Báo cáo "Stability Radius of Linear Dynamic Equations with Constant Coefficients on Time Scales " pdf

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Stability Radius of Linear Dynamic Equations

with Constant Coefficients on Time Scales Le Hong Lan!*, Nguyen Chi Liem?

| Department of Basic Sciences, University of Transport and Communication, Hanoi, Vietnam 2 Department of Mathematics, Mechanics and Informatics,

University of Science, VNU, 334 Nguyen Trai, Hanoi, Vietnam

Received 10 August 2010

Abstract This paper considers the exponential stability and stability radius of time-invarying dynamic equations with respect to linear dynamic perturbations on time scales A formula for the stability radius is given

Keywords and phrases : time scales, exponential function, linear dynamic equation, exponentially stable, stability radius

1 Introduction

In the last decade, there have been extensive works on studying of robustness measures, where one of the most powerful ideas is the concept of the stability radii, mtroduced by Hinrichsen and Pritchard [1] The stability radius is defined as the smallest (in norm) complex or real perturbations destabilizing the system In [2], if a = Aa is the nominal system they assume that the perturbed system can be represented in the form

va =(A+BDC)a, (1)

where D is an unknown disturbance matrix and B, C’ are known scaling matrices defining the “structure” of the perturbation The complex stability radius is given by

—1

max Cứ 4) 'B|| Eu - (2)

If the nominal system is the difference equation 2,4; = Aa, in [3] they assume that the perturbed system can be represented in the form

Int = (A+ BDC)2y (3)

Then, the complex stability radius is given by

max ||C(wl —A)-'BI|| - (4)

we :|wl=1 * Corresponding authors E-mail: honglanle229@gmail.com

This work was supported by the project B2010 - 04

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Earlier results for time-varying systems can be found, e.g., in [4, 5] The most successful attempt for finding a formula of the stability radius was an elegant result given by Jacob [5] Using this result, the notion and formula of the stability radius were extended to linear time-invariant differential-algebraic

systems [6, 7]; and to linear time-varying differential and difference-algebraic systems [8, 9]

On the other hand, the theory of the analysis on time scales, which has been received a lot of attention, was introduced by Stefan Hilger in his Ph.D thesis in 1988 (supervised by Bernd Aulbach) [10] in order to unify the continuous and discrete analyses By using the notation of the analysis on

time scale, the equations (1) and (3) can be rewritten under the unified form

a = (A+ BDC)zx, (5)

where A is the differentiable operator on a time scale T (see the notions in the section )

Naturally, the question arises whether, by using the theory of analysis on time scale, we can express the formulas (2) and (4) in a unified form The purpose of this paper is to answer this question The difficulty we are faced when dealing with this problem is that although A, B, C are constant matrices but the structure of a time scale is, perhaps, rather complicated and the system (5) in fact is an time-varying system Moreover, so far there exist some concepts of the exponential stability which have not got a unification of point of view In [11], author used the classical exponent function to define the asymptotical stability meanwhile the exponent function on time scale has been used in [12] The first obtained result of this paper is to show that two these definitions are equivalent To establish

a unification formula for computing stability radii of the system (1) and (3) which is at the same time

an extention to (5), we follow the way in [12] to define the so-called domain of the exponential stability of a time scale By the definition of this domain, the problem of stability radius for the equation (5) deduces to one similar to the autonomous case where we know how to solve it as in [13]

This paper is organized as follows In the section , we summarize some preliminary results on time scales Section 3 gives a definition of the stability domain for a time scale and find out some its properties The last section deals with the formula of the stability radius for (5)

2 Preliminaries

A time scale is a nonempty closed subset of the real numbers R, and we usually denote it by the symbol T The most popular examples are T = R and T = Z We assume throughout that a time scale T has the topology that inherits from the standard topology of the real numbers We define the forward jump operator and the backward jump operator o, p:T > T by o(t) = inf{s € T: s > t}

(supplemented by inf @ = sup T) and p(t) = sup{s € T: s < t} (supplemented by sup 9 = inf T) The graininess 4 : T > Rt U{0} is given by u(t) = o(t)—t A point t € T is said to be right-dense if a(t) = t, right-scattered if a(t) > t, left-dense if p(t) = t, left-scattered if p(t) < t, and isolated if t

is right-scattered and left-scattered For every a, b € T, by [a, b], we mean the set {¢ € T:a <t < bd}

