Boundary control systems in factor form: Transfer functions and input-output maps

Integr. equ. oper. theory 41 (2001) 1-37 0378-620X/01/010001-37 $1.50+0.20/0 �9 Birkhfiuser Verlag, Basel, 2001

I Integral Equations and Operator Theory

B O U N D A R Y C O N T R O L S Y S T E M S I N F A C T O R F O R M : T R A N S F E R F U N C T I O N S A N D I N P U T - O U T P U T M A P S

P I O T R GRABOWSKI AND FRANK M. CALMER

Some facts from the theory of boundary control systems in factor form are recalled. The existence of a transfer function modulo a compatibili ty- and H ~ assumption is proved. General forms (standard and distributional) of the input-output map are given. Two detailed transmission line examples illustrate how the theory can be applied. The general form of the Hankel operator is given as well as its relation to the input-output map. It is shown that boundary control systems in factor form lead naturally to regular Salamon-Weiss abstract linear systems with a well- defined state-space description.

i. INTRODUCTION

In this section we present some facts of the theory of boundary control systems in factor form; for more detail see [16].

In a Hilbert space H with a scalar product {., ")H consider the SIS0 model of boundary control in factor form [16],

(1.1) { :~(t) = A[x(t)+du(t)] } y(t) : c#x(t)

We assume that A : (D(A) c H) ~ g generates a linear exponentially stable (EXS), Co-semigroup {S(t)}t>_o on H, d E H is a factor control vector, u E L2(0, c~) is a scalar control function, y is a scalar output defined by an A-bounded linear observation functional c # with D(A) C D(c#). Such a functional can be represented on D(A) as

(1.2) c#z = (Ax, h)H = h*Ax, x e D(A)

for h E H, h* = c#A -1.

Def in i t ion 1.1. The linear A-bounded observation functional c # is called admissible if there exists "y > 0 such that

o ~ <_ ~/ VXo E D(A) le s(t) xol dt Ilxoll

2 Grabowski, CaIIier

i.e. the observability map P : D(A) 3 x0 ~ > c#S(')x0 E L2(0, oo) has a bounded extension to H, denoted by P.

The following result is proved in [16, Theorem 4.1]. Here L and R = L* denote the generators of the semigroups of left- and right-shifts on L2(0, oe), viz.

{ L f = f ' , D(L)=Wl '2(O, oo) } (1.3) n f = - f ' , D(/~) W~'2(0, c~)

where W1'2(0, oo) = {f E L2(0, c@ : f is absolutely continuous, f ' ~ L2(0, oo)}, and Wo,t2(0, cc) = {f E W1'2(0, oo) : f(0) = 0}. In the sequel T(t) and T*(t) with t > 0 are respectively the right-shift- and left-shift operator given for any function f defined on IR+ by

{ 0, s~[0 , t) } (1.4) (T(t) f)(s) = f ( s - t), s E [t,oe) ' (T*(t)f)(s) = f ( s + t) .

V E L(H, L2(0, oo)) is given by

(Vz)(t) = h*S(t)x < > V*f = S*(t)hf( t)dt ,

where < > means "with corresponding Hilbert adjoint given by".

Theorem 1.1. Let c # be admissible. Then

{ } (1.5) R(V) C D(L), P = LV ( > P*lD(n) = V*R

P = V A < > R(V*) c D(A*), P * = A*V*

d [h*S(t)xo] with Laplace transtbrm (~Zo)(s) = In particular for x0 E H, (-Pxo)(t) =

c#(sI - A)-lxo.

The role of d in (1.1) motivates the next definition.

Definition 1.2. The factor control vector d E H is called admissible if

If/ ] (1.6) S(t)du(t)dt e D(A) Vu e L2(0, oo) .

The operator W given by

// (1.7) Wu := S(t)du(t)dt

belongs to L(L2(O, ec), H). Since A is closed and (1.6) is equivalent to the inclusion R(W) c D(A) then applying the closed graph theorem we get

(1.8) Q = A W E L(L2(0, co), H) .

