DSpace at VNU: On The Convergence of Stability Domain on Time Scales

1 On The Convergence of Stability Domain on Time Scales Nguyen Thu Ha1, Khong Chi Nguyen2,*, Le Hong Lan3 1 Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet,

1

On The Convergence of Stability Domain on Time Scales

Nguyen Thu Ha1, Khong Chi Nguyen2,*, Le Hong Lan3

1

Department of Basic Science, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

2 Tan Trao University, Trung Mon Affairs, Yen Son, Tuyen Quang, Vietnam 3

Faculty of Basic Sciences, Hanoi University Transportation, Hanoi, Vietnam

Received 18 August 2015 Revised 28 September 2015; Accepted 20 November 2015

Abstract: This paper studies convergence of the stability domains for a sequence of time scales It

is proved that if the sequence of time scales(Tn)converges to a time scale Tin Hausdorff topology then their stability domains UTn will converge to the stability domain UTof T

Keywords: Implicit dynamic equations, time scales, convergence, stability domain, stability radius.

1 Introduction∗∗∗∗

It is known that the linear systemx t
( )=Ax t t( ); ∈+ =[0, )∞ is exponentially stable if and only if the spectrum ( )σ A lies within the half plan ⊂  of the complex numbers  Also, if we consider the − different systemx n+1=x n+hAx n n, ∈  , then this system is exponentially stableh + ⇔σ( )A lies in the disk h: { :|z z 1| 1}

= + <

T

U Where h> and 0 h+h ={0, , 2 , }h h

In view of theory of time scales, the sets   are time scales and +, +h  and − UTh are respectively their stability domains Further, when h→ , the sequence of time scales (0 +h) is “close” to  and +

the set of the disks UTh will “enlarge” to the set  in some sense Indeed, the Hausdorff distance − ( , h) sup{ ( , h) : }

d  + + = d x + x∈+ of  and + +h

 is h and

0

h h

+

>

= 

The question rises here if we can generalize this idea to an arbitrary set of time scales (T ? That n)

is if the sequence (T tends to the time scale T in Hausdorff topology, can we conclude that their n) respective stability domain converges to one of T

_

∗

Corresponding author Tel.: 84-984732576

Email: nguyenkc69@gmail.com

This paper concerns with such problem Firstly, we consider the continuous dependence of the exponential functions to time scales Then, we show that the stability domain UTn corresponding to the time scale T converges when n T tends to T n

The paper is organized as follows Section 2 summarizes some preliminary results on time scales, property of exponential functions on time scales and characterizes the stability domain of a time scale The main results of the paper are derived in Section 3 We study the convergence of the stability domains here The last section deals with some conclusions and open problems

2 Preliminaries

2.1 Time scales

Let T be a closed subset of  , enclosed with the topology inherited from the standard topology

on  Let ( )σ t =inf{s∈T:s>t}, ( )µ t =σ( )t −t and ρ( )t = sup{s∈ T :s<t}, ( )ν t = −t ρ( )t

(supplemented by sup ∅ = inf T ,inf ∅ = sup T) A point t ∈ T is said to be right-dense if ( )σ t = , t right-scattered if σ( )t > , left-dense if t ρ( )t = , left-scattered if t ρ( )t < and isolated if t is t simultaneously right-scattered and left-scattered A function f defined on T is regulated if there exist

the left-sided limit at every left-dense point and right-sided limit at every right-dense point A

regulated function is called rd-continuous if it is continuous at every right-dense point, and ld-continuous if it is continuous at every left-dense point It is easy to see that a function is continuous if

and only if it is both rd-continuous and ld-continuous A function f from T to  is positively regressive if 1+µ( ) ( )t f t > for every t ∈ T We denote by 0 +

