http://www.BillHowell.ca/Neural nets/Paper reviews/171127 journal paper review - math only.txt
15Nov2017 initial, ddmmmyyyy final
Bill@BillHowell.ca
IMPORTANT! For proper alignment of math expressions, 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
- window width to give >=122 characters
- ensure that long text lines "wrap" to a new line.
I use the text editor "kwrite" in Linux, which is my favourite. There are many, many others! (gedit,I forget many others- especially those geared to programming). One of these days, I need to stop fooling around and start with emacs...
Suggestion : Copy ["Flat-liner" UNICODE nomenclature, Symbols, Summary of reviewer's assumptions] to a separate text file for easy reference while viewing the derivations.
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TABLE OF CONTENTS (select-find to go to the section desired)
>>>>> Objectives of these step-by-step math checks
>>>>> "Flat-liner" UNICODE nomenclature :
>>>>> Symbols
>>>>> Summary of reviewer's assumptions
>>>>> Unfinished checks
>>>>> Summary of [errors, missing terms, variable nomenclature, etc]
>>>>> Background
>>>>> check (5b) :
>>>>> check (9a) :
>>>>> check (9b) :
>>>>> check (9c) :
>>>>> check (9d) :
>>>>> check (10a) :
>>>>> check (10b) :
>>>>> PROOF OF THEOREM 1
>>>>> check (p5L19a) :
>>>>> check (p5L19b) :
>>>>> check (p5L19c) :
>>>>> check (p5L21) :
>>>>> check (16a) :
>>>>> check (16b) :
>>>>> check (16c) :
>>>>> check (17a-d) :
>>>>> check (17e) :
>>>>> check (17f) :
>>>>> PROOF OF THEOREM 2
>>>>> check (30a) :
>>>>> check (30b) :
>>>>> check (30c) :
>>>>> check (30d)
>>>>> check (30e)
>>>>> check (31) :
>>>>> check (32) :
[Quick, short] checks (eg by inspection) are not included in the list above.
Note that Sections "TABLE OF CONTENTS" through "Summary of [errors,...]" are maintained in a separate file for easy reference while doing the work. They are copied into "filepath journal paper review - math only.txt" when finished.
table of contents is generated by :
$ grep ">>>>> " "filepath journal paper review - math only.txt"
copy-paste output as list above
**********************
>>>>> Math Checks [completed, unfinished, not done] and questions
Note that I have read through the entire paper in detail. This section ONLY refers to areas that I have NOT done a step-by=step re-derivation of the authors' results.
I have ONLY checked the derivations for "2 Model formulation and preliminaries", and Theorems 1&2.
I did NOT check :
- derivations for Lemma 1, Theorem 3, Corollaries [1-6]
- numerical results
I did NOT do a detailed analysis of the [consistency, coherence] across the various components of the paper, although the "Symbols" section provides part of that in addition to providing a few comparisons to symbol usage in other papers..
NOTE : Authors' convention for summation of terms - It is now apparent that the summation symbol (capital sigma) applies ONLY to the first term following the symbol. Parenthesis are used to sum over a complex expression of symbols. On several occasions I mistakenly included several other terms. Some of those were back-corrected, but in particular (30) I did not, having to finish this review.
List of [unfinished checks, questions] :
>> ???? check (10b),(R.10b.1) do C&D expressions work? use Lenovo -> qnial -> symbolic math, to show that they do
>> ???? p4L52 Eq (11b) - It would be more clear to put V(tk) = V(tk+) ≤ vk*V(tk-), k ∈ ℤ
>> ???? p5L11 Expr (14) AMBIGUOUS Expression. Parenthesis should be added to clarify (14) >> ???? (p5L19) - what guarantees that ω(t) ∈ K? I didn't attempt to check...
>> ???? (p5L19) - what guarantees that ω(t) ∈ K? I didn't attempt to check...
>> ???? p5L21 "... Suppose that the above assertion for ι = 1 is false ..." Authors need to be clear about WHICH assumption : (14b) or (15)? Reviewer assumes (14b)
>> ???? check (16a) my check seems unconvincing, probably erroneous
>> ???? check (16c) unfinished, D^+[dt: W(t)] ≥ 0 at some point 0 <= t <= t*, but what about at t*?
>> ???? check (16c) I need to show that V(t) ≥ 0 for 0 <= t <= t*
>> ???? check (16a) to (16c) review is not really confident in his derivations for these relations
>> ???? checks (16a) to (16c) - are the inequalities "too" restrictive and [conflicting,incoherent]? This is a general issue for me with the approach
>> ???? check (17e),(R.17e.1) - I'm not fully comfortable with my assumption that "W(t* -τ(t*))/W(t*) <= 1", as that is stated p5L47 AFTER this proof is carried out (yet it still applies). Also - we don't know the signs of the other terms.
>> ???? check (R.17f.1) contradicts the counter-proof constraint (i.e. confirms the authors' result) if W(t*) > 0, but what if W(t*) < 0 or = 0 ??
>> ???? check (22d) ≤ vk*Π[k=1 to K-1: vk ]*V_til(0) OK based partly on (19), but it is to early to invoke this, as (19) is being proved!?
>> ???? check (p6L53),Π(1,t) - is the indexing correct for c(j,i,t)?
>> ???? check (30c),(R.30.16) unfinished
>> ???? check (30d) - unfinished
>> ???? check (32) separate term "sup[s ∈ [t_til(0),0]: V_Lyap(s) ]" essentially a constant with respect to Π[k=1 to ∞: ??
>> ???? check (32) This may have been the authors' original intent, but sometimes their notation is AMBIGUOUS
>> ???? check (32) - I must prove (R.32.1) vk ≤ ω_ovr(k) !!
list of [unfinished checks, questions] is generated by :
$ grep "???" "filepath journal paper review - math only.txt"
copy-paste output as list above
**************************
>>>>> Objectives of these step-by-step math checks
This text document is a record of this reviewer's step-by-step check over parts of the paper. It is too lengthy for a read-through by the authors, but if they want to scrutinize specific parts of my checks, there are provided in detail. My [style,nomenclature] is distracting, but quick and easy to use for a review.
As a reviewer, I find that a step-by-step re-typing of a part of the paper as I have done below forces me to :
- pay attention to details that I might otherwise skim over,
- learn & remember the author's detailed notations.
This is too time intensive to apply to the full paper, but by doing so over part of the authors' work, it gives me far greater confidence in the rest of the paper, which is read, but not analysed step-by-step. It also gives the authors a better idea of the weaknesses of the reviewer!
Reviewer names and the paper title have been removed from the text file, and the obscure posting of it on my zero-traffic website should provide the necessary privacy (I hope). If the authors want me to remove it after they have a chance to look at it, simply email me (my email address is at the top of this file).
*****************************
"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
||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 :
(p5L42) 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"
***************************
Symbols
+-----+
special contexts :
* asterix 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
- 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?
(p3L0) 𝕁 subset of ℝ^n
(p2L58) ℝ, real numbers
(p2L58) ℝ+, non-negative real numbers
(p2L59) ℝ^n n-dimensional Euclidean spaces
(p2L59) ℝ^(n×n) n×n real matrix
(p3L0) 𝕊 subset of ℝ
(p2L58) ℤ positive integers
+--+
Latin symbols sorted in Latin alphabetical order
parameters of inertial NN (1), real-valued continuous functions, i ∈ N
(p3L9) a(i,t) > 0
(p3L9) b(i,t) > 0
(p3L9) c(i,j,t) connection weights without delays
(p3L9) d(i,j,t) connection weights with delays
(p4L26) A(t) = diag[α(1,t),α(2,t),...,α(n,t)]_T
(p4L26) B(t) = diag[β(1,t),β(2,t),...,β(n,t)]_T
(p4L26) C(t) = ( c(i,j,t) )^(n*n)
(p4L26) D(t) = ( d(i,j,t) )^(n*n)
(pp3L0) 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
(p3L0) 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
* f = (f1,···,fn) ∈ C(ℝn,ℝn) is the activation function
(p3L9) f(j,ξ(j,t)) represents the neuron activation function of the jth neuron at time t
(p3L18) |f(j,u) - f(j,v)| ≤ l(j)*|u - v| ∀u,v ∈ R, j ∈ N maximum |"gradient"| sort-of
(p3L11) J(i,t) denotes the neuron of an external input on the ith neuron
(p3L0) J ⊆ ℝ^n interval
* k ∈ N0
(p3L1) K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
* L = diag(l1,···,ln)
(p3L25) L = diag{l,l,...,l}, where l = max{1≤j≤n: l(j) }
(p3L22) l(j) are positive constants
(p3L25) l = max[1≤j≤n: l(j) ]
* M ≥ 0 constant
(p4L5) M > 0 constant Eq (8)
* N = {1,2···},
(p2L60) 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
(p3L1) 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) }
* R(l) I, l ≥ 1, is impulsive matrix
(p4L26) R(t) = diag[γ(1,t),γ(2,t),...,γ(n,t)]_T but γ(1,t) = γ(1t)!?!
(p3L18) s ∈ [t_til(0),0], for Eq (3), (p3L0) S ⊆ ℝ interval
* t ≥ t0 , denotes the reset rate of neurons
(p3L14) 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 )
(p5L21) t* = ∈ [0,t1)
(p6,L0) t∗∗ ∈ [tK,t(K+1))
* {tk}_sup(∞)_sub(k=1) is the impulsive time sequence
(p3L55) 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
(p4L54) 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
(p6L60) V_Lyap(t) Lyapunov function Eq (29)
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_ovr(t) ≠ 0, for Eq (14)
(p4L52) vk ≥ 1 constant between impulses for Eq (11)
(p5L11) 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
(p3L61) X(t) = (ξ(t)_T, η(t)_T)_T solution of (7)
(p4L8) 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
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
(p4L26) x(t) = (x(1,t),x(2,t),...,x(n,t))_T
(p4L26) y(t) = (y(1,t),y(2,t),...,y(n,t))_T
(p4L26) z(t) = (z(1,t),z(2,t),...,z(2*n,t))_T = (x(t)_T,y(t)_T] )_T
Greek symbols sorted in greek alphabet order
(variables for which I used similar Latin symbols instead of Greek are listed under Latin symbols)
(p3L55) α(i,t) = γ(i)*{a(i,t) - γ(i,t)} - b(i,t) Eq (7b)
(p3L55) β(i,t) = a(i,t) - γ(i,t) Eq (7b)
* γk positive constant
(p3L30) γ(i,t) -> seems to be an incorrect notation (1st used Eq (4))
(p3L30) γ(i) - constant term (reviewer substitute for γ(i,t) : 1st used Eq (4))
(p3L59) δk constant
(p3L59) ζk ∈ [−2,0] represent the strength of impulses
(p3L60) η(t) = { η(1,t),η(2,t),...,η(i,n) }_T
(p3L30) η(i,t) expression related to d[dt: ξ(i,t)] and ξ(i,t) used to simplify expressions, defined by (4)
(p4L57) 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?