For our purpose, we will assume that the time scale T is unbounded above, 1.¢., sup T = oo Let f

be a function defined on T We say that f is delta differentiable (or simply: differentiable) at t € T provided there exists a number, namely {4(t), such that for all c > 0 there is a neighborhood V around

t with | f(o(t)) — f(s) — f4()(e(4) — s)| < elo(t) — s[ for all s EV If f is differentiable for every

t¿€TT, then / is said to be differentiable on T If T = R then delta derivative is f (¢) from continuous

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function f : T — R is called regulated provided its right-sided limits (finite) at all right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T A function f defined on T is rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point The set of all rd-continuous function from T to R is denoted by C,q(T, R) A function

ƒ from T to R is regressive (resp positively regressive) if 1+ u(t) f(t) A 0 (esp 1+ p(t) f(t) > 0)

for every t € T We denote R (resp R*) the set of regressive functions (resp positively regressive) from T to R The space of rd-continuous, regressive functions from T to R is denoted by C,qR(T, R)

and, C.qR*(T,R) := {f C;a#X(T,R):1+”@)ƒ() >0 forall ¢ € T} The circle addition

® is defined by (P ì oe ) = p(t) + q(t) + w(t)p(Hq(t) For p € R, the inverse element is given by (Sp)() = oa and if we define circle subtraction © by (p © q)(t) = (p @ (Gq))(E) then

(p©4)0) = Bw

Let s € T and let (A(¢)), , be a d x d rd-continuous function The initial value problem

aS — A(t), x(s) = a (6)

has a unique solution a(t, s) defined on ¢ > s For any s € TT, the unique matrix-valued solution,

namely ®4(t, s), of the initial value problem X“ — A(t)X,X(s) = I, is called the Cauchy operator of (6) It is seen that ® 4(t, s) = @4(t,7)®a(7, s) for all t > 7 > s

When d = 1, for any rd-continuous function q(-), the solution of the dynamic equation x

q(t)x, with the initial condition x(s) = 1 defined a so-called exponential function (defined on the time scale T if g(-) is regressive; defined only t > s if q(-) is non-regressive) We denote this exponential

function by e,(t, s) We list some necessary properties that we will use later

A _

Theorem 2.1 Assume p,q :T — R are rd-continuous, then the followings hold

ii) ep(t, 8)ep(s,r " ae r),

iv) ep(t, 8)eg(t, 8) = cpog(t 3),

v) care } Cndq(t, 8) if q is regressive, vi) ene then e,(t, s) > 0 for all t, s € T,

vii) fp p(s)ep(e, a(s))As = ep(e, a) — cp(e, b) for all a,b,c € T,

viii) If p € R* and p(t) < q(t) for all t > s then e,(t, s) <e(t,s) forall t>s Proof See [14], [15] and [16]

The following relation is called the constant variation formula

Theorem 2.2 [See [17], Definition in 5.2 and Theorem 6.4] Jf the right-hand side of two equations

a® = A(t)a and x = A(t)a+ f(t, ax) is rd-continuous, then the solution of the initial value problem a = A(t)x + f(t, x), x(to) = ao is given by

x(t) = ®y(t, to) xo +f eate a(s))f(s,a(s))As, t> to

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Lemma 2.3 [Gronwall’s Inequality] Let u,a,b € Cra(T, R), b(¢) > 0 for all t € T The inequality t u(t) < a(t) + / b(s)u(s) As for all t > to to implies Corollary 2.4 t 1 If wu € Cra(T, R), b(t) = L > 0 and u(t) < a(t) + Lf u(s) As for all t > to implies to u(t) < a(t) + Lƒ c;(t, a(s))a(s) As for all t > to t 2 If u,b € Cra(T, R), b(t) > 0 for all ¢ € T and u(t) < uo + f b(s)u(s) As for all ¢ > to then to u(t) < uoes(t, to) for all ý > to

To prove the Gronwall’s inequality and corollaries, we can find in [14] For more information on the analysis on time scales, we can refer to [17, 18, 19, 20]

3 Exponential Stability of Dynamic Equations on Time Scales

Denote T* = [to,00) NT We consider the dynamic equation on the time scale T

a = f(t,x), (7)

where f : T x R¢ — R¢ to be a continuous function and f(t, 0) = 0

For the existence, uniqueness and extendibility of solution of initial value problem (7) we can refer to [15] On exponential stability of dynamic equations on time scales, we often use one of two following definitions