R the set of positively regressive functions from T to 

Definition 2.1 (Delta Derivative) A function f:T→ d is called delta differentiable at t if there exists a vector f∆( )t such that for all ε>0

f σ t f s f∆ t σ t s σ t s

for all s∈ −(t δ,t+δ)∩ T and for some δ >0. The vector f∆( )t is called the delta derivative of f at

t

If T=  then the delta derivative is f t′( ) from continuous calculus; if T=  then the delta derivative is the forward difference, ∆f t( )= f t( +1)− f t( ), from discrete calculus

Let f be a rd-continuous function and , a b∈ T Then, the Riemann integral b ( )

a f s ∆ s

∫ T exists (see [1]) In case a b, ∈/T , writing b ( )

a f s ∆ s

a f s ∆ s

a = t>a t∈ T b =max{t<b t: ∈ T If there is no confusion, we write simply } b ( )

a f s ∆s

∫

(resp b ( ) n

a f s ∆ s

a f s ∆ s

∫ T (resp ( )

n

b

a f s ∆ s

∫ T )

Fix t0∈  Let T be the set of all time scales with bounded graininess such that t0∈ T for all

∈

T T We endow T with the Hausdorff distance, that is Hausdorff distance between two time scales

1

T and T is defined by 2

( , ) : max{sup ( , ), sup ( , )},

H

=

where

( , ) inf | | and ( , ) inf | |

For properties of the Hausdorff distance, we refer the interested readers to [2, 3]

2.2 Exponential Function

Let T be an unbounded above time scale, that is sup = ∞T

Definition 2.2 (Exponential stability) Let p:T→  is regressive, we define the exponential function by

0

Ln(1+hp(s))

h

µ

where Lna is the principal logarithm of the number a

Theorem 2.3 (see [4]) If p is regressive and t0∈ T , then e p(., )t0 is a unique solution of the initial value problem

0

( ) ( ) ( ), ( ) 1

y∆ t = p t y t y t =

When p t( )=λ, where λ is a constant in  , we write e t sλ( , ) for e p(·)( , )t s

Theorem 2.4 (Properties of the Exponential Function) If p q, :T→  are regressive, rd-continuous functions and t r s, , ∈ T then the following hold:

1 e t s0( , )=1,and e t t p( , )=1;

2 e p( ( ), )σ t s =(1+µ( ) ( ))t p t e t s p( , );

p

4 e t s e t s p( , ) ( , )q =e p q+ ( , )t s

5 e t s e s r p( , ) p( , )=e t r p( , )

6 ( , ) ( , )

( , )

p

p q q

e t s

e t s = −

0

0, ( )

k> f t ∈  Assume that ( ) f t satisfies the inequality

0

( ) t ( ) , for all ,

t

f t ≤ f +k∫ f s∆s t∈ Tt≥t (2.2)

Then, the following relation holds,

( ) ( , ), for all k ,

We see that

0

0

( )

ln |1 |

| ( , ) | exp{ t lim }

t h s

h

h

λ

µ

λ +

By using the notation

if 0

ln | 1 |

if 0,

s h

s

λ

λ λ

+

≠









we can rewrite

0

0

| ( , ) | exp{ t ( ( )) }

t

We note that |e t sλ( , ) | |= e t sλ( , ) | for any λ∈  Further, it is easy to see that ζλ( ) |x ≤ λ| for all 0

x≥ For the properties of exponential function e t sλ( , ) the interested readers can refer to [5]

2.3 Exponential stability

Let

t = t∈ t≥t

T T Consider a dynamic equation

0

( , ),

We assume that the function

0

f T × → satisfies conditions such that Equation (2.5) has a unique solutionx t s x( , , 0),t≥ with the initial condition s x s s x( , , 0)=x0 for any

0

t

s∈ T and 0

m

x ∈ 

Definition 2.1 (Exponential stability) The dynamic equation (2.5) is called uniformly

exponentially stable if there exist constants α > with 0 α +

− ∈ R and K > such that for every 0

0

, , t

s<t s t ∈ T , the inequality

x t s x ≤K x e−α t s

holds for any x0∈  m

In the linear homogeneous case, i.e., ( , )f t x =Ax we have the equation

0

It is known that Equation (2.7) is uniformly exponentially stable if and only if so is for the scalar equation