(p3L61) θ(s) = { φ(s)_T, ψ(s)_T }
(p4L37) 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)
(p5L9) ι ∈ ℤ, eg t ∈ [0,tι)
(p4L5) λ > 0 constant Eq (8)
(p3L60) ξ(t) = { ξ(1,t),ξ(2,t),...,ξ(i,n) }_T
(p3L8) 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)
(p4L56) 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
(p3L51) ∆ξ(i,tk) = ξ(i,tk+) − ξ(i,tk-) = δk*x(i,tk) Eq (7c)
(7e) ξ(i,s) = φ(i,s), s ∈ [t_til(0),0], i ∈ N
(7f) η(i,s) = ψ(i,s), s ∈ [t_til(0),0], i ∈ N
(p6L53) 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) ] }
(p6L53) Π(2,t) = { sum[i=1 to n: max[1≤j≤n: |d(j,i,t)| ] ] }*l
(p4L52) ρ(t) ≥ 0 continuous function for Eq (11)
(p4L52) ς(t) ≥ 0 continuous function for Eq (11)
* τ(t) is the time-varying delay satisfying τ(t) ≤ τ_ovr with τ_ovr > 0,
(p3L12) 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
(p3L60) 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
(3a) Φ(i,s) = ξ(i,s), s ∈ [t_til(0),0], i ∈ N p3L18
(3b) φ(i,s) = d[ds : ξ(i,s)], s ∈ [t_til(0),0], i ∈ N p3L18
(p4L37) 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)
(p4L38) φ_cup(s) = { φ_cup(1,s),φ_cup(2,s),...,φ_cup(n,s) }_T for Eq (10)
+--+
(p3L61) ψ(t) = { ψ(1,t), ψ(2,t),...,ψ(i,n) }_T for (7)
(p4L8) ψ(s) = { Φ_ovr(s)_T, ψ_ovr(s)_T }_T initial conditions for (8)
(p4L38) ψ_cup(s) = { ψ_cup(s),ψ_cup(s),...,ψ_cup(s) }_T initial conditions for (10)
(p4L58) ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞, Eqs (11)(12)
(p6L55) ω_ovr(k) = { max |δk| + 1, |ζk| + 1 } Eq (28)
**********************
>>>>> Summary of reviewer's assumptions
(2) t_til(0) = inf{over t ∈ [0,∞): t − τ(t) }, satifies t − τ(t) → ∞ as t → ∞ (p3L14)
often use s ∈ [t_til(0),0] -> 1st appears in Eq (2)
Here, t(0) is taken to be at the time of the first impulse t(k1), following initial conditions as in Eq (3). For any inter-impulse interval, this interval does not refer to t_til(0) = t_til(tk) = inf{over t ∈ [tk,∞): t − τ(t) }.
Therefore, t_til(0) is the same constant for all inter-pulse intervals.
Time-independent γ(i,t) = γ(i) :
(R.5b.4) γ(i,t) = γ(i), d[dt: γ(i,t)] = d[dt: γ(i)] = 0
This assumption is REQUIRED to make Eq (5b) work
"_ovr" notation : The "over bar" is NOT explicitly defined by the authors
Using the following as a template :
p5L14 V_ovr(t) = sup[s ∈ [t_til(0),0]: V(s)], where V_ovr(t) ≠ 0,
BUT - isn't ξ_ovr(i,t) = constant? So it should be denoted ξ_ovr(i)?
Following the above :
(R.8.1a) ξ_ovr(i,t) = ξ_ovr(i) = sup[s ∈ [t_til(0),0]: ξ(i,s)], where ξ_ovr(i) ≠ 0, d[dt: ξ_ovr(i)] = 0
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
(R.8.2a) φ_ovr(i,t) = φ_ovr(i) = sup[s ∈ [t_til(0),0]: φ(i,s)], where φ_ovr(i) ≠ 0, d[dt: φ_ovr(i)] = 0
(R.8.2a) ψ_ovr(i,t) = ψ_ovr(i) = sup[s ∈ [t_til(0),0]: ψ(i,s)], where ψ_ovr(i) ≠ 0, d[dt: ψ_ovr(i)] = 0
(R.Dini) assumption - Dini derivatives can be used in the same way as conventional derivatives
... [distribution,chain] rules apply
... D^+[dt: x(t)] => d[dt: x(t)] where x(t) is continuous
... ...except at discontinuities of function and first derivative d[dt: x(t)] undefined (eg at impulse points)
***********************
>>>>> Summary of [errors, missing terms, variable nomenclature, etc]
>>** p3L35 Eq (5a) WRONG? Although my result is ALMOST the same as the authors, I have a d[dt: γ(i,t)] term, whereas they have nothing for that.
>>** p3L35 Eq (5a) If I ASSUME that γ(i,t) is a constant γ(i) rather than a function of time as their notation suggests, then the results are the same.
>>** p3L30 Eq (4) : γ(i,t) appears to be a constant γ(i) rather than a function of time!! see Section above "Reviewer's assumptions"
>>** p3L45 Eq (7a) NOTATION ERROR : as with other expressions, should γ(i,t) be γ(i)?
>>** p3L59 QUESTION WHAT exactly is unbounded? must mean infinite delay, not impulse amplitude (see Abstract as stated)
>>** p4L14 Eq (9a) NOTATION ERROR : as with other expressions, should γ(i,t) be γ(i)?
>>** p4L14 Eq (9a) MISSING TERM? my result (R.9a.6) has the extra term "+ η_ovr(i,t)",
>>** p4L14 Eq (9a) shouldn't η_ovr(i,t) be a constant iwith respect to time? -> η_ovr(i)
>>** p4L16 Eq (9b) MISSING TERMS? Although (R.9b.2) is substantially the same as author's (9b), this reviewer has extra terms η_ovr(i),ξ_ovr(i),J(i,t)
>>** p4L39 Eq (10a) MISSING TERMS? Although (R.10a.1) is similar to (10a), I have extra terms "- R(t)*ξ_ovr(i,t)) + η_ovr(i)"
>>** p4L39 Eq (10a) Although R(t) is defined (p4L26) as the TRANSPOSE of diag[γ(1,t),γ(2,t),...,γ(n,t)], in this derivation it seems to me that it should NOT be transposed, even though it doesn't hurt to do so!
>>** p4L39 Eq (10b) MISSING TERMS? Although (R.10b.0) is similar to (10b), I still have the extra terms η_ovr(i),ξ_ovr(i),J(i,t) carried over from (R.9b.2)
>>** p4L52 Eq (11) Reviewer emphasis - This is the key inequality referred to by the authors throughout the text?
>>** check (16a) - note that I use ">=" for (R.16a.2), and NOT ">"!!, which is not strictly correct
>>** p5L39 Ineq (18), p5L44 Ineq (19), p5L50 Ineq (20a),(20b) QUESTION : It's strange seeing t ∈ [0,tK) rather than t ∈ [tk,t(K+1)), as this would seem to make the approach more conservative thatn it has to be?
>>** p6L29 Ineq (26) But is this unnecessarily conservative?
>>** p7L2 Expr (30) NOTE : As the author's expressions were used, the checks below do NOT include the "missing terms" : η_ovr(i),ξ_ovr(i),J(i,t)
>>** (30) 3rd expression (inequality),(R.30.16) HIGHLY CONSERVATIVE! Simply take norm of sign() terms in first & last two expressions of (R.30.14)
>>** check (30e) 5th expression (inequality) - I cannot just replace max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] with Π(1,t) !!
>>** ... AMBIGUITY -instead of expression given, I must assume max[1≤j≤n: (1 - β(i,t)) ]*|y(i,t)|
>>** check (30e),(R.30e.2) -WRONG!
>>** ... I get the term "Π(2,t)*sum[j=1 to n: |x(j,t-τ(t))| ]",
>>** ... whereas the authors have "Π(2,t)*V_Lyap(t-τ(t))"
>>** (33) does NOT require implication via (32), as it is simply a re-arrangmeent of (28)
This list is generated from the math file by :
$ grep ">>\*\*" "filepath journal paper review - math only.txt"
copy-paste output as list below
*****************
>>>>> Background
Consider the following inertial neural network with unbounded time-varying delay, for all i ∈ N
(1) d^2[dt^2 : ξ(i,t)] =
-a(i,t)*d[dt: ξ(i,t)] - b(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
.
(p3L2) {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
(2) t_til(0) = inf{over t ∈ [0,∞): t-τ(t) }, satifies t - τ(t) → ∞ as t → ∞
The initial conditions of inertial neural networks (1) are :
(3a) Φ(i,s) = ξ(i,s), s ∈ [t_til(0),0], i ∈ N p3L18
(3b) φ(i,s) = d[ds : ξ(i,s)], s ∈ [t_til(0),0], i ∈ N p3L18
Assumption 1. f(j,·) satisfies the condition that there exist some positive constants l(j) such that
(p3L23) |f(j,u) - f(j,v)| ≤ l(j)*|u - v| ∀u,v ∈ R, j ∈ N maximum |"gradient"| sort-of
(p3L22) l(j) are positive constants
(p3L25) L = diag{l,l,...,l}, where l = max{1≤j≤n: l(j) }
(p3L25) l = max[1≤j≤n: l(j) ]
By using suitable variable substitution, the second-order system can be transformed into a first-order system.
