confirmation of a usage of variable symbols that is similar to other papers in the area
(*) taken from earlier papers, mostly for NN stability Theorems
p2h0.8 (example) - taken from "180207 conference paper review - math only.txt"
kwrite editor, regular expression -> Search : ^\(p[0-9]*L[0-9]*\), Replace : \(171127\)
special contexts :
- normally, I will use a Latin letter in place of Greek or some Unicode if that letter is not used elsewhere
(for ease of typing)
- Note that the authors use symbols in a manner consistent with other papers in the area, of course making this paper easier to follow, and to identify subtle differences in assumptions etc.
+-----+
General list : normally in alphabetical order, except some "collections" of themes (start/end with +--+)
+--+
Metric spaces, sets (hexa refers to U+hexa)
LibreWrite (Ctrl+Shift)+U+hexa
kwrite text editor enter into LibreWrite, copy to kwrite?
* 𝕁 subset of ℝ^n
* ℝ, real numbers
* ℝ+, non-negative real numbers
p3h0.25 ℝ denotes the real number set
p3h0.25 ℝ+ denotes the nonnegative real number set
* ℝ^n n-dimensional Euclidean spaces
* ℝ^(n×n) n×n real matrix
p2h0.75 𝓡 and 𝓡+ denote the set of regressive and positive regressive functions;
* 𝕊 subset of ℝ
p2h0.33 𝕋 represents a time scale which is an arbitrary nonempty closed subset of the real set ℝ.
p3h0.3 min{𝕋} = 0 Without other statements, we always assume this
p3h0.75 𝕋aux ∆= {t(k,2)}|k=1 to ∞ an auxiliary discrete time scale
where
p3h0.75 t(i,2) is the i-th right-scattered point of 𝕋.
* ℤ positive integers
p3h0.25 ℤ denotes the integer number set
+--+
Latin symbols sorted in Latin alphabetical order
parameters of inertial NN (1), real-valued continuous functions, i ∈ N
* a(i,t) > 0
* b(i,t) > 0
* c(i,j,t) connection weights without delays
* d(i,j,t) connection weights with delays
* A(t) = diag[α(1,t),α(2,t),...,α(n,t)]_T
* B(t) = diag[β(1,t),β(2,t),...,β(n,t)]_T
* C(t) = ( c(i,j,t) )^(n*n)
* D(t) = ( d(i,j,t) )^(n*n)
p4h0.65 A = max[t≤T,t ∈ ℝ+(𝕋) : {G(t)} ]
p4h0.65 B = max{M, A},
* C^+(S,J) = {z : S → J is continuous and its upper right Dini derivative D+z exists }
... for any interval S ⊆ ℝ, J ⊆ ℝ^n
* D^+[dz: f(z)] ∆= lim[sup[h -> 0+: (f(z+h) - f(z))/h], Dini derivative of z
... https://en.wikipedia.org/wiki/Dini_derivative
p3h0.0 c a regressive constant
p2h0.75 Crd = Crd(𝕋) = Crd(𝕋,ℝ) denotes the set of rd-continuous functions
p2h0.85 e(p,t,s) = exp( ∫[∆τ from s to t: ξ(µ(τ),p(τ)) ] ) for p ∈ 𝓡, s,t ∈ 𝕋^k
... exponential function definition
p3h0.0(a) e(c,t,0) = exp(c*t) for any t ∈ ℝ, 𝕋 = ℝ timescale-type exponential function
p3h0.0(b) e(c,t,0) = (1 + c*h)^(t/h) for any t ∈ h*ℤ, 𝕋 = h*ℤ timescale-type exponential function
p2h0.85 e(p,t,s) = exp( ∫[∆τ from s to t: ξ(µ(τ),p(τ)) ] ) for p ∈ 𝓡, s,t ∈ 𝕋^k exponential function definition
* f = (f1,···,fn) ∈ C(ℝn,ℝn) is the activation function
* f(j,ξ(j,t)) represents the neuron activation function of the jth neuron at time t
p1h0.75 f : 𝕋 × ℝ^n → ℝ^n is a rd-continuous function satisfying f(t,0) = 0 for all t ∈ 𝕋.