Let w(t) = a(t,7, 20) be a solution of (7) with the initial condition x(7) = xo, 7 > to, where #o € Rẻ,

Definition 3.1 [See S Hilger [10, 17], J J DaCunha [11], .] The solution « = 0 of the dynamic

equation (7) is said to be exponentially stable if there exists a positive constant a with —a~ € R*

such that for every 7 € T* there exists a N = N(r) > 1, the solution of (7) with the initial condition x(T) = xo satisfies ||œ(f; 7, øo)|| < N||œolle_„(£,7) for all £ > 7, € TT

Definition 3.2 [See C Podtzsche, S Siegmund, F Wirth [12|, | The solution z #= 0 Of (7) 1s

called exponentially stable if there exists a constant a > 0 such that for every 7 € T* there exists

a N = N(r) > 1, the solution of (7) with the initial condition x(7) = apo satisfies ||a(t;7, 20) || < NIlsolle %É—”), for all £ > r„,£€ TT

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Note that when applying Definition , the condition —a € RT is equivalent to p(t) < +, This means that we are working on time scales with bounded graininess

Beside these definitions, we can find other exponentially stable definitions in [21] and [22]

Theorem 3.3 7wo definitions and are equivalent on time scales with bounded graininess t Proof If -a € R*,t > 7 then e_,(t, 7) = exp { f lim maul a s| where 7 Uns) lim In|1 — aul _ tui " if p(s) = 0, us) u “a if p(s) > 0 So lim In| = awl < —a, forall s € TT uu(s) u

Therefore, e „(f,7) < c~aŒ~?) for all a > 0,-a € Rt and t >7 Henee, the stability due to Definition implies the one due to Definition

Conversely, with a > 0 we put

_ ¬ ee] —œ if u(t) = 0, a(t) = lim —————= Á 4 au@ 1,

s\ u(t) S ult) if u(t) > 0

It is obvious that ä(-) € 7#” and ca(3(f,7) —= et) Let M := supye + p(t) If M = 0, ic, u(t) = 0 for all ¢ € T, then a(t) = —a When M > 0 we consider the function y = ——

with 0 < u < M It is easy to see that this function is increasing In both two cases we have

a(t) < B= Jim, e“— for allt € TY

Therefore, eg )(¢, 7) = e tT) < ea(f,7) for all £ > 7 By noting that —3 > 0 and 8 € RT

we conclude that Definition implies Definition The proof is complete

By virtue of Theorem , in this paper we shall use only Definition to consider the exponential stability

We now consider the condition of exponential stability for linear time-invariant equations

a* = Ax, (8)

where A € K¢*4 (K = R or K=C) We denote o( A) = {\ € C, ) is an eigenvalue of A}

Theorem 3.4 The trivial solution x = 0 of the equation (8) is uniformly exponentially stable if and only if for every € o( A), the scalar equation #&^ = zx is uniformly exponentially stable

Proof

“ => ” Assume that the trivial solution = 0 of the equation (8) is uniformly exponentially stable

and \ € o(A) with its corresponding eigenvector v € C% \ {0} It is easy to see that cA(f,7)0

is a solution of the equation (8) Therefore, there are N > 1 and a > 0,-a € R* such that

le,(t, T)v| < Ne_a(t,7)|lv||, > 7 Hence, |e, (¢,7)| < Ne_g(t,T), Đ Sr

<= Let (đA(f,7)); ; be the Cauchy operator of the equation (8) We consider the Jordan form

of the matrix A

Ji 0

S 'AS = me

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where J; € C%*% is a Jordan block A 1 0 0 À¡¿ 1 0 J; = ) À¿ and À;¿€ ơ(4), dị + dạ + + dạy = đ,1 SŠ?<n << đ Since ® 7, (Â,7) đA(f,7) = S " st, it suffices to prove the reverse relation with A 1 0 - 0 À 1 0 A= , 0

where the equation a“ = \x is uniformly exponentially stable Let a = (wa, wa,

A — Ax can be rewritten as follows , Xa) The equation

at = = rx, + #2 ap = = À#a2 + #3

(9)

z2 = À#g,

with the mitial conditions #+(7) = axe, k = I, -,d The assumption that the equation 2 = dx is

uniformly exponentially stable implies |ea(f,7)| < Ne_a(t,7), with N,a > 0,-a € Rt and t > 7 The last equation of (9) gives xq = e,(t,7)x9 So

