Stability Radius of Linear Dynamic Equations with Constant Coefficients on Time Scales Le Hong Lan!*, Nguyen Chi Liem?. This paper considers the exponential stability and stability radi
Trang 1Stability 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, expo-
nentially 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
where D is an unknown disturbance matrix and B, C’ are known scaling matrices defining the “struc- ture” of the perturbation The complex stability radius is given by
—1
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
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
163
Trang 2Earlier 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
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
calculus; if T = Z then the delta derivative is the forward difference, Af, from discrete calculus A
Trang 3function 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
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
to
Trang 4Lemma 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
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
If the constant NV can be chosen independent from 7 € TT" then the solution 2 = 0 of (7) is called uniformly exponentially stable
Trang 5Note 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
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), § Sr
“<=” Let (®A(f,7));¿ ; be the Cauchy operator of the equation (8) We consider the Jordan form
of the matrix A
Ji 0
Trang 6where J; € C%*% is a Jordan block
À¡¿ 1 0
À¿
and À;¿€ ơ(4), dị + dạ + + dạy = đ,1 SŠ?<n << đ
Since
® 7, (¢,7)
it suffices to prove the reverse relation with
A 1 0 - 0
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 „
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
< Nlz9 ;|e sứ, ch a ° 3 z1 — Sus)
Trang 7Since —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
x(t) = eA(f,7)#o vf ey(t, a(s))(u— A)a(s)As
This implies
Ie0)| Š N|eo|e—a( nf Nee_a(t, o(s))|a(s)|As
= N\aole_a(t, 7 ị Tang) + , s)|a(s)|As,
Trang 8or
Applying the Gronwall’s inequality, we have
I=0)|
NAN e_„(É, 7) #o|e xe ( 7) c (E7),
or
|a(t)| < N|wole_ ene _(t,7) = N|@ole_(a_ne(t, T) for all £ >7
l-ap(-)
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 =
eMrin|I+A)(m—m) Thus, A € Sif and only if RA + In|1 +A] < 0 If s,¢ € T such that 2m < s<2m+ and 2n <t <2n +1 Since, |e,(t, s)| = |[AC™*! Ye, (Qn, 2m + 2A 291 +d) <
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
is uniformly exponentially stable, where A ¢ K?%? (K = R or K= C)
Consider the perturbed equation
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
r)(A; D; E) := sup{r > 0, € p(A+ DAE) for all A € K'*4 with |[Al] <r}
Trang 9For 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,
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
"¬
We have the domain of exponential stability of this time scale is
Trang 10|
KEC, RA + ail + SÀ + ĐÌ,
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
|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
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
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
In this paper we have considered the exponential stability and given a formula for the stability radius of time-invarying linear dynamic equations with linear disturbance on time scales by giving the domain of exponential stability and showing the existence of a ’bad” perturbation whose norm is