p2h0.75 fσ(t) = f(σ(t))
p2h0.75 f∆(t) denotes the ∆-derivative of f at t ∈ 𝕋;
* |f(j,u) - f(j,v)| ≤ l(j)*|u - v| ∀u,v ∈ R, j ∈ N maximum |"gradient"| sort-of
p4h0.6 G(t) = ∫[∆τ, from 0 to t : g(τ) ]
p3h0.5 g : ℝ+ → ℝ a regressive function
* J(i,t) denotes the neuron of an external input on the ith neuron
* J ⊆ ℝ^n interval
* k ∈ N0
* K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
p1h0.95 k ∈ ℤ discrete time index
* L = diag(l1,···,ln)
* L = diag{l,l,...,l}, where l = max{1≤j≤n: l(j) }
* l(j) are positive constants
* l = max[1≤j≤n: l(j) ]
* M ≥ 0 constant
* M > 0 constant Eq (8)
p4h0.65 M < 0 constant
* N = {1,2···},
* N = {1,2,···,n} total number of neurons
* N0 = N ∪ {0}
* PC(J,_ovr,∆) = {φ : J_ovr → ∆ is continuous everywhere except at finite
... number of points t, at which φ(t+), φ(t−) exist and φ(t+) = φ(t)}.
... assume "PC" stands for "Pulse Condition" or something like that
* PC(S,J) = {z : S → J is continuous everywhere except at finite number of points t
... at which z(t+), z(t−) exist and z(t−) = z(t) }
p2h0.8 ⊖p = -p(t) /(1+µ(t)p(t)) p ∈ 𝓡+
p2h0.8 p⊖q = (p(t)−q(t))/(1+µ(t)q(t)) p, q ∈ 𝓡+
* R(l) I, l ≥ 1, is impulsive matrix
* R(t) = diag[γ(1,t),γ(2,t),...,γ(n,t)]_T but γ(1,t) = γ(1t)!?!
p2h0.75 𝓡 and 𝓡+ denote the set of regressive and positive regressive functions
p2h0.75 rd-continuous = right dense continuous (http://campus.mst.edu/adsa/contents/v6n2p1.pdf)
* s ∈ [t_til(0),0], for Eq (3), (p3L0) S ⊆ ℝ interval
* t ≥ t0 , denotes the reset rate of neurons
* t_til(0) = inf{over t ∈ [0,∞): t − τ(t) } often use s ∈ [t_til(0),0] Eq (2)
* t∗ = t∗ + l*T + θ, where θ ∈ [0, T )
* t* = ∈ [0,t1)
* t∗∗ ∈ [tK,t(K+1))
* {tk}_sup(∞)_sub(k=1) is the impulsive time sequence
* tk impulse time, satisfy 0 < t1 < t2 < ... < tk → ∞ as k → ∞
* T is the length of a time sub-interval within t* -> t**
... such that T(1) ≤ t(k+1) − t(k) ≤ T(2) with T(2) ≥ T(1) > 0
p3h0.65 UUB ultimate upper-bound
p3h0.65 UUB = α^(−1)(V(0,x0)*e^λ ) by Theorem 1
* V(t) impulsive differential delay inequality function Eq (11a)
... while not introduced as a Lyapunov functional, as below with V_Lyap(t), that is what is used
p3h0.5 V : ℝ+(𝕋) × ℝ^n → ℝ+ with V (t,0) = 0 for all t ∈ ℝ+(𝕋)
... an rd-continuous differentiable function
p3h0.6(ii) V∆(t,x) ≤ g(t)*V(t,x) for t ∈ ℝ+(𝕋), with ∫[∆t, from 0 to ∞: g(t)*∆t] ≤ λ
* V_Lyap(t) Lyapunov function Eq (29)
* V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_ovr(t) ≠ 0, for Eq (14)
* vk ≥ 1 constant between impulses for Eq (11)
* W(t) == V(t)*ω(t) ≤ Π[k=1 to ι-1: vk*V_til(0) ] Eq (14)
... note "==" is used for U+2250, which doesn't display in this text editor
* X(t) = (ξ(t)_T, η(t)_T)_T solution of (7)
* Y(t) = (ξ_ovr(t)_T, η_ovr(t)_T)_T solution of (7)
* x = (x1,···,xn) T ∈ ℝ^n is the state vector of neurons
p1h0.75 x(t) ∈ ℝ^n is the state of the system,
p1h0.6(1) x∆(t) = f (t,x(t)), t ∈ 𝕋, timescale-type time-varying system