(4) η(i,t) = d[dt: ξ(i,t)] + γ(i,t)*ξ(i,t), i ∈ N
then system (1) can be rewritten as
(5a) d[dt: ξ(i,t)] = -γ(i,t)*ξ(i,t) + η(i,t)
>> OK by easy inspection of (4)
(5b) d[dt: η(i,t)]
= - { b(i,t) + γ(i,t) *(γ(i,t) - a(i,t) }*ξ(i,t)
- { a(i,t) - γ(i,t) }*η(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t)) ]
+ J(i,t)
>>>>> check (5b) :
Differentiating (4) :
d[dt: η(i,t)] = d[dt: d[dt: ξ(i,t)] + γ(i,t)*ξ(i,t) ]
d[dt: η(i,t)] = d[dt: d[dt: ξ(i,t)]] + d[dt: γ(i,t)*ξ(i,t) ]
Rearranging :
(R.5b.1) d[dt: d[dt: ξ(i,t)] = d[dt: η(i,t)] - d[dt: γ(i,t)*ξ(i,t)]
Look at d[dt: γ(i,t)*ξ(i,t)] = d[dt: γ(i,t)]*ξ(i,t) + γ(i,t)*d[dt: ξ(i,t)]
Subbing from (5a) for d[dt: ξ(i,t)]
(R.5b.2) d[dt: γ(i,t)*ξ(i,t)] = d[dt: γ(i,t)]*ξ(i,t) + γ(i,t)*(-γ(i,t)*ξ(i,t) + η(i,t))
Subbing (R.5b.2) into (R.5b.1)
(R.5b.3) d[dt: d[dt: ξ(i,t)] = d[dt: η(i,t)] - d[dt: γ(i,t)]*ξ(i,t) + γ(i,t)*(-γ(i,t)*ξ(i,t) + η(i,t))
>> Note that I lack any relations for d[dt: γ(i,t)]
Substituting (R.5b.3) into (1)
d[dt: η(i,t)] - d[dt: γ(i,t)]*ξ(i,t) + γ(i,t)*(-γ(i,t)*ξ(i,t) + η(i,t))
= -a(i,t)*d[dt: ξ(i,t)] - b(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
Rearranging
d[dt: η(i,t)]
= + d[dt: γ(i,t)]*ξ(i,t) - γ(i,t)*(-γ(i,t)*ξ(i,t) + η(i,t))
- a(i,t)*d[dt: ξ(i,t)] - b(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
Again sub for d[dt: ξ(i,t)] using (5a)
d[dt: η(i,t)]
= + d[dt: γ(i,t)]*ξ(i,t) + γ(i,t)^2*ξ(i,t) - γ(i,t)*η(i,t)
- a(i,t)*{- γ(i,t)*ξ(i,t) + η(i,t)} - b(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
= + {+ a(i,t)*γ(i,t) - b(i,t) + d[dt: γ(i,t)] + γ(i,t)^2}*ξ(i,t)
+ {- a(i,t) - γ(i,t)}*η(i,t))
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
Putting signs in form of authors
(R.5b.3) d[dt: η(i,t)]
= - {- a(i,t)*γ(i,t) + b(i,t) - d[dt: γ(i,t)] - γ(i,t)^2}*ξ(i,t)
- {+ a(i,t) + γ(i,t)}*η(i,t))
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
>>** p3L35 Eq (5a) WRONG? Although my result is ALMOST the same as the authors, I have a d[dt: γ(i,t)] term, whereas they have nothing for that.
>>** p3L35 Eq (5a) If I ASSUME that γ(i,t) is a constant γ(i) rather than a function of time as their notation suggests, then the results are the same.
>>** p3L30 Eq (4) : γ(i,t) appears to be a constant γ(i) rather than a function of time!! see Section above "Reviewer's assumptions"
Also added to the Section "Summary of reviewer's assumptions" :
(R.5b.4) γ(i,t) = γ(i), d[dt: γ(i,t)] = d[dt: γ(i)] = 0
<<<<< end check.
and the initial conditions of system (1) can be written as, for s ∈ [t_til(0),0], i ∈ N
(6a) ξ(i,s) = φ(i,s)
(6b) η(i,s) = γ(i,s)*φ(i,s) + φ(i,s) = ψ(i,s)
(6c)
>> I didn't check this as seems straightforward
Considering the impulses effects, the impulsive inertial neural networks with unbounded time-varying delay can be obtained in the following form
(7a) d[dt: ξ(i,t)] = -γ(i,t)*ξ(i,t) + η(i,t)
>>** p3L45 Eq (7a) NOTATION ERROR : as with other expressions, should γ(i,t) be γ(i)?
>> same as (5a), should be for limits as approach an impulse, but this is addresed later
(7b) d[dt: η(i,t)]
= - β(i,t)*η(i,t) + α(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
for t != tk
where :
(p3L55a) α(i,t) = γ(i)*{ a(i,t) - γ(i,t) } - b(i,t)
(p3L55b) β(i,t) = a(i,t) - γ(i,t)
>> OK by inspection from (5b)
(7c) ∆ξ(i,tk) = ξ(i,tk+) - ξ(i,tk-) = δk*x(i,tk-)
>> Taken as stated
(7d) ∆η(i,tk) = η(i,tk+) - η(i,tk-) = ζk*y(i,tk-), k ∈ Z
>> Taken as stated
>>** p3L59 QUESTION WHAT exactly is unbounded? must mean infinite delay, not impulse amplitude (see Abstract as stated)
>> OK & YES - I address this elsewhere...
(7e) ξ(i,s) = φ(i,s), s ∈ [t_til(0),0], i ∈ N
(7f) η(i,s) = ψ(i,s), s ∈ [t_til(0),0], i ∈ N
Definition 1 The system (7) is said to be global asymptotically stable, if it is locally stable in the sense of
Lyapunov and is globally attractive. In addition, the system (7) is said to be globally exponential stable if
for any two solutions X(t), Y(t) of system (7) with initial values Θ(s), Ψ(s), s ∈ [̃t_til(0),0], there exist constants M > 0 and λ > 0 such that :
(8) ||X(t) - Y(t)|| ≤ M*e^(-λ*t)*sup[s ∈ [t_til(0),0]: ||Θ(s) - Ψ(s)|| ] , ∀t ≥ 0
>> Also added to the Section "Summary of reviewer's assumptions" :
(R.8.1a) ξ_ovr(i,t) = ξ_ovr(i) = sup[s ∈ [t_til(0),0]: ξ(i,s)], where ξ_ovr(i) ≠ 0, d[dt: ξ_ovr(i)] = 0
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
(R.8.2a) φ_ovr(i,t) = φ_ovr(i) = sup[s ∈ [t_til(0),0]: φ(i,s)], where φ_ovr(i) ≠ 0, d[dt: φ_ovr(i)] = 0
(R.8.2a) ψ_ovr(i,t) = ψ_ovr(i) = sup[s ∈ [t_til(0),0]: ψ(i,s)], where ψ_ovr(i) ≠ 0, d[dt: ψ_ovr(i)] = 0
For the purpose of simplicity, letting
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
the impulsive inertial neural networks with unbounded time-varying delay can be obtained in the following form
(9a) d[dt: x(i,t)] = -γ(i,t)*x(i,t) + y(i,t)
(p4L26) R(t) = diag[γ(1,t),γ(2,t),...,γ(n,t)]_T
>>** p4L14 Eq (9a) NOTATION ERROR : as with other expressions, should γ(i,t) be γ(i)?
>> should be for limits as approach an impulse, but that is in more complete, later expressions
>>>>> check (9a) :
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
(R.9a.0) d[dt: x(i,t)] = d[dt: ξ(i,t)] - d[dt: ξ_ovr(i,t)]
using :
(7a) d[dt: ξ(i,t)] = -γ(i,t)*ξ(i,t) + η(i,t)
(R.9a.1) d[dt: x(i,t)] = -γ(i,t)*ξ(i,t) + η(i,t) - d[dt: ξ_ovr(i,t)]
using
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
(R.9a.2) η(i,t) = y(i,t) + η_ovr(i,t)
(R.8.1a) ξ_ovr(i,t) = ξ_ovr(i) = sup[s ∈ [t_til(0),0]: ξ(i,s)], where ξ_ovr(i) ≠ 0, d[dt: ξ_ovr(i)] = 0
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
(R.5b.4) γ(i,t) = γ(i), d[dt: γ(i,t)] = d[dt: γ(i)] = 0
subbing into (R.9a.1) :
(R.9a.3) d[dt: x(i,t)] = -γ(i)*ξ(i,t) + y(i,t) + η_ovr(i)
>>** p4L14 Eq (9a) MISSING TERM? my result (R.9a.6) has the extra term "+ η_ovr(i,t)",
>>** p4L14 Eq (9a) shouldn't η_ovr(i,t) be a constant iwith respect to time? -> η_ovr(i)
<<<<< end check.
(9b) d[dt: y(i,t)]
= - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
for t ≠ tk
>>>>> check (9b) :
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
(R.9b.0) d[dt: y(i,t)] = d[dt: η(i,t)] - d[dt: η_ovr(i,t)],
(7b) d[dt: η(i,t)]
= - β(i,t)*η(i,t) + α(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
From the Section "Summary of reviewer's assumptions" :
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
sub (7b),(R.8.2a) into (R.9b.0) :
(R.9b.2) d[dt: y(i,t)]
= - β(i,t)*η(i,t) + α(i,t)*ξ(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
using :
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
(R.9a.2) η(i,t) = y(i,t) + η_ovr(i,t)
then :
(R.9b.4) β(i,t)*η(i,t) = β(i,t)*{ y(i,t) + η_ovr(i,t) }
(R.9b.5) α(i,t)*ξ(i,t) = α(i,t)*{ x(i,t) + ξ_ovr(i,t) }
From the Section "Summary of reviewer's assumptions" :
(R.5b.4) γ(i,t) = γ(i), d[dt: γ(i,t)] = d[dt: γ(i)] = 0
(R.8.1a) ξ_ovr(i,t) = ξ_ovr(i) = sup[s ∈ [t_til(0),0]: ξ(i,s)], where ξ_ovr(i) ≠ 0, d[dt: ξ_ovr(i)] = 0
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
then :
(R.9b.7) β(i,t)*η(i,t) = β(i,t)*{ y(i,t) + η_ovr(i) }
(R.9b.8) α(i,t)*ξ(i,t) = α(i,t)*{ x(i,t) + ξ_ovr(i) }
sub (R.9b.7),(R.9b.8) into (R.9b.2) :
(R.9b.2) d[dt: y(i,t)]
= - β(i,t)*{ y(i,t) + η_ovr(i) }
+ α(i,t)*{ x(i,t) + ξ_ovr(i) }
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
for t ≠ tk
>>** p4L16 Eq (9b) MISSING TERMS? Although (R.9b.2) is substantially the same as author's (9b), this reviewer has extra terms η_ovr(i),ξ_ovr(i),J(i,t)
<<<<< end check.
(9c) ∆x(i,tk) = x(i,tk+) - x(i,tk-) = δk*x(i,tk-)
>>>>> check (9c) :
starting with
(7c) ∆ξ(i,tk) = ξ(i,tk+) - ξ(i,tk-) = δk*x(i,tk-)
using
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
(R.9b.6) ξ_ovr(i,t) = ξ_ovr(i), d[dt: ξ_ovr(i,t)] = d[dt: ξ_ovr(i)] = 0
subbing
(R.9c.0) ∆ξ(i,tk)
= ξ(i,tk+) - ξ(i,tk-)
= (x(i,tk+) + ξ_ovr(i)) - (x(i,tk-) + ξ_ovr(i))
= x(i,tk+) - x(i,tk-)
which is denoted as δk*x(i,tk-)
>> OK (R.9c.0) is same as (9c)
<<<<< end check.