Iza)| = |eAU,z)zj| € N|edle-a(t,z) < N|leolle-a(t,7), for allt > 7

By the constant variation formula, we have the representation,

t

ag_1(t) = ex(t, ray „

Therefore, :

Iza-i(0)| € Ne-a(f,7)|#2 3| ff N?e_a(t,o(s))e—a(s,7)|aglAs

< Nls8 ¡|e-a(1,7) + N?aÐl [ ©_32(1,Ø(5))€ 2 (s,T)As

= Na ¡|e_a(,7) + NP|a9| [x Gay ele eles 3 3

As

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Since —a € Rt, we have 1 —ay(s) > 0 which is equivalent to 1 — Ÿ#⁄/„(s) > š for all s € T Hence, va) < Nia leg (t,7) + 3N7ag|(t— T)e_ 20 (4,7)

Further, from the relation (—¥)6(—$)(t) = —22+(—22)? u(t) > —22, it follows that e_a(t,7).e_a(t,7) = e(-a)@(-2)(47 7) > €_z.(, 7) On the other hand, e_a(t,7) < exp(—#E—”)) for any t > T Therefore, ( — 7)e_ s (f,7) < (ứ — 7)exp(—#E—) < = Thus,

Jza (9| < K}llzol|c- zŒ, 7),

where K! = N + aNTexp (1) |

Continuing this way, we can find K > 0 and 6 > 0 with 6 € R* such that ll+z|| < K|lolle-aŒ,7) for allt >7

The theorem 1s proved

Remark 3.5 It is easy to give an example where on the time scale T, the scalar dynamic equation

a2 — Àø is exponentially stable but it is not exponentially uniformly stale Indeed, denote ((a,b)) = {n€Ñ:a< n < db} Consider the time scale

T— J2” gent) Lor, 2212),

n

Let \ = —2 and 7 € T, says 2” < 7 < 2*!, We can choose a = —1 and N = 2”°*! to obtain le,(t, T)| < Ne_1(t, 7) However, we can not choose N to be independent from 7

4 The domain of exponential stability of a time scale We denote

S={XEC, the scalar equation a = dx is uniformly exponentially stable}

The set S' is called the domain of exponential stability of the time scale T By the definition,

if \ € S, there exist a > 0,-a € R* and N > 1 such that |eA(f,7)| < NWe_„(,7) for all £ > 7

Theorem 4.1 S is an open set in C Proof

Let \ € S There are a > 0,-a € Rt and N > 1 such that Je)(t,7)| < Ne_a(t,7) for all t >

7 and assume that € €,| — Al < €, where 0 < € < 4 We consider the equation x = pe =

Ax + (wu — r)x with the initial condition x(7) = xo

By the formula of constant variation, we obtain

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or

t ‘ oN

OL whey) + ff Ye BOL 3,

e_a(, 7) x1 — O(5) C a(8, 7)

It is obvious that ~a—Ne > 0 and —(a—Ne) € R*) This relation says that {42 € C : |uw—A| < e} CS,

i.e., S is an open set in C The proof is complete Example 4.2

1 When T= R then S= {AX € C, RA < 0}

2 When T = hZ (h > 0) then S = {A € C, |1 + Ah| < 1}

3 When T = U9 [2k, 2k + 1] then S = {A € C, RA+F In|1 + Al < OF

Indeed, if \ = —1 then for all T € T there exists t € T,t > T such that 1 + Au(t) = 0, this implies (o(t)) = 0 Therefore, in this case the equation «~ = zx is (uniformly) exponentially stable Now assume \ 4 —1 When 2m = s < t = 2n we have |ea(, s)| = e*Œ—m)|1 + A|?—m =

Nepy+in|i+al)/2(t, 8) we have the proof

4 Similarly, if T= UP olk, k + al, a € (0,1) then S = {A € C,aRA + Injl + (1—a)A] <

0}, where we use the convention In0 = —oo

5 Stability radius of linear dynamic equations with constant coefficients on time scales Assume that the nominal equation

a® = Aa (10)

is uniformly exponentially stable, where A ¢ K?%? (K = R or K= C) Consider the perturbed equation

a* = Ax + DA Ba, (11)

with D e K“ề%!, E e K*?*#, and A € K'4 is an unknown time-invariant linear parameter disturbance