p1h0.8(3a) d[dt: x(t)] = f(t,x(t)), t ∈ ℝ,
p1h0.95(4a) x(k+1) = x(k) + f(k,x(k)), k ∈ ℤ,
* x(i,t) = ξ(i,t) - ξ_ovr(i,t)
* y(i,t) = η(i,t) - η_ovr(i,t)
* x(t) = (x(1,t),x(2,t),...,x(n,t))_T
* y(t) = (y(1,t),y(2,t),...,y(n,t))_T
* z(t) = (z(1,t),z(2,t),...,z(2*n,t))_T = (x(t)_T,y(t)_T] )_T
p3h0.3 = inf{ x_til ∈ 𝕋|x_til > x} For any x ∈ ℝ
Greek symbols sorted in greek alphabet order
(variables for which I used similar Latin symbols instead of Greek are listed under Latin symbols)
* α(i,t) = γ(i)*{a(i,t) - γ(i,t)} - b(i,t) Eq (7b)
p3h0.5 α a K-class function
* β(i,t) = a(i,t) - γ(i,t) Eq (7b)
* γk positive constant
* γ(i,t) -> seems to be an incorrect notation (1st used Eq (4))
* γ(i) - constant term (reviewer substitute for γ(i,t) : 1st used Eq (4))
p2h0.75 Δ derivative; f∆(t) denotes the ∆-derivative of f at t ∈ 𝕋;
* δk constant
* ζk ∈ [−2,0] represent the strength of impulses
* η(t) = { η(1,t),η(2,t),...,η(i,n) }_T
* η(i,t) expression related to d[dt: ξ(i,t)] and ξ(i,t) used to simplify expressions, defined by (4)
* η(i,tk+) = lim[t→tk+0 : η(i,t)], i.e., the solution η(i,t) is right continuous at impulse point tk
* θ ∈ [0,T) (p14L5), and as ε≥0 (p4L37), this implies that θ≥0?
* θ(s) = { φ(s)_T, ψ(s)_T }
* Θ(s) = { φ(1,s),φ(2,s),...,φ(2*n,s) }_T = (φ_cup(s)_T , ψ_cup(s)_T)_T for s ∈ [t_til(0),0] for Eq (10)
* ι ∈ ℤ, eg t ∈ [0,tι)
* λ > 0 constant Eq (8)
p2h0.75 µ denotes the graininess function
* ξ(t) = { ξ(1,t),ξ(2,t),...,ξ(i,n) }_T
* ξ(i,t) denotes the state vectors of the ith neurons at time t,
... solution is right continuous at impulse point tk
... second-order derivative of ξ(i,t) is called an inertial term of system (1)
* ξ(i,tk+) = lim[t->tk+0: ξ(i,t)] = ξ(i,tk),
... i.e., the solution ξ(i,t) is right continuous at impulse point tk
p2h0.9 ξ(h,z) = { log(1+zh)/h for h ≠ 0; z for h = 0 } cylinder transformation definition
* ∆ξ(i,tk) = ξ(i,tk+) − ξ(i,tk-) = δk*x(i,tk) Eq (7c)
* ξ(i,s) = φ(i,s), s ∈ [t_til(0),0], i ∈ N
* η(i,s) = ψ(i,s), s ∈ [t_til(0),0], i ∈ N
* Π(1,t) = max{ max[1≤i≤n: sum[j=1 to n: |c(j,i,t)|*l] + |α(i,t)| - γ(i,t) ], max[1≤i≤n: 1 - β(i,t) ] }
* Π(2,t) = { sum[i=1 to n: max[1≤j≤n: |d(j,i,t)| ] ] }*l
* ρ(t) ≥ 0 continuous function for Eq (11)
p2h0.75 ρ backward jump operator
* ς(t) ≥ 0 continuous function for Eq (11)
p2h0.75 σ denotes the forward jump operator
* τ(t) is the time-varying delay satisfying τ(t) ≤ τ_ovr with τ_ovr > 0,
* τ(t) ∈ PC(R+,R+) time delay satisfying t − τ(t) → ∞ as t → ∞ for Eq (1)
* φ(s) ∈ PC([-τ_ovr,0],ℝ^n) is the initial condition
* Φ(t) = { Φ(1,t),Φ(2,t),...,Φ(i,n) }_T
+--+
initial conditions of inertial NN (1), real-valued continuous functions, s ∈ [t_til(0),0], i ∈ N
* Φ(i,s) = ξ(i,s), s ∈ [t_til(0),0], i ∈ N p3L18
* φ(i,s) = d[ds : ξ(i,s)], s ∈ [t_til(0),0], i ∈ N p3L18
* Φ(s) = { φ(1,s),φ(2,s),...,φ(2*n,s) }_T = (φ_cup(s)_T , ψ_cup(s)_T)_T for s ∈ [t_til(0),0] for Eq (10)
* φ_cup(s) = { φ_cup(1,s),φ_cup(2,s),...,φ_cup(n,s) }_T for Eq (10)