(9d) ∆η(i,tk) = η(i,tk+) - η(i,tk-) = ζk*y(i,tk-), k ∈ Z
>>>>> check (9d) :
starting with
(7d) ∆η(i,tk) = η(i,tk+) - η(i,tk-) = ζk*y(i,tk-), k ∈ Z
using
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t)
(R.8.2a) η_ovr(i,t) = η_ovr(i) = sup[s ∈ [t_til(0),0]: η(i,s)], where η_ovr(i) ≠ 0, d[dt: η_ovr(i)] = 0
subbing
(R.9d.0) ∆η(i,tk)
= η(i,tk+) - η(i,tk-)
= (y(i,tk+) + η_ovr(i)) - (y(i,tk-) + η_ovr(i))
= y(i,tk+) - y(i,tk-)
which is denoted as ζk*x(i,tk-)
>> OK (R.9d.0) is same as (9d)
<<<<< end check.
then the system (9) transforms to the following matrix form :
(10a) d[dt: x(t)] = -R(t)*x(t) + y(t)
>>>>> check (10a) :
using :
(R.9a.3) d[dt: x(i,t)] = -γ(i)*ξ(i,t) + y(i,t) + η_ovr(i)
(p4L12a) x(i,t) = ξ(i,t) - ξ_ovr(i,t)
therefore
(R.10a.0) d[dt: x(i,t)] = -γ(i)*(x(i,t) + ξ_ovr(i,t)) + y(i,t) + η_ovr(i)
using
(p4L26) x(t) = (x(1,t)],x(2,t)],...,x(n,t)])_T
(p4L26) y(t) = (y(1,t)],y(2,t)],...,y(n,t)])_T
(p4L26) R(t) = diag[γ(1,t),γ(2,t),...,γ(n,t)]_T
then
d[dt: x(t)] = -R(t)*(x(t) + ξ_ovr(i,t)) + y(i,t) + η_ovr(i)
(R.10a.1) d[dt: x(t)] = -R(t)*x(t) + y(t) - R(t)*ξ_ovr(i,t)) + η_ovr(i)
>>** p4L39 Eq (10a) MISSING TERMS? Although (R.10a.1) is similar to (10a), I have extra terms "- R(t)*ξ_ovr(i,t)) + η_ovr(i)"
>>** p4L39 Eq (10a) Although R(t) is defined (p4L26) as the TRANSPOSE of diag[γ(1,t),γ(2,t),...,γ(n,t)], in this derivation it seems to me that it should NOT be transposed, even though it doesn't hurt to do so!
<<<<< end check.
(10b) d[dt: y(t)] = -B(t)*y(t) + A(t)*x(t) + C(t)*f(x(t)) + D(t)*f(x(t-τ(t))), t tk
>>>>> check (10b) :
using
(R.9b.2) d[dt: y(i,t)]
= - β(i,t)*{ y(i,t) + η_ovr(i) }
+ α(i,t)*{ x(i,t) + ξ_ovr(i) }
+ sum[j=1 to n: c(i,j,t)*f(j,ξ(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,ξ(j,t-τ(t))) ]
+ J(i,t)
(p4L26) A(t) = diag[α(1,t),α(2,t),...,α(n,t)]_T
(p4L26) B(t) = diag[β(1,t),β(2,t),...,β(n,t)]_T
(p4L26) C(t) = ( c(i,j,t) )^(n*n)
(p4L26) D(t) = ( d(i,j,t) )^(n*n)
(p4L12b) y(i,t) = η(i,t) - η_ovr(i,t) (p4L12b)
therefore
(R.10b.0) d[dt: y(i,t)]
= - B(t)*y(i) + B(t)*η_ovr(i)
+ A(t)*x(i) + A(t)*ξ_ovr(i)
+ C(t)*f(x(t))
+ D(t)*f(x(t-τ(t)))
+ J(i,t)
>>** p4L39 Eq (10b) MISSING TERMS? Although (R.10b.0) is similar to (10b), I still have the extra terms η_ovr(i),ξ_ovr(i),J(i,t) carried over from (R.9b.2)
>> ???? check (10b),(R.10b.1) do C&D expressions work? use Lenovo -> qnial -> symbolic math, to show that they do
<<<<< end check.
(10c) ∆x(tk) = δ(k)*x(tk-)
>> This is the same as (9c), so no check is required
(10d) ∆y(tk) = ζ(k)*y(tk-), k ∈ Z
>> This is the same as (9c), so no check is required
The initial condition of (10) is given by :
(p4L37) 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)
(p4L38) φ_cup(s) = { φ_cup(1,s),φ_cup(2,s),...,φ_cup(n,s) }_T
>> Seems straightforward, so I didn't check these.
Lemma 1 For any vectors a, c ∈ ℝ^n , ε > 0 , R is a diagonal positive definite matrix with appropriate dimensional,
then the following inequality holds
(p4L43) 2*a_T*c ≤ ε*a_T*R*a + ε^(-1)*c_T*R^(-1)*c
>> I did not check Lemma 1 (p4L43)
**************************
>>>>> PROOF OF THEOREM 1
>> Reviewer comment : For the proofs below, I simply use the authors "Background expressions", and i do NOT carry forward the extra terms arising from my checks [ξ_ovr(i,t)),η_ovr(i)]. This keeps the checks simple, but I loose the possible carry-through of the extra terms.
Theorem 1 Let continuous functions ρ(t) ≥ 0, ς(t) ≥ 0, constant vk ≥ 1, τ(t) ≥ 0 and t - τ(t) → +∞ as t → +∞, for the following impulsive differential delay inequality
(11a) D^+[dt: V(t)] ≤ -ρ(t)*V(t) + ς(t)*V(t-τ(t)), t ≠ tk, t ≥ 0
(11b) V(tk) ≤ vk*V(tk-), k ∈ ℤ
(R.11b) V(tk) = V(tk+) ≤ vk*V(tk-), k ∈ ℤ
>>** p4L52 Eq (11) Reviewer emphasis - This is the key inequality referred to by the authors throughout the text?
>> ???? p4L52 Eq (11b) - It would be more clear to put V(tk) = V(tk+) ≤ vk*V(tk-), k ∈ ℤ
if there exists a function ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞ such that
(12) ρ(t) > ς(t)*ω(t)/ω(t-τ(t)) + D^+[dt: ω(t)]/ω(t), t ≥ 0
and the condition Π[k=1 to ∞: vk] < ∞ holds, then
(13) V(t) ≤ Π[k=1 to ∞: vk] /ω(t)*sup[s ∈ [t_til(0),0]: V(s)], t ≥ 0
Proof To prove this theorem, we now show that the following claim holds:
For t ∈ [0,tι), ι ∈ ℤ,
(14a) W(t) == V(t)*ω(t)
(14b) [W(t), V(t)*ω(t)] ≤ Π[k=1 to ι-1: vk ]*V_til(0) to be proven
where
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_til(0) ≠ 0
(p5L19) W(0) ≤ V(0) ≤ sup[s ∈ [t_til(0),0]: V(s)] = V_til(0), to be proven
>> ???? p5L11 Expr (14) AMBIGUOUS Expression. Parenthesis should be added to clarify (14)
We take the mathematical induction to prove this claim. When ι = 1, i.e., t ∈ [0,t1), we show that
(15) W(t) ≤ V_til(0) to be proven
In fact, it is easy to find the fact
(p5L19) ω(t) ∈ K
>> ???? (p5L19) - what guarantees that ω(t) ∈ K? I didn't attempt to check...
that
(p5L19a) W(0) ≤ V(0) ω(t) ∈ K
(p5L19b) V(0) ≤ sup[s ∈ [t_til(0),0]: V(s)], ω(t) ∈ K
(p5L19c) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], ω(t) ∈ K
>>>>> check (p5L19a) :
start with, at t = 0
(14a) W(t) == V(t) *ω(t)
using
(p3L2) K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
therefore, as t > 0
(R.p5L19a.0) ω(0) <= 1
therefore
(R.p5L19a.1) W(0) <= V(0)
compare to
(p5L19a) W(0) ≤ V(0) ω(t) ∈ K
>> OK - (R.p5L19a.1) is the same as (p5L19a)
<<<<< end check.
>>>>> check (p5L19b) :
given the definition that this reviewer assumes for V_til(0)
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_til(0) ≠ 0
Then this is simply the value of V at t = 0, as there has been no change (other than first impulse?)
compared to
(p5L19b) V(0) ≤ sup[s ∈ [t_til(0),0]: V(s)] ω(t) ∈ K
>> OK - (p5L19b) is simply the value of V at t = 0, as there has been no change (other than first impulse?)
<<<<< end check.
>>>>> check (p5L19c) :
given the definition that this reviewer assumes for V_til(0)
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_til(0) ≠ 0
but this is the same as
(p5L19c) sup[s ∈ [t_til(0),0]: V(s)] = V_til(0), ω(t) ∈ K
>> OK - (R.p5L19c) is simply a re-statement of the definition V_til(0)
<<<<< end check.
which implies (reviewer - this refers to (p5L19a,b,c) )
(p5L21a) W(0) ≤ V_til(0).
>>>>> check (p5L21) :
Given
(p5L19a) W(0) ≤ V(0) ω(t) ∈ K
(p5L19b) V(0) ≤ sup[s ∈ [t_til(0),0]: V(s)] ω(t) ∈ K
(p5L19c) sup[s ∈ [t_til(0),0]: V(s)] = V_til(0), ω(t) ∈ K
Then (p5L21a) follows simply
>> OK given (p5L19a,b,c) then (p5L21) follows simply
<<<<< end check.