0

x t∆ λx t t t

for any λ σ∈ ( )A (see [6, 7])

Denote by TU the set of complex values λ such that (2.8) is uniformly exponentially stable We call TU the domain of uniformly exponential stability (or stability domain for short) of the time scale

T Denote

1 ( ) : limsup t ( ( )) 0

s

t s

L

− →∞

Propotion 2.7 Let λ∈  , then λ∈ TU if and only if

1 ( ) : limsup t ( ( )) 0

s

t s

− →∞

Proof Assume that (2.7) is uniformly exponentially stable It implies that there exist constants

0,K 0

α > > such that

( )

s h

h

h

α λ

µ τ

λ

+

( )

t

s h

h

h

µ τ

λ

+

∫ 

Therefore, we have

1

s

t s

− →∞

∆ < − <

− ∫

Assume that ( ) :L λ = − < Then there is an integer number N large enough such that α 0

1

2

t

t s λ

α

Thus |e t s( , ) | |e2 (t s) |, t s N

α λ

−

< − ∀ − >

By virtue of the inequality ζλ( ) |x ≤ λ| for any x≥ , we have 0

| |

| ( , ) | exp{ t ( ( )) } N,

s

e t sλ = ∫ζ µλ s ∆s ≤eλ ∀ − <t s N

Hence

( ) 2

t s

α λ

− −

with : | | 2 1

N N

α

λ +

= > The proposition is proved □

Further, it is easy to verify that if |b1| |≥ b2| then

a ib x a ib x

ζ + ≥ζ + for any a∈  and x≥0 So T

U is symmetric with respect to the real axis of the complex plan  This means that λ∈ TU implies the segment [ , ]λ λ ⊂ TU

open set in 

Proof Let λ∈ TU ,then there are K> and 0 λ +

∈ R such that

|e t sλ( , ) |≤Ke−αt s− , forallt≥s (2.11)

We now proof that there exists ε>0 such that the equation

x∆ βx

is also uniformly exponentially stable for allβ∈  with |β−λ|≤ ε We rewrite Equation (2.12) under the form x∆ λx (β λ) x

= + − Using variation of constants formula yields

( , ) ( , ) t ( , ( ))( ) ( , )

s

e t sβ =e t sλ +∫ e tλ σ u β−λ e u sβ ∆u (2.13)

This implies that

( )

| ( , ) | t s | | t| ( , ( )) ( , ) |

s

e t sβ Ke−α − K β λ e tλ σ u e u sβ u

Hence,

s

eα − e t sβ K K β λ eαµ eα − e u sβ u

Let f t( )=eα(t s− )|e t sβ( , ) |, we have

s

f t ≤K+K β−λ eα ∫ f u ∆u

Using Gronwall's inequality (with ( ) 1f s = ) obtains

f t ≤Ke t s t≥s

where M =K|β−λ|eαH.Thus

e t sβ ≤Ke− +α − t≥s

By choosing

Keα

α

=

t s

α

β

≤ ≥ for any β such that |β−λ|≤ ε The proof is complete □

3 Main results

In this section, we consider a sequence { }Tn n∈ ⊂T of time scales satisfying:

lim n

n→∞T =T

Denote byµn( )t (resp ( )µ t ) the graniness of T (resp T ) at time t Since n T T ∈ , sup{ ( ) :µ t t∈T}< ∞ Therefore, it is easy to prove that if lim n

n→∞T =T then sup{µn( ) :t t∈Tn,n∈}< ∞,{sup{ ( ) :µ t t∈T,n∈}< ∞ Denote

*

First, we need the following lemmas to derive some characteristics of stability domains UTn when

n

T tends to T

( ) 0 if and only if

Proof Denote µ*=sup{ ( ) :µ t t∈ T and let } λ∈U T  Then, there is a sequence { }λn ⊂ TU such that lim n