+--+
* ψ(t) = { ψ(1,t), ψ(2,t),...,ψ(i,n) }_T for (7)
* ψ(s) = { Φ_ovr(s)_T, ψ_ovr(s)_T }_T initial conditions for (8)
* ψ_cup(s) = { ψ_cup(s),ψ_cup(s),...,ψ_cup(s) }_T initial conditions for (10)
* ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞, Eqs (11)(12)
* ω_ovr(k) = { max |δk| + 1, |ζk| + 1 } Eq (28)
*******************************************
>>>>> MATH DEFINITIONS
+-----+
Definition 1. System (1) is said to be:
1. Bounded if for any initial value x0 , there exists a positive scalar δ = δ(x0 ) such that kx(t)k ≤ δ for all t ∈ T.
2. Lyapunov stable if for an arbitrary ε > 0, there exists δ(ε, t0 ) such that kx(0)k < δ ⇒ kx(t)k < ε, ∀t ∈ T.
3. Asymptotically stable if it is Lyapunov stable, and there is a constant c = c(x0 ) > 0 such that x(t) → 0 as t → ∞, for all kx0 k < c.
inf (infinum) inf = inf{ x_tild ∈ T| x_tild > x}
*******************************
>>>>> 1 Introduction
In the paper, we are concentrated on the timescale-type time-varying system with its boundedness and asymptotic stability
p1h0.6(1) x∆(t) = f (t,x(t)), t ∈ 𝕋, timescale-type time-varying system
with its initial value
p1h0.7(2) x(0) = x0 , x0 ∈ ℝ^n ,
p1h0.75 x(t) ∈ ℝ^n is the state of the system,
p1h0.75 f : 𝕋 × ℝ^n → ℝ^n is a rd-continuous function satisfying f(t,0) = 0 for all t ∈ 𝕋.
Let 𝕋 = ℝ, system (1)-(2) represents the following ordinary differential system
p1h0.8(3a) d[dt: x(t)] = f(t,x(t)), t ∈ ℝ,
p1h0.8(3b) x(0) = x0 .
Let 𝕋 = ℤ, then system (1)-(2) reduces to the following ordinary difference
system with its initial value
p1h0.95 k ∈ ℤ discrete time index
p1h0.95(4a) x(k+1) = x(k) + f(k,x(k)), k ∈ ℤ,
p1h1.0(4b) x(0) = x0 .
>>>>> 2 Preliminaries
p2h0.33 𝕋 represents a time scale which is an arbitrary nonempty closed subset of the real set ℝ.
p2h0.75 σ denotes the forward jump operator
p2h0.75 ρ backward jump operator
p2h0.75 µ denotes the graininess function
p2h0.75 Crd = Crd(𝕋) = Crd(𝕋,ℝ) denotes the set of rd-continuous functions
p2h0.75 𝓡 and 𝓡+ denote the set of regressive and positive regressive functions;
p2h0.75 fσ(t) = f(σ(t)); f∆(t) denotes the ∆-derivative of f at t ∈ 𝕋; In general, for a function f
p2h0.8 ⊖p = -p(t) /(1+µ(t)p(t)) p ∈ 𝓡+
p2h0.8 p⊖q = (p(t)−q(t))/(1+µ(t)q(t)) p, q ∈ 𝓡+
p2h0.85 e(p,t,s) = exp( ∫[∆τ from s to t: ξ(µ(τ),p(τ)) ] ) for p ∈ 𝓡, s,t ∈ 𝕋^k exponential function definition
p2h0.9 ξ(h,z) = { log(1+z*h)/h for h ≠ 0; z for h = 0 } cylinder transformation definition
Remark 1. Given two special time scales ℝ and h*ℤ(h ≠ 0), and
p3h0.0 c a regressive constant
p3h0.0(a) e(c,t,0) = exp(c*t) for any t ∈ ℝ, 𝕋 = ℝ timescale-type exponential function
p3h0.0(b) e(c,t,0) = (1 + c*h)^(t/h) for any t ∈ h*ℤ, 𝕋 = h*ℤ timescale-type exponential function
Definition 1. System (1) is said to be:
p3h0.1 1. Bounded if for any initial value x0, there exists a positive scalar δ = δ(x0)
... such that ||x(t)|| ≤ δ for all t ∈ 𝕋.