Suppose that the above assertion for ι = 1 is false, then there exists
(p5L21b) t* = ∈ [0,t1)
>> ???? p5L21 "... Suppose that the above assertion for ι = 1 is false ..." Authors need to be clear about WHICH assumption : (14b) or (15)? Reviewer assumes (14b)
such that
(16a) W(t*) = V_til(0) if (15) is false
>>>>> check (16a) :
The assumption to disprove in this counter-proof is
(14b) [W(t), V(t)*ω(t)] <= Π[k=1 to ι-1: vk ]*V_til(0) to be proven
The counter-proof assumed falsehood becomes, for t ∈ [0,t1)
(R.16a.0) [W(t), V(t)*ω(t)] >= Π[k=1 to ι-1: vk ]*V_til(0) assumed for counter-proof
>>** check (16a) - note that I use ">=" for (R.16a.2), and NOT ">"!!, which is not strictly correct
but this occurs for t ∈ [0,t1), and after the first impulse at t=0, k=0 (I start k index at 0, not 1 for clarity). Using
(p3L2) K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
(p4L58) ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞, Eqs (11)(12)
therefore, for tk = t0 = 0 < t* < t1
(R.16a.1) ω(t) >= 1, t ∈ [0,t1)
using
(R.11b) V(tk) = V(tk+) ≤ vk*V(tk-), k ∈ ℤ
for k=0
(R.16a.2) V(0-) = V_til(0)
(R.16a.3) V(0) = V(0+) ≤ v0*V(0-), k ∈ ℤ (again, I start k index at 0, not 1, for clarity)
(R.16a.4) V(0) = V(0+) ≤ v0*V_til(0), k ∈ ℤ
oops, this is the same as previous results, combining (p5L19b),(p5L19c)
(R.16a.5) V(0) ≤ V_til(0), ω(t) ∈ K
even though
(p4L52) vk ≥ 1 constant between impulses for Eq (11b)
therefore vk is not sufficiently large to make V(0) >= V_til(0)
using
(R.16a.6) Π[k=1 to ι-1: vk ] = v0
then subbing (R.16a.6) into (R.16a.0)
(R.16a.7) [W(t), V(t)*ω(t)] >= v0*V_til(0) from assumption for counter-proof, t ∈ [0,t1)
at t = 0
(R.16a.8) [W(0), V(0)*ω(0)] >= v0*V_til(0) from assumption for counter-proof, t ∈ [0,t1)
from
(p5L21a) W(0) ≤ V_til(0).
Reconciling (R.16a.8) & (p5L21a), considering (p4L52)
(R.16a.9) W(0) = V_til(0)
(R.16a.10) v0 = 1 = Π[k=1 to ι-1: vk ] for k=0
Then for t = t*, (R.16a.0) becomes
(R.16a.11) W(t*) >= Π[k=1 to ι-1: vk ]*V_til(0) from assumption for counter-proof
(R.16a.12) W(t*) >= V_til(0) from assumption for counter-proof
but at t = t1-
from
(11a) D^+[dt: V(t)] ≤ -ρ(t)*V(t) + ς(t)*V(t-τ(t)), t ≠ tk, t ≥ 0
(12) ρ(t) > ς(t)*ω(t)/ω(t-τ(t)) + D^+[dt: ω(t)]/ω(t), t ≥ 0
>> ???? check (16a) my check seems unconvincing, probably erroneous
compare to
(16a) W(t*) = V_til(0) if (14b) is false
>> OK - (R.16a.4) is the same as (16a)
<<<<< end check.
(16b) W(s) ≤ V_til(0) for s ∈ [t_til(0),0) if (14b) is false
>>>>> check (16b) :
start with
(14a) W(t) == V(t)*ω(t)
using
(p4L58) ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞, Eqs (11)(12)
(p3L2) K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
therefore
(R.16b.0) ω(s) <= 1 for s ≤ 0
therefore
(14a) W(s) <= V(s)
given the definition that this reviewer assumes for V_til(0) or (p5L19c)
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_til(0) ≠ 0
(p5L19c) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], ω(t) ∈ K
then
(R.16b.1) V(s) <= V_til(0)
and subbing (R.16b.1) into (14a)
(R.16b.2) W(s) <= V_til(0)
compare to
(16b) W(s) ≤ V_til(0) for s ∈ [t_til(0),0) if (14b) is false
>> OK - (R.16b.0) is the same as (16b)
<<<<< end check.
(16c) D^+[dt, t=t*: W(t)] ≥ 0 if (14b) is false
>>>>> check (16c) :
start with
(11a) D^+[dt: V(t)] ≤ -ρ(t)*V(t) + ς(t)*V(t-τ(t)), t ≠ tk, t ≥ 0
using
(14a) W(t) == V(t)*ω(t)
then for t ≠ tk, t ≥ 0
(R.16c.0) D^+[dt: V(t)] ≤ -ρ(t)*W(t)/ω(t) + ς(t)*W(t-τ(t))/ω(t-τ(t))
If
(16a) W(t*) = V_til(0) if (14b) is false
(R.16a.1) ω(t*) >= 1
then given that
(14a) W(t) == V(t) *ω(t)
(R.16c.0) W(t*) = V(t*) *ω(t*)
so
(R.16c.1) W(t*) ≤ V(t*)
such that it must be the case that
(R.16c.2) D^+[dt: W(t)] ≥ 0 at some point 0 <= t <= t*
(16b) W(s) ≤ V_til(0) for s ∈ [t_til(0),0) if (14b) is false
>> ???? check (16c) unfinished, D^+[dt: W(t)] ≥ 0 at some point 0 <= t <= t*, but what about at t*?
>> ???? check (16c) I need to show that V(t) ≥ 0 for 0 <= t <= t*
<<<<< end check.
>> ???? check (16a) to (16c) review is not really confident in his derivations for these relations
>> ???? checks (16a) to (16c) - are the inequalities "too" restrictive and [conflicting,incoherent]? This is a general issue for me with the approach
Moreover, according to condition (12), one obtain that
(17) D^+[t=t*: W(t)]
(17a) = D^+[t=t*: V(t)]*ω(t*) + D^+[t=t*: ω(t)]*V(t*)
(17b) ≤ { - ρ(t*)*V(t*) + ς(t*)*V(t* -τ(t*)) }*ω(t*) + D^+[t=t*: ω(t)]*V(t*)
(17c) = - ρ(t*)*V(t*)*ω(t*) + ς(t*)*V(t* -τ(t*)) *ω(t*) + D^+[t=t*: ω(t)]*V(t*)
(17d) = - ρ(t*)*W(t*) + ς(t*)*W(t* -τ(t*)) *ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]*W(t*) /ω(t*)
(17e) ≤ { - ρ(t*) + ς(t*) *ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
(17f) < 0
which contradicts equation (16).
>>>>> check (17a-d) :
using
(p3L0) D^+(z) ∆= lim[sup[h -> 0+: (f(z+h) - f(z))/h], Dini derivative of z
(11a) D^+[V(t)] ≤ -ρ(t)*V(t) + ς(t)*V(t-τ(t)), t ≠ tk, t ≥ 0
(11b) V(tk) ≤ vk*V(tk-), k ∈ ℤ
(p4L58) ω(t) ∈ K satisfying ω(t) → ∞ as t → ∞, Eqs (11)(12)
(14a) W(t) == V(t)*ω(t)
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where ,
(p5L14) V_til(0) ≠ 0
(p5L21) t* = ∈ [0,t1)
(p5L52) vk ≥ 1 constant for Eq (11)
then
(R.17a-c.0) D^+[t=t*: W(t)]
= D^+[t=t*: V(t) *ω(t)]
= D^+[t=t*: V(t)]*ω(t*) + V(t*)*D^+[t=t*: ω(t)]
≤ { - ρ(t*)*V(t*) + ς(t*)*V(t*- τ(t*)) }*ω(t*) + V(t*)*D^+[t=t*: ω(t)]
= - ρ(t*)*V(t*)*ω(t*) + ς(t*)*V(t* -τ(t*)) *ω(t*) + D^+[t=t*: ω(t)]*V(t*)
>> OK - (R.17a-c.0) sequence of derivations is the same as authors (17a) through (17c)
multiply 1st&3rd terms by 1 == W(t*)/V(t*)/ω(t*)
multiply 2nd term by 1 == W(t* - τ(t*))/V(t* - τ(t*))/ω(t* - τ(t*))
(R.17d.0) D^+[t=t*: W(t)]
= - ρ(t*)*W(t*) + ς(t*)*W(t* - τ(t*))*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]*W(t*)/ω(t*)
>> OK - (R.17d.0) is the same as (17d)
<<<<< end check.
>>>>> check (17e) :
starting with (17d), pull out W(t*) as factor
(R.17e.0) D^+[dt, t=t*: W(t)]
= { - ρ(t*) + ς(t*)*W(t* -τ(t*))/W(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)] /ω(t*) }*W(t*)
>> p5L32 Ineq (17e) is MISSING a TERM! I have the ratio W(t* -τ(t*))/W(t*) which does NOT appear in (17e).
>> However, I still have an equality, whereas (17e) is an inequality, implying
(R.17e.1) ς(t*)*W(t* -τ(t*))/W(t*)*ω(t*)/ω(t* - τ(t*))*W(t*)
<= ς(t*) *ω(t*)/ω(t* - τ(t*))*W(t*)
thereby implying
(R.17e.2) W(t* -τ(t*))/W(t*) <= 1
p5L47 states that Π[k=1 to ι-1: vk ] "is monotonous nondecreasing on K ∈ Z", then so is W(t)
>> ???? check (17e),(R.17e.1) - I'm not fully comfortable with my assumption that "W(t* -τ(t*))/W(t*) <= 1", as that is stated p5L47 AFTER this proof is carried out (yet it still applies). Also - we don't know the signs of the other terms.
now (R.17e.0) becomes
(R.17e.3) D^+[dt, t=t*: W(t)] ≤ { - ρ(t*) + ς(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
compare to
(17e) D^+[dt, t=t*: W(t)] ≤ { - ρ(t*) + ς(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
>> OK - (R.17e.3) is the same as the authors' (17e)
<<<<< end check.
(17f) D^+[t=t*: W(t)] < 0
>>>>> check (17f) :
Proceding, using authors' p5L32 (17e)
(17e) D^+[dt, t=t*: W(t)]
≤ { - ρ(t*) + ς(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
using
(12) ρ(t) > ς(t)*ω(t)/ω(t-τ(t)) + D^+[dt: ω(t)]/ω(t), t ≥ 0
then
(R.17f.0) 0 > - ρ(t*) + ς(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*)
for W(t*) > 0
(R.17f.1) D^+[dt, t=t*: W(t)] < 0
for W(t*) < 0
(R.17f.2) 0 < D^+[dt, t=t*: W(t)] < { - ρ(t*) + ς(t*)*ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
BUT, (R.17f.1) contrdicts the counter-proof constraint :
(16c) D^+[t=t*: W(t)] ≥ 0 if (14b) is false
>> ???? check (R.17f.1) contradicts the counter-proof constraint (i.e. confirms the authors' result) if W(t*) > 0, but what if W(t*) < 0 or = 0 ??
>> OK - (R.17f.1) contrdicts the counter-proof constraint, matching the authors' result
<<<<< end check.
Now, we assume that the claim holds for all ι ≤ K , K ∈ ℤ, i.e.,
(18) W(t) ≤ Π[k=1 to K-1: vk*V_til(0)], t ∈ [0,tK)
>>** p5L39 Ineq (18), p5L44 Ineq (19), p5L50 Ineq (20a),(20b) QUESTION : It's strange seeing t ∈ [0,tK) rather than t ∈ [tk,t(K+1)), as this would seem to make the approach more conservative thatn it has to be?