→∞ = Let λ= +a ib with b≠ and 0 λn =a n+ib n Using Lagrange finite increments formula, for all x> we have 0

(| | | | ) 2( )

n

Since 0 1 2< + xa+x2|λ|2 for all x≥ , we can choose a 0 n0∈  and a constant c1> such that 0

1 1 2 ( ( n )) (| | (| n| | | )

c < + x a+θ a −a +x λ +θ λ − λ for all 0≤ ≤x µ* and n>n0 Thus, for any ε>0, there exists n1>n0 satisfying

n

ζ −ζ <ε ∀ ≤ ≤µ ∀ >

This implies that

n

sζ µ τλ ∆ <τ sζλ µ τ ∆ +τ t−s < t−s ∀t ≤ ≤ ∀ >s t n n

Hence

1

s

− →∞

∆ < ∀ >

Thus

1

s

− →∞

∆ ≤

− ∫

Conversely, let λ= +a ib∈  such that

1

s

− →∞

∆ ≤

− ∫

For any ε>0, let λε =aε+ibε be chosen such that 0 |< bε| | |;< b aε<a and |λ| |> λε| |> λ|−ε Since

2 2

0 1 2< + xa+x |λ| , we can choose aε and bε such that

2

0<2(1 2 (+ x a+θ(aε−a))+x (|λ| +θ λ(| ε| −|λ| ))<c for all 0≤ ≤x µ*

Thus,

* 2

( )x ( )x a a, 0 x

c

ε

ε

This implies that

0 2

c

ζ µ τ ∆ <τ ζ µ τ ∆ +τ − − ∀ ≤ ≤

ε

Hence,

2

1

s

t s

− →∞

−

∆ ≤ <

− ∫ ε

ε

which follows that λε∈ TU This means that λ∈ TU The proof is completed □

Lemma 3.2 Let K ⊂  be a compact set Suppose that lim n ,

n→∞T =T then for any ε>0, there are δ >0, n0∈  such that

t s

δ

−

for n>n0,λ∈K and

*

, [0, ]

sup | ( ) |

K x

ζ

∈ ∈

Proof Since K⊂  is a compact set, M < ∞ First, assume that Tn⊂T We see that the function

1

2

x

dx

x

λ

=











is continuous in ( , ),x λ provided ℑ ≠λ 0 Therefore, the family of functions (ζλ( ))u λ∈K is

equi-continuous in u on [0,µ*], i.e., for any ε>0, there exists δ =δ( )ε >0 such that if |u− <v| δ then

|ζλ( )u −ζλ( ) |v < ε for any λ∈K Since lim n

n→∞T =T , we can choose n0 such that ( , )

2

<

T T when n>n0

Fix t0≤ <s t s t; , ∈[0, )∞ and n>n0 Denote

1 { n [ , ] : n( ) },

A = h∈T ∩ s t µ h ≥δ A2 ={h∈Tn∩[ , ] :s t µn( )h <δ}

The assumption Tn⊂T implies that 0≤µ( )h ≤µn( )h for all h∈ T If n h∈A2 then ( )h n( )h ,

µ ≤µ <δ which implies |ζ µλ( ( ))h −ζ µλ( n( )) |h < ε On the other hand, the cardinal of A1,

say r, is finite and r t s

δ

−

  Thus, we can write A1={s1<s2< … <s r}

Denote sequence τi by

( )

2

Since d H( ,T Tn)≥max{|τi−s i|,|σn( )s i −τi|}, it follows that

i s i δ n s i i δ

τ − < σ −τ <

Therefore,

| ( )µ τi −µn( ) |s i =µn( ) |s i − τi−τi| |=τi−s i|+|σn( )s i −τi|<δ,

which implies

|ζ µ τλ( ( ))i −ζ µλ( n( )) |s i < ε

For any h∈ T , there exists a unique u∈ T , say n u=γT T , n( )h , such that either h= or u