p3h0.15 2. Lyapunov stable if for an arbitrary ε > 0, there exists δ(ε,t0)
... such that ||x(0)|| < δ ⇒ ||x(t)|| < ε, ∀t ∈ 𝕋.
p3h0.15 3. Asymptotically stable if it is Lyapunov stable, and there is a constant c = c(x0) > 0
... such that x(t) → 0 as t → ∞, for all kx0 k < c.
Notations. Throughout this paper,
p3h0.25 ℝ denotes the real number set
p3h0.25 ℝ+ denotes the nonnegative real number set
p3h0.25 ℤ denotes the integer number set
p3h0.25 ||·|| denotes the Euclidean norm.
p3h0.3 = inf{ x_til ∈ 𝕋|x_til > x} For any x ∈ ℝ
p3h0.3 min{𝕋} = 0 Without other statements, we always assume this
>>>>> 3 Boundedness and stability conditions for time-varying systems on time scales
Theorem 1. Consider system (1). Given x0 ≠ 0. Suppose that there exist
p3h0.5 V : ℝ+(𝕋) × ℝ^n → ℝ+ with V (t,0) = 0 for all t ∈ ℝ+(𝕋)
... an rd-continuous differentiable function
p3h0.5 α a K-class function
p3h0.5 g : ℝ+ → ℝ a regressive function
such that
p3h0.55(i) α(||x(t)||) ≤ V(t,x) for t ∈ ℝ+(𝕋)
p3h0.6(ii) V∆(t,x) ≤ g(t)*V(t,x) for t ∈ ℝ+(𝕋), with ∫[∆t, from 0 to ∞: g(t)] ≤ λ
then the solution of system (1) is ultimately bounded above, and the ultimate upper-bound (UUB) is
p3h0.65 UUB ultimate upper-bound
p3h0.65 UUB = α^(−1)(V(0,x0)*e^λ )
Moreover, if λ = −∞, then it is asymptotically stable.
>>>>> Theorem 1 Proof.
From condition (ii), one gets
p3h0.7(5) V(t,x(t)) ≤ V(0,x0)*e(g(t),t,0)
... = V(0,x0)*exp( ∫[∆τ, from 0 to t: ξ(µ(τ),g(τ)) ] )
For any point t on any time scale 𝕋^k , t is either right-dense or right-scattered.
In what follows, we introduce
p3h0.75 𝕋aux ∆= {t(k,2)}|k=1 to ∞ an auxiliary discrete time scale
where
p3h0.75 t(i,2) is the i-th right-scattered point of 𝕋.
Unless 𝕋 is ℝ, then 𝕋aux is non-empty, and
p3h0.8 𝕋 = [0,t(1,2)] ∪ k=1 to ∞ [σ(t(k,2)),t(k+1,2) ].