Next, we need to prove that
(19) W(t) ≤ Π[k=1 to K: vk*V_til(0)], t ∈ [0,t(K+1))
Since
(p4L52) vk ≥ 1,
noting that
(p5L47b) Π[k=1 to K: vk] is monotonous nondecreasing on K ∈ Z,
>> OK - (19) obvious given (p4L52),(p5L47b)
then
(20a) W(t) ≤ Π[k=1 to K-1: vk*V_til(0)], t ∈ [0,tK)
implies that
(20b) W(t) ≤ Π[k=1 to K : vk*V_til(0)], t ∈ [0,tK)
>> OK - this is also obvious, as an extra term is involved, vk >=1
We only need to prove that
(21) W(t) ≤ Π[k=1 to K : vk*V_til(0)], t ∈ [tk,t(K+1))
Note that, when t = tK, ι ≤ K, K ∈ ℤ
(22a) W(tK) = V(tK) *ω(tK)
>> OK from
(14a) W(t) == V(t) *ω(t)
continuing
(22b) W(t) ≤ vK*V(tK-)*ω(tK)
>> OK from (11b),(14a) - same as authors' (22b)
(11b) V(tk) ≤ vk*V(tk-), k ∈ ℤ
continuing
(22c) vK*V(tK-)*ω(tK) ≤ vK*W(tK-)
>> OK from (22b), using (14a) again, and note that :
(14a) W(t) == V(t)*ω(t)
Since
(p4L52) vk ≥ 1,
noting that
(p5L47b) Π[k=1 to K: vk] is monotonous nondecreasing on K ∈ Z,
then
(22d) vK*W(tK-) ≤ vK*Π[k=1 to K-1: vk ]*V_til(0)
>> OK - (22d) makes sense given (19),(p4L52),(p5L47b)
>> ???? check (22d) ≤ vk*Π[k=1 to K-1: vk ]*V_til(0) OK based partly on (19), but it is to early to invoke this, as (19) is being proved!?
(19) W(t) ≤ Π[k=1 to K: vk*V_til(0)], t ∈ [0,t(K+1))
(22e) vK*Π[k=1 to K-1: vk ]*V_til(0) ≤ Π[k=1 to K-1: vk ]*V_til(0)
>> OK - (22e) is a straightforward rearrangement of product series in (22d)
We claim that (21) holds, otherwise, we can choose some t** ∈ [tK,t(K+1) ), such that
(23a) W(t**) ≤ Π[k=1 to K: vk*V_til(0)]
(23b) W(s) ≤ Π[k=1 to K: vk*V_til(0)] s ∈ [t_til(0),t**)
(23c) D^+[t=t*: W(t)] ≥ 0
Moreover, according to condition (12), one obtains
(24) D^+[t=t**: W(t)]
= D^+[t=t**: V(t)]*ω(t^*) + D^+[t=t**: ω(t)]*V(t**)
≤ { -ρ(t**)*V(t**) + ς(t**)*V(t** -τ(t**)) }*ω(t**) + D^+[t=t**: ω(t)]*V(t**)
= -ρ(t**)*V(t**)*ω(t**) + ς(t**)*V(t** -τ(t**)) *ω(t**) + D^+[t=t**: ω(t)]*V(t**)
= -ρ(t**)*W(t**) + ς(t**)*W(t** -τ(t**)) *ω(t**)/ω(t** - τ(t**)) + D^+[t=t**: ω(t)]*W(t**) /ω(t**)
≤ { -ρ(t**) + ς(t**)*ω(t**)/ω(t** - τ(t**)) + D^+[t=t**: ω(t)]/ω(t**) }*W(t**)
< 0
>> OK By comparison, this is the same derivation as for (17), only with t** rather than t* :
(17) D^+[t=t*: W(t)]
= D^+[t=t*: V(t)]*ω(t*) + D^+[t=t*: ω(t)]*V(t*)
≤ { - ρ(t*)*V(t*) + ς(t*)*V(t* -τ(t*)) }*ω(t*) + D^+[t=t*: ω(t)]*V(t*)
= - ρ(t*)*V(t*)*ω(t*) + ς(t*)*V(t* -τ(t*)) *ω(t*) + D^+[t=t*: ω(t)]*V(t*)
= - ρ(t*)*W(t*) + ς(t*)*W(t* -τ(t*)) *ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]*W(t*) /ω(t*)
≤ { - ρ(t*) + ς(t*) *ω(t*)/ω(t* - τ(t*)) + D^+[t=t*: ω(t)]/ω(t*) }*W(t*)
< 0
which contradicts equation (23). Hence, by mathematical induction, we can obtain that the following inequality holds:
(25) V(t) ≤ Π[k=1 to ι-1: vk*V_til(0)]/ω(t), t ∈ [0,tι), ι ∈ ℤ
>> OK follows from sustituting W(t) = V(t)*ω(t) from (14a)
which implies
(26) V(t) ≤ Π[k=1 to ∞: vk*sup[s ∈ [t_til(0),t**): V(s)]]/ω(t), t ≥ 0
>> OK follows from definition of V_til(0) :
(p5L14) V_til(0) = sup[s ∈ [t_til(0),0]: V(s)], where V_ovr(t) ≠ 0, for Eq (14)
>> plus as vk >= 1, extra terms to k = ∞ also make Right Hand Side larger than in (25)
>>** p6L29 Ineq (26) But is this unnecessarily conservative?
The proof is completed.
Remark 1 Note that the function ω(t) is to ensure that the relationship between unbounded time-varying
delay τ(t) and system parameters ρ(t) and ς(t), which is important to overcome the effect of unbounded
time-varying delay. In other words, the developed approach takes fully account of the relationship between
system parameters, unbounded time-varying delay and impulses, which can be seen as one of the novelties
of our work. In addition, new analysis techniques can effectively avoid the difficulties caused by unbounded
time-varying, and we obtain the condition which are simple and easy to verify by using algebraic method.
**************************
>>>>> PROOF OF THEOREM 2
Theorem 2 Suppose the Assumption 1 holds. If there exist function γi(t) > 0 , i ∈ N , function ω(t) ∈ K
satisfying ω(t) → ∞ as t → ∞ such that
(27) Π(1,t) > Π(2,t)*ω(t)/ω(t - τ(t)) + D^+[ω(t)]/ω(t), t ≥ 0
and the condition Π[k=1 to ∞: vk] < ∞ holds. Then, system (7) is globally asymptotically stable, i.e.,
(28) ||z(t)||_1 ≤ Π[k=1 to ∞: ω_ovr(k)]/ω(t)*sup[s ∈ [t_til(0),0]: ||Φ(s)||_1], t > 0
where
(p6L53) Π(1,t) = max{ max[1≤i≤n: sum[j=1 to n: |c(j,i,t)|*l + |α(i,t)| - γi(t) ],
max[1≤i≤n: (1 - βi(t))]
} for Eq (28)
>> ???? check (p6L53),Π(1,t) - is the indexing correct for c(j,i,t)?
(p6L53) Π(2,t) = { sum[i=1 to n: max[1≤j≤n: |d(j,i,t)| ] ] }*l, for Eq (28)
(p6L55) ω_ovr(k) = { max |δk| + 1, |ζk| + 1 } for Eq (28)
>> Reviwer note : I make the Lyapunov nature explicity by using V_Lyap in this particular section, which is a departure from normal practice and notation elsewhere in this review. But this helps me focus after breaks in working on this review.
Proof Consider the following Lyapunov function:
(29) V_Lyap(t) = sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ]
Then take the derivative of V_Lyap(t) along the trajectories of the system (9) can be obtained as follow, for t ≠ tk,
(30) D^+[dt: V_Lyap(t)]
(30a) = sum[i=1 to n: D^+[dt: |x(i,t)|] ] + sum[i=1 to n: D^+[dt: |y(i,t)|] ]
(30b) = + sum[i=1 to n: sign(x(i,t))*{ - γ(i,t)*x(i,t) + y(i,t) }
+ sum[i=1 to n: sign(y(i,t))*{ + α(i,t)*x(i,t) - β(i,t)*y(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t )) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}
]
(30c) ≤ + sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*l + |α(i,t)| - γ(i,t) ]*|x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*l *|x(j,t-τ(t))] ]
+ sum[i=1 to n: (1 - β(i,t))*|y(i,t)| ]
(30d) ≤ + sum[i=1 to n: max[1≤i≤n: sum[j=1 to n: |c(i,j,t)| *l + |α(i,t)| - γ(i,t) ] ]*|x(i,t)| ]
+ sum[i=1 to n: max[1≤j≤n: |d(i,j,t)| ]*l *sum[j=1 to n: |x(j,t-τ(t)) ] ]
+ sum[i=1 to n: max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] ]
(30e) ≤ Π(1,t)*V_Lyap(t) + Π(2,t)*V_Lyap(t-τ(t))
>>>>> check (30a) :
>>** p7L2 Expr (30) NOTE : As the author's expressions were used, the checks below do NOT include the "missing terms" : η_ovr(i),ξ_ovr(i),J(i,t)
starting with
(29) V_Lyap(t) = sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ]
(R.30.0) D^+[dt: V_Lyap(t)]
= D^+[dt: { sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ] }
= D^+[dt: sum[i=1 to n: |x(i,t)| ] ] + D^+[dt: sum[i=1 to n: |y(i,t)| ] ]
= sum[i=1 to n: D^+[dt: |x(i,t)| ] ] + sum[i=1 to n: D^+[dt: |y(i,t)| ] ]
compare to
(30a) = sum[i=1 to n: D^+[dt: |x(i,t)|] ] + sum[i=1 to n: D^+[dt: |y(i,t)|] ]
>> OK - same as authors' (30a)
<<<<< end check.
>>>>> check (30b) :
+--+
taking the expression for a conventional derivative of an absolute function :
https://proofwiki.org/wiki/Derivative_of_Absolute_Value_Function
Let |x| be the absolute value of x for real x.
Then: d(dx : |x|) = x/|x| for x≠0.