Denote V = {A € K'*4,0(A+ DAE) € S}

Definition 5.1 The structured stability radius of the dynamic equation (10) is defined by

r(A; D; E) := inf{||A]] the solution of (11) is not uniformly exponentially stable} By the assumption on (10) and due to Theorem , we have o(A) C S and

r(A; D; E) = inf{||A]| : A € NV} = sup{r > 0,0(A+ DAE) CSVA €R™?, |All <r} Let X € p(A) := C \ (A), we define

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For a subset 2 C p(A), we define

rq(A; D; E) := sup{r > 0,0 C p(A+ DAB) for all A € K'*? with ||Al] <r}

1

Theorem 5.2 [See [13]] For all \ € p(A) we have r)(A; D; FE) = ]EOT- AD)’ where I is

the identity matrix

Corollary 5.3 [See [13]] If Q C p(A) then rQ(A; Dd; E) = inf IFAT By E — Applying this result with Q = CS = C\ Ø we have, Theorem 5.4 1 A: D: EB = A: D: EB = i f — r( T2) ) ral T2) ) ye #øJEOTF— 4)1DỊ Denote G(A) := E(AI — A4) 1D By virtue of the properties dim G(A) = 0 and CS to be CO

closed, we see that ||G(\)|| reaches its maximum value on CS Moreover, since the function G() is analytic, the maximum value of ||G(\)|| over 9 can be achieved on the boundary 00S = 0S Thus,

Theorem 5.5 -1

r(A; D; B) = ro( A; D; B) = { max ||G()||}

We now construct a destabilizing perturbation whose norm is equal (A; D; £) Since ||G(A)]||

reaches its maximum value on CS, by the theorem , there exists a Ap € OS such that r(A; D; F) =

|GQo) ||

Let u € C! satisfying ||G(Ao)u|| = |lG(A)|l, |lu|| = 1 Applying the Hahn-Banach theorem, there exists a linear functional y* defined on K? such that y*(G(Ao)u) = ||@(Ao) ull = ||@(Ao) || and

\|y*|| = 1 Putting A := ||G(Ao)||~!uy* we get

IAlI < EQo) ˆlellly = [EQo) I~

From

AG (Ao) = ||GQo) || uy*GOo)u = u,

we have

|All > IEQo) I~

Combining these inequalities we obtain

|All = EQ)

Furthermore, let a = (Ag! — A)~!Du and from

(Aol — A — DAE)a# = (Aol — A)(ol — A)! Du — D||G(o) || 7 !wy* Eo — A)! Du

= Du —D||GQo)||'uy*G(o)u = 0, 1t follows that Ào € o(A + DAE) NOES This means A € NV and it is a destabilizing perturbation Example 5.6 Let T = Ue o[k, k + 4] and

0 —2 1 1 02

"¬

We have the domain of exponential stability of this time scale is

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| KEC, RA + ail + SÀ + ĐÌ, tee!

Fig 1 The domain of stability

It is easy to see that o(A) = {-1,-2} C S The boundary of S is the set 0S = {A € C, š#A +In |1 + $À| = 0} and ( 2(A+1) 5 2 G(Q)= | M2 AT AI? |, 0 N24+BAF2 With the maximum norm of R?, ie., ||(a, y)|| = max{|a|, |y|} we have 2|A + 1] +2 |X + 2| 1 = ase) |A2+3A+2||A2+3A+2|J |A2+3A+2| |GO)|| = max { =ms*parn) — |A+2| |A+1|2 Put A = a + yi From À € ØS we have (2œ + 3)? + 4y? = 95“ and ø <0 Then |CO)|= F@) = li —— <2 = F(0) for all x <0 fe 8% +a44 fe 3% a — 8 || = |G(0)|| = 2 and r(A; D; E) = 5 1,1) it yields || O)ul] = ]@(0)|] = 2 Take the functional y* = (1,0), we have y*(G(0)u) = ||G(0)u|| = ||G@(0)|] = 2 and ||y*|| = 1 Let A = |G) ay" = (3 5) Therefore max ||G(A NEOS —~ ~_~

With the vector u =

We see that o(A + DAE) = {0,—2} ¢ S' which implies A € NV and r(A; D; E) = 5 = |All 6 Conclusion

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equal to the stability radius In the theory of stability radii, the investigation whenever the real stability radius and complex one are equal is very important Since the structure of the stability set is rather complicated, so far we have to leave it as open question

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