Therefore,
p3h0.85 exp( ∫[∆τ, from 0 to t : ξ(µ(τ),g(τ)) ] )
= exp( ∫[∆τ, from 0 to t(1,2) : ξ(µ(τ),g(τ)) ]
+ ∫[∆τ, from t(1,2) to σ(t(1,2)) : ξ(µ(τ),g(τ)) ]
+ sum[k=1 to ∞ :
∫[∆τ, from σ(t(k,2)) to t(k+1,2) : ξ(µ(τ),g(τ)) ]
∫[∆τ, from t(k+1,2) to σ(t(k+1,2)): ξ(µ(τ),g(τ)) ]
]
)
>>
<<
For any t(2,k) ∈ 𝕋aux,
p4h0.0 ∫[∆τ, from t(k,2) to σ(t(k,2)) : ξ(µ(τ),g(τ)) ]
= µ(t(k,2))*ξ(µ(t(k,2)),g(t(k,2)))
= Log( 1 + g(t(k,2)*µ(t(k,2)) )
≤ g(t(k,2)*µ(t(k,2))
= ∫[∆τ, from t(k,2) to σ(t(k,2)) : g(τ)) ]
>>
<<
and
p4h0.35 ∫[∆τ, from σ(t(k,2)) to t(k+1,2) : ξ(µ(τ),g(τ)) ]
= ∫[∆τ, from σ(t(k,2)) to t(k+1,2) : g(τ) ]
>>
<<
Then we have
p4h0.5(6) V(t,x(t)) ≤ V(0,x0)*e(g(t),t,0)
= V(0,x0)*exp( ∫[∆τ, from 0 to t : g(τ) ] )
>>
<<
Therefore,
p4h0.55 V(t,x(t)) ≤ V(0,x0)*e^λ
i.e., the solution of system (1) is ultimately bounded above, and
p4h0.55 UUB = α^(−1)( V(0,x0)*e^λ )
For the case of λ = −∞, we perform the proof with two steps.
Step 1: Lyapunov stable. Let
p4h0.6 G(t) = ∫[∆τ, from 0 to t : g(τ) ]
From the definition of g and condition (ii) that G(t) is rd-continuous and G(t) → −∞ as t → +∞, which means that for any M < 0, there exists a T (T ≥ 0, T ∈ 𝕋) such that G(t) ≤ M for all t ≥ T, t ∈ ℝ+(𝕋).
p4h0.65 M < 0 constant
Let
p4h0.65 A = max[t≤T,t ∈ ℝ+(𝕋) : {G(t)} ]
p4h0.65 B = max{M, A},
then for any t ≥ 0, t ∈ ℝ+(𝕋), one has
p4h0.66(7) V(t,x) ≤ V(0,x0)*exp(G(t)) ≤ V(0,x0)*exp(B)
Given an arbitrary ε > 0, it follows from the definition of V and B that there exists a scalar δ = δ(ε)
such that for any ||x0|| ≤ δ
p4h0.8(8) V(0,x0)*exp(B) ≤ α(ε).
Together with (7)-(8) and condition (i), one gets
p4h0.85 ||x(t)|| ≤ ε, t ∈ 𝕋
which implies the Lyapunov stability of system (1).
Step 2: Asymptotical convergence. It can be directly deduced from (6) and condition (ii) that
p4h0.9 V(t,x) → 0, as t → +∞, i.e., x → 0 as t → +∞,
which implies the asymptotical convergence of system (1).
Remark 2. In Theorem 1, the function V(t,x) is not bound to be monotone decreasing, which is different from most of the works before.
Here, we have the following two special cases for systems (3) and (4).
Corollary 1. Consider system (3). Given x0 ≠ 0. Suppose that there exists a continuous differentiable function V : ℝ+ × ℝ^n → ℝ+ with V (t, 0) = 0 for all t ∈ ℝ+ , a K-class function α, and a continuous function g : ℝ+ → ℝ such that
p5h0.33(i) α(kx(t)k) ≤ V (t, x) for t ∈ ℝ+
p5h0.33(ii) V(t,x) ≤ g(t)*V(t,x) with ∫[∆τ, from 0 to ∞ : g(τ) ] ≤ λ, for t ∈ ℝ+
then the solution of system (3) is ultimately bounded above, and
p5h0.4 UUB = α^(−1)( V(0,x0)*e^λ )
Moreover, if λ = −∞, then it is asymptotically stable.
Corollary 2. Consider system (4). Given x0 ≠ 0. Suppose that there exist a function V : ℤ+ × ℝ^n → ℝ+ with V(k,0) = 0 for all k ∈ ℤ+ , a K-class function α, and a function g : ℤ+ → ℝ such that
p5h0.5(i) α(kx(k)k) ≤ V (k, x), for k ∈ ℤ+
p5h0.5(ii) V(k+1,x) ≤ [1 + g(k)]*V(k,x) with sum[0 to ∞ : g(τ) ] ≤ λ, for k ∈ ℤ+
then the solution of system (4) is ultimately bounded above, and the
p5h0.6 UUB = α^(−1)( V(0,x0)*e^λ )
Moreover, if λ = −∞, then it is asymptotically stable.
# enddoc