At x=0, |x| is not differentiable.
http://www.eecs.berkeley.edu/~wkahan/MathH110/NormOvrv.pdf
Kahan's formulation
p21h0.2
d||z|| = u_T dotProd dz / ||z||
where u_T is the linear functional dual to z wrt ||...||
z,u are vectors, and p16h0.7
||x||2 = (sum(|xi|^2))^(1/2)
or Kreyszig 1972 p200 Eqn (2)
|x| = (x dotProd x)^(1/2)
+--+
starting with first term of (R.30.0) : D^+[dt: |x(i,t)| ], applying chain rule : f(g(x)) is f'(g(x))⋅g'(x)
(R.30.1) D^+[dt: |x(i,t)| ] = D^+[dx: |x(i,t)| ] * D^+[dt: x(i,t) ]
where, from "conventional derivative of an absolute function" above
(R.30.2) D^+[dx: |x(i,t)| ] = x(i,t) / |x(i,t)| = sign(x(i,t))
NOTE : CAUTION : component-by-component derivatives are NOT the same as full-vector derivative!!
looking at
(R.30.3) D^+[dx: |x(i,t)| ] = sign(x(i,t))
using
(9a) d[dt: x(i,t)] = -γ(i,t)*x(i,t) + y(i,t)
and making the assumption (see "Summary of reviewer's assumptions" section above)
(R.Dini) assumption - Dini derivatives can be used in the same way as conventional derivatives
... [distribution,chain] rules apply
... D^+[dt: x(t)] = d[dt: x(t)] where x(t) is continuous
... ...except at discontinuities of function and first derivative d[dt: x(t)] undefined (eg at impulse points)
subbing (R.30.3),(9a) into (R.30.1)
(R.30.5) D^+[dt: |x(i,t)| ]
= D^+[dx: |x(i,t)| ] * D^+[dt: x(i,t) ]
= sign(x(i,t)) * { -γ(i,t)*x(i,t) + y(i,t) }
+--+
now looking at second term of (R.30.0) : D^+[dt: |y(i,t)| ]
(R.30.6) D^+[dt: |y(i,t)| ]
= D^+[dy: |y(i,t)| ] * D^+[dt: y(i,t) ]
where, from "conventional derivative of an absolute function" above
(R.30.7) D^+[dy: |y(i,t)| ] = sign(y(i,t))
using (9b) - which does NOT carry the "Missing terms"
(9b) d[dt: y(i,t)]
= - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
and making the assumption (see "Summary of reviewer's assumptions" section above)
(R.Dini) assumption - Dini derivatives can be used in the same way as conventional derivatives
... [distribution,chain] rules apply
... D^+[dt: x(t)] = d[dt: x(t)] where x(t) is continuous
... ...except at discontinuities of function and first derivative d[dt: x(t)] undefined (eg at impulse points)
(R.30.8) D^+[dt: y(i,t)]
= - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
subbing (R.30.7),(R.30.9) into (R.30.6)
(R.30.6) D^+[dt: |y(i,t)| ] = D^+[dy: |y(i,t)| ] * D^+[dt: y(i,t) ]
(R.30.9) D^+[dt: |y(i,t)| ]
= sign(y(i,t))*
{ - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}
+--+
using
(R.30.0) D^+[dt: V_Lyap(t)] = sum[i=1 to n: D^+[dt: |x(i,t)| ] ] + sum[i=1 to n: D^+[dt: |y(i,t)| ] ]
(R.30.5) D^+[dt: |x(i,t)| ] = sign(x(i,t))*{ -γ(i,t)*x(i,t) + y(i,t) }
(R.30.9) D^+[dt: |y(i,t)| ] = sign(y(i,t))*
{ - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}(30) D^+[dt: V_Lyap(t)]
(30a) = sum[i=1 to n: D^+[dt: |x(i,t)|] ] + sum[i=1 to n: D^+[dt: |y(i,t)|] ]
(30b) = + sum[i=1 to n: sign(x(i,t))*{ - γ(i,t)*x(i,t) + y(i,t) }
+ sum[i=1 to n: sign(y(i,t))*{ + α(i,t)*x(i,t) - β(i,t)*y(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t )) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}
]
(30c) ≤ + sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*l + |α(i,t)| - γ(i,t) ]*|x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*l *|x(j,t-τ(t))] ]
+ sum[i=1 to n: (1 - β(i,t))*|y(i,t)| ]
(30d) ≤ + sum[i=1 to n: max[1≤i≤n: sum[j=1 to n: |c(i,j,t)| *l + |α(i,t)| - γ(i,t) ] ]*|x(i,t)| ]
+ sum[i=1 to n: max[1≤j≤n: |d(i,j,t)| ]*l *sum[j=1 to n: |x(j,t-τ(t)) ] ]
+ sum[i=1 to n: max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] ]
(30e) ≤ Π(1,t)*V_Lyap(t) + Π(2,t)*V_Lyap(t-τ(t))
subbing (R.30.5),(R.30.9) into (R.30.0)
(R.30.10) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: sign(x(i,t))*{ -γ(i,t)*x(i,t) + y(i,t) } ]
+ sum[i=1 to n: sign(y(i,t))*
{ - β(i,t)*y(i,t) + α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}
= + sum[i=1 to n: sign(x(i,t))*(-1)*γ(i,t)*x(i,t) ] + sum[i=1 to n: sign(x(i,t))*y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*(-1)*β(i,t)*y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*α(i,t)*x(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
= + sum[i=1 to n: sign(x(i,t))*(-1)*γ(i,t)*x(i,t) ]
+ sum[i=1 to n: sign(x(i,t))*y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*(-1)*β(i,t)*y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*α(i,t)*x(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
collect terms of sign(x(i,t)),sign(y(i,t))
(R.30.11) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: sign(x(i,t))*{ - γ(i,t)*x(i,t) + y(i,t) } ]
+ sum[i=1 to n: sign(y(i,t))*
{ - β(i,t)*y(i,t)
+ α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
}
]
compare to
(30b) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: sign(x(i,t))*{ - γ(i,t)*x(i,t) + y(i,t) }
+ sum[i=1 to n: sign(y(i,t))*
{ + α(i,t)*x(i,t)
- β(i,t)*y(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t )) ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ]
}
]
>> OK - (R.30.11) this is the same as (30b)
<<<<< end check.
>>>>> check (30c) :
starting with
(R.30.11) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: sign(x(i,t))*{ - γ(i,t)*x(i,t) + y(i,t) } ]
+ sum[i=1 to n: sign(y(i,t))*
{ - β(i,t)*y(i,t)
+ α(i,t)*x(i,t)
+ sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
}
= + sum[i=1 to n: sign(x(i,t))*-γ(i,t)*x(i,t) ]
+ sum[i=1 to n: sign(x(i,t))* y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*-β(i,t)*y(i,t) ]
+ sum[i=1 to n: sign(y(i,t))* α(i,t)*x(i,t) ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
I want to substitute for sign(x), sign(y)
(R.30.12) x(i,t) = sign(x(i,t))*|x(i,t)|
(R.30.13) y(i,t) = sign(y(i,t))*|y(i,t)|
Subbing (R.30.12),(R.30.13) into (R.30.11) :
(R.30.14) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: sign(x(i,t))*-γ(i,t)*sign(x(i,t))*|x(i,t)| ]
+ sum[i=1 to n: sign(x(i,t)) *sign(y(i,t))*|y(i,t)| ]
+ sum[i=1 to n: sign(y(i,t))*-β(i,t)*sign(y(i,t))*|y(i,t)| ]
+ sum[i=1 to n: sign(y(i,t))* α(i,t)*sign(x(i,t))*|x(i,t)| ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
knowing sign^2(x) = 1
(R.30.15) D^+[dt: V_Lyap(t)]
= + sum[i=1 to n: (-1)*γ(i,t)*|x(i,t)| ]
+ sum[i=1 to n: |y(i,t)| ]
+ sum[i=1 to n: (-1)*β(i,t)*|y(i,t)| ]
+ sum[i=1 to n: sign(y(i,t)) *α(i,t)*sign(x(i,t))*|x(i,t)| ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
= + sum[i=1 to n: { sign(y(i,t))*α(i,t)*sign(x(i,t)) - γ(i,t) }*|x(i,t)| ]
+ sum[i=1 to n: { 1 - β(i,t) }*|y(i,t)| ]
+ sum[i=1 to n: |x(i,t)| ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: c(i,j,t)*f(j,x(j,t)) ] ]
+ sum[i=1 to n: sign(y(i,t))*sum[j=1 to n: d(i,j,t)*f(j,x(j,t-τ(t))) ] ]
I want to substitute for f(j,x(j,t)), f(j,x(j,t-τ(t)))
>>** (30) 3rd expression (inequality),(R.30.16) HIGHLY CONSERVATIVE! Simply take norm of sign() terms in first & last two expressions of (R.30.14)
(p3L2) K = {f ∈ C+(ℝ,ℝ+) | f(t) ≤ 1, if t ≤ 0; f(t) ≥ 1, if t > 0 }
(R.30.16) D^+[dt: V_Lyap(t)]
≤ + sum[i=1 to n: { |α(i,t)| - γ(i,t) }*|x(i,t)| ]
+ sum[i=1 to n: { 1 - β(i,t) }*|y(i,t)| ]
+ sum[i=1 to n: |x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*|f(j,x(j,t))| ] ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*|f(j,x(j,t-τ(t)))| ] ]
use definition of l
(p3L22) l(j) are positive constants
(p3L25) l = max[1≤j≤n: l(j) ]
(p3L18) |f(j,u) - f(j,v)| ≤ l(j)*|u - v| ∀u,v ∈ R, j ∈ N maximum |"gradient"| sort-of
(R.30.17) Applying (p3L18) to t=0
(R.30.17) |f(j,x(j,t)) - f(j,x(j,0))| ≤ l*|t - 0|
(R.30.18) |f(j,x(j,t-τ(t)))| ≤ l
>> ???? check (30c),(R.30.16) unfinished
compare to
(30c) ≤ + sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*l + |α(i,t)| - γ(i,t) ]*|x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*l *|x(j,t-τ(t))] ]
+ sum[i=1 to n: (1 - β(i,t))*|y(i,t)| ]
<<<<< end check.