( , n( ))

h∈ uσ u It is easy to check that the function γT T, n( )h

is rd-continuous on T By definition of

integral on time scales we have

,

sζ µλ h ∆ h= sζ µ γλ h ∆h

Therefore,

,

sζ µλ h ∆ h− sζ µλ h ∆ ≤h sζ µ γλ h −ζ µλ h ∆h

| ( ( )) ( ( n( ))) |

s

n

1

| ( ( )) ( ( ( ))) |

( i

n i

r t

n s

i

τ

∧

=

| ( ( )) ( ( n( ))) |

s

n

1

( i

n i

t

n s

r

i

τ

∧

=

,

| ( ( )) ( ( ( ))) |

i

n i

t

n

∧

∧

n i

t s

n

t σ λ h λ h h

∧

∧

1

1

, ( )

1

| ( ( )) ( ( ( ))) |

i

n

r s

n s

i

−

=

where a∧ =b min{ , }a b

Since | ( )µ h −µ γ( T T, n( )) |h <δ for all h∈[ , )t s0 1 ∪[ ( ),σ s i s i+1∧t),1≤ ≤ − i r 1,

1

| ( ( )) ( ( n( ))) | ( )

s

n

s ζ µλ h −ζ µ γλ h ∆ <h τ −s

1

i

n

s

+

On the other hand, for i=1, 2, ,r we have

,

| ( ( )) ( ( ( ))) |

i

n i

t

n

∧

(t τi t τi) |ζ µ τλ( ( ))i ζ µλ( n( )) |s i (t τi t τi),

and

,

i

n i

t

s∧τ ζ µλ h −ζ µ γλ h ∆ ≤h M t∧τ −s ≤ Md

T T

n i

t s

∧

T T

Thus, we obtain

,

t

n s

1

1

i

=

<ε − +∑ ∧ − ∧

1 1 1

r

i

− +

=

1

r

i

=

+ ∑ T T <ε − 4Mrd H( , n) (t s) 4M t s d H( , n)

δ

−

Therefore,

( ( )) ( ( )) ( ) 4 ( , ).

t s

δ

−

If Tn⊂/ T , we put Tn=Tn∪T It is easy to see that

( , ) max{ ( n, ), ( n, )}

By the above proof, we have



n

t s

δ

−



n

t s

δ

−

This implies that

t s

δ

−

The proof is complete □

Denote by UTn (resp TU ) the domain of stability of the time scale T (resp T ) n

Proposition 3.3 Suppose that lim n

n→∞T =T Then, for any λ∈ TU we can find a neighborhood

( , )

B λ δ of λ and nλ > such that ( , )0 n

n n

B

λ

λ δ

>

⊂UT∩ UT .

Proof Firstly, we prove the proposition with λ∈U T  Following the proof of Lemma 3.1 and

by Proposition 2.8, there exists a δ1> satisfying 0 B( ,λ δ1)⊂ TU and

( )

4

L

λ

Hence, by (2.10)

λ ≤ λ +− = for any λ∈B( ,λ δ1) (3.3)

By choosing : min{ ,1 | |} 0

3

λ

λ

δ = δ ℑ > we see that B( ,λ δλ)⊂U T  Using Lemma 3.2 with

( , )

K=B λ δλ and ( )

8

L λ

−

=

ε we can find a δ2> and 0 n0 such that

2

( )

4

λ

δ

for n>n0 and λ∈B( ,λ δλ) We choose nλ >n0 such that 2 ( )

( , )

32

L d

M

−

<

T T for any n>nλ From (3.4) we get

1

s

t s

h h

t s ζ µλ

− →∞

∆

s

t s

h h

t s λ

ζ µ

− →∞