>>>>> check (30d)
starting with (30c)
(30) D^+[dt: V_Lyap(t)]
(30c) ≤ + sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*l + |α(i,t)| - γ(i,t) ]*|x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*l *|x(j,t-τ(t))] ]
+ sum[i=1 to n: (1 - β(i,t))*|y(i,t)| ]
using
subbing
(R.30c.0) D^+[dt: V_Lyap(t)]
(30c) ≤ + sum[i=1 to n: sum[j=1 to n: |c(i,j,t)|*l + |α(i,t)| - γ(i,t) ]*|x(i,t)| ]
+ sum[i=1 to n: sum[j=1 to n: |d(i,j,t)|*l *|x(j,t-τ(t))] ]
+ sum[i=1 to n: (1 - β(i,t))*|y(i,t)| ]
(30d) ≤ + sum[i=1 to n: max[1≤i≤n: sum[j=1 to n: |c(i,j,t)| *l + |α(i,t)| - γ(i,t) ] ]*|x(i,t)| ]
+ sum[i=1 to n: max[1≤j≤n: |d(i,j,t)| ]*l *sum[j=1 to n: |x(j,t-τ(t)) ] ]
+ sum[i=1 to n: max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] ]
(R.32.0) V_Lyap(t) ≤ Π[k=1 to ∞: vk] *sup[s ∈ [t_til(0),0]: V_Lyap(s) ] ]/ω(t), t ≥ 0
using
(11a) D^+[V(t)] ≤ -ρ(t)*V(t) + ς(t)*V(t-τ(t)), t ≠ tk, t ≥ 0
(11b) V(tk) ≤ vk*V(tk-), k ∈ ℤ
(13) V(t) ≤ Π[k=1 to ∞: vk]/ω(t)*sup[s ∈ [t_til(0),0]: V(s)], t ≥ 0
(30e) D^+[dt: V_Lyap(t)] ≤ Π(1,t)*V_Lyap(t) + Π(2,t)*V_Lyap(t-τ(t))
(31c) V_Lyap(tk) ≤ ω_ovr(k)*V_Lyap(tk-)
>> ???? check (30d) - unfinished
compare to
(30d) ≤ + sum[i=1 to n: max[1≤i≤n: sum[j=1 to n: |c(i,j,t)| *l + |α(i,t)| - γ(i,t) ] ]*|x(i,t)| ]
+ sum[i=1 to n: max[1≤j≤n: |d(i,j,t)| ]*l *sum[j=1 to n: |x(j,t-τ(t)) ] ]
+ sum[i=1 to n: max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] ]
<<<<< end check.
>>>>> check (30e)
starting with (30d)
(30) D^+[dt: V_Lyap(t)]
(30d) ≤ + sum[i=1 to n: max[1≤i≤n: sum[j=1 to n: |c(i,j,t)| *l + |α(i,t)| - γ(i,t) ] ]*|x(i,t)| ]
+ sum[i=1 to n: max[1≤j≤n: |d(i,j,t)| ]*l *sum[j=1 to n: |x(j,t-τ(t)) ] ]
+ sum[i=1 to n: max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] ]
using
(p6L53) 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) ] }
(p6L53) Π(2,t) = { sum[i=1 to n: max[1≤j≤n: |d(j,i,t)| ] ] }*l
subbing into (30d)
>>** check (30e) 5th expression (inequality) - I cannot just replace max[1≤j≤n: (1 - β(i,t)) *|y(i,t)| ] with Π(1,t) !!
>>** ... AMBIGUITY -instead of expression given, I must assume max[1≤j≤n: (1 - β(i,t)) ]*|y(i,t)|
(R.30e.0) D^+[dt: V_Lyap(t)]
≤ + sum[i=1 to n: Π(1,t) ]*|x(i,t)| ]
+ Π(2,t)*sum[j=1 to n: |x(j,t-τ(t))| ]
+ sum[i=1 to n: Π(1,t) ]*|y(i,t)| ]
collecting terms, Π(1,t),Π(2,t) can come out of the summation expression
(R.30e.1) D^+[dt: V_Lyap(t)]
≤ + Π(1,t)*{ sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ] }
+ Π(2,t)* sum[j=1 to n: |x(j,t-τ(t))| ]
subbing
(29) V_Lyap(t) = sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ]
into (R.30e.1) yeilds
(R.30e.2) D^+[dt: V_Lyap(t)] ≤ Π(1,t)*V_Lyap(t) + Π(2,t)*sum[j=1 to n: |x(j,t-τ(t)) ] ]
compare to
(30e) D^+[dt: V_Lyap(t)] ≤ Π(1,t)*V_Lyap(t) + Π(2,t)*V_Lyap(t-τ(t))
>>** check (30e),(R.30e.2) -something is WRONG?
>>** ... I get the term "Π(2,t)*sum[j=1 to n: |x(j,t-τ(t))| ]",
>>** ... whereas the authors have "Π(2,t)*V_Lyap(t-τ(t))"
>>** ... did I drop a term, or was it dropped from the paper?
<<<<< end check.
Next, considering when t = tk , we can obtain
(31) V_Lyap(tk)
(31a) = sum[i=1 to n: |x(i,tk)| ] + sum[i=1 to n: |y(i,tk)| ]
(31b) = sum[i=1 to n: (|δk| + 1)*|x(i,tk-)| ] + sum[i=1 to n: (|δk| + 1)*|y(i,tk-)| ]
(31c) ≤ ω_ovr(k)*V_Lyap(tk-)
>>>>> check (31) :
starting with
(29) V_Lyap(t) = sum[i=1 to n: |x(i,t)| ] + sum[i=1 to n: |y(i,t)| ]
repace t with tk
(R.31.0) V_Lyap(tk) = sum[i=1 to n: |x(i,tk)| ] + sum[i=1 to n: |y(i,tk)| ]
>> OK - same as authors' (31a)
using
(9c) ∆x(i,tk) = x(i,tk+) - x(i,tk-) = δk*x(i,tk-)
(9d) ∆η(i,tk) = η(i,tk+) - η(i,tk-) = ζk*y(i,tk-), k ∈ Z
(R.31.1) x(i,tk) = x(i,tk+) = (1 + δk)*x(i,tk-)
(R.31.2) y(i,tk) = y(i,tk+) = (1 + ζk)*y(i,tk-)
yields
V_Lyap(tk) = sum[i=1 to n: |(1 + δk) * x(i,tk-)| ] + sum[i=1 to n: |(1 + ζk) * y(i,tk-)| ]
(R.31.3) V_Lyap(tk) = sum[i=1 to n: (1 + |δk|)*|x(i,tk-)| ] + sum[i=1 to n: (1 + |ζk|)*|y(i,tk-)| ]
>> OK - same as authors' (31b)
using
(p6L55) ω_ovr(k) = { max |δk| + 1, |ζk| + 1 } Eq (28)
yields
V_Lyap(tk) = sum[i=1 to n: ω_ovr(k)*|x(i,tk-)| ] + sum[i=1 to n: ω_ovr(k)*|y(i,tk-)| ]
I can move ω_ovr(k) outside of summation
(R.31.4) V_Lyap(tk) = ω_ovr(k)*sum[i=1 to n: |x(i,tk-)| ] + sum[i=1 to n: |y(i,tk-)| ]
using (29)
(R.31.5) V_Lyap(tk) = ω_ovr(k)*V_Lyap(tk-)
compare to
(31c) V_Lyap(tk) ≤ ω_ovr(k)*V_Lyap(tk-)
>> OK - my (R.31.5) is the same as the authors' (31c)
<<<<< end check.
Combining (30),(31) and by employing Theorem 1, one obtains that
(32) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k) *sup[s ∈ [t_til(0),0]: V_Lyap(s) ] ]/ω(t), t ≥ 0
>>>>> check (32) :
using V(t) = V_Lyap(t) in this context
Noting that sup[s ∈ [t_til(0),0]: V_Lyap(s) ] is essentially a constant with respect to Π[k=1 to ∞:
>> ???? check (32) separate term "sup[s ∈ [t_til(0),0]: V_Lyap(s) ]" essentially a constant with respect to Π[k=1 to ∞: ??
>> ???? check (32) This may have been the authors' original intent, but sometimes their notation is AMBIGUOUS
(R.32.0) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k) ]*sup[s ∈ [t_til(0),0]: V_Lyap(s) ] /ω(t), t ≥ 0
starting with
(11b) V(tk) ≤ vk*V(tk-), k ∈ ℤ
(31c) V_Lyap(tk) ≤ ω_ovr(k)*V_Lyap(tk-)
then
(R.32.1) vk ≤ ω_ovr(k)
>> ???? check (32) - I must prove (R.32.1) vk ≤ ω_ovr(k) !!
using
(13) V(t) ≤ Π[k=1 to ∞: vk ]/ω(t)*sup[s ∈ [t_til(0),0]: V(s)], t ≥ 0
subbing (R.32.1) into (13)
(R.32.2) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k)]/ω(t)*sup[s ∈ [t_til(0),0]: V_Lyap(s)], t ≥ 0
rearranging
(R.32.3) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k)] *sup[s ∈ [t_til(0),0]: V_Lyap(s)]/ω(t), t ≥ 0
compare to
(R.32.0) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k)] *sup[s ∈ [t_til(0),0]: V_Lyap(s) ] /ω(t), t ≥ 0
(32) V_Lyap(t) ≤ Π[k=1 to ∞: ω_ovr(k) *sup[s ∈ [t_til(0),0]: V_Lyap(s) ] ]/ω(t), t ≥ 0
>> OK - (R.32.3),(R.32.0),(32) are all the same
<<<<< end check.
which implies
(33) ||z(t)||_1 ≤ Π[k=1 to ∞: ω_ovr(k)] *sup[s ∈ [t_til(0),0]: ||Φ(s)||_1] ]/ω(t), t ≥ 0
(28) ||z(t)||_1 ≤ Π[k=1 to ∞: ω_ovr(k)]/ω(t)*sup[s ∈ [t_til(0),0]: ||Φ(s)||_1], t > 0
>> OK - by simple re-arrangement of (28)
>>** (33) does NOT require implication via (32), as it is simply a re-arrangmeent of (28)
The proof is completed.
Remark 2 Comparing with previous works [9, 11, 27–32], few results dealt with the global asymptotical sta-
bility of nonautonomous impulsive system with unbounded time-varying delays. Therefore, based on a new
impulsive differential delay inequality and some analysis techniques, several conditions of global asymptotical
stability of the considered system are obtained.
Remark 3 In fact, it is relatively difficult to deal with this class of the unbounded time-varying delays because
none of any other assumptions are imposed on it, compared with other unbounded
R ∞ time-varying delays such
as unbounded distributed delays often require that the delay kernel functions 0 e μs k ij ( s ) ds < ∞ in [10,
15]. It should be mentioned that the assumption conditions t - τ ( t ) → ∞ as t → ∞ and τ ( t ) ≥ 0 are the
least conservative restriction for unbounded time-varying delay. It should be noted that we do not have any
other restrictions on τ ( t ), such as boundedness, differentiability, monotonicity and so on.
Remark 4 In [29], the global asymptotic stability and global exponential stability of neural networks with
unbounded time-varying delays were obtained. But the boundedness of activation functions was required.
In [31], by applying a generalized Halanay inequality, the authors studied the stability of system with
unbounded time-varying delays. But the boundedness of a i ( t ) was required. Therefore, the sufficient condi-
tions introduced in Theorem 2 are less conservative, which not require the boundedness of the coefficients
functions and activation functions.
enddocc