The operator Q is called the reaehability map. Moreover the following is known ([16, Theorem 4.2]) with the reflection operator Rt given for any function u defined on R+ by

{ U(t--T), TE [0, t] } (1.9) (Rtu)(T) :-= 0, ~- > t "

Grabowski, Callier 3

(1.1o)

is a weak solution of

L e m m a 1.1. Let d E H be an admissible factor control vector and let u E L2(O, o0). Then the function

x(t) := QRtu = A S(t - 7)du(T)dT

= A[x(t) + } x(0) = 0

i.e. it is a continuous H-valued function of t such that for all y E D(A*), t ~ (x(t),y) d

is absolutely continuous and for almost all t and all y E D(A*) we have ~-~ (x(t), Y)H =

(x(t) + du(t), A*y)H.

The duality theory presented in [16, Section 4] can be abbreviated as follows

L e m m a 1.2. The observation functional c # in the observed system

{ ~:(t) : Ax(t) y(t) c#x(t) }

is admissible iff h, h* = c#A -~ is an admissible factor control vector for the dual controlled system

{ z(O)~(t) == oA*[z(t)+hf(t)] }

and similarily, d is an admissible factor control vector for the controlled system

{ == d< )l } iff d'A* is the admissible observation functional for the dual observed system

q(t) = d*A*z(t) "

A few words on perspective are now in order. For the boundary control system in factor form given by (i.i) we study successively representations of the transfer function, the input-output map, the impulse response and some transmission line examples.

In Section 2 we collect some results concerning the observation functional c #. In Section 3 we discover the transfer function ~ and (using distribution theory) the associated impulse response 9 and unit step response hs, and make contact with transfer function spaces. This leads to transfer function Theorem 3.1, where in particular we get by restricting the range of c # that hs has an initial Lebesgue value hs (0+) -- 0 (see also Lemma 3. I)(" regularity with zero feedthrough'). In Section 4 we show then that, despite the lack of admissibility of the control factor vector d, our boundary control system in factor form with an admissible observation operator, a compatibility condition and a transfer function in H~(Fi +) has a

well-defined input-output map (Theorem 4.1). We study also its input-output map as a convolution operator in the sense of distributions (Corollary 4.1), giving meaning to the impulse response and unit step response. This leads to input-output map Theorem 4.2. The

4 Grabowski, Callier

transmission line examples of Section 5 show that if the H ~ assumption is satisfied then usually more structure is available e.g. transfer function in the Callier-Desoer class.

We obtain also that our system with in addition an admissible factor control vector is an abstract linear system in the sense of Weiss [28, Definition 2.1] (Theorem 7.1) which is regular, and get its state-space- and frequency-domain description (Theorem 7.2). Prepara- tory properties of the input-output map and the Hankel operator are displayed (Lemmas 6.1 and 6.2).

We establish hereby links with established results (by e.g. Weiss, Lasiecka, Callier- Desoer and Grabowski) and get some interesting applications (e.g. impulse response descrip- tions of tranmission lines).

In what fbllows, for all T _> 0 and t > 0, P~ and P~,T+t are truncation operators given for any function f defined on P~ by

(I.II)

and

(1.12) 0, e [0, u + t,

All other notions will be declared along the way. In Sections 6 and 7 assumptions valid throughout are stated at the beginning of a section and not repeated in the statements of results.

2. OBSERVATION REVISITED

In this section we discuss the observation functional c # described on D(A) by (1.2) and assume that it is admissible. The objective is to study its properties as an extension of c#1D(A) = h*A for motivating a reasonable assumption on its range.

Recall here (e.g. [25, p. 168]) that any function f in L~oc(0 , oo) is such that

I ft+r lim - - ] I f ( t ) - - f (~ - ) ]dT=O for almost every t > O

r--+O+ 2r Jt-r whence for such t

f ( t ) = lira 1 f+rf(~)dr r-+0+ 2r jt_r

Note that for t = 0+ one must have

f (0+) = lim 1 fo~f(7_)dT r-+O+ r

Points t of this type are called Lebesgue points and the corresponding values are then naturally the Lebesgue values of f. Since such functions are defined almost everywhere they are defined by their Lebesgue values (which agree with the values of f at all points of continuity). This will be assumed from now on.

Now for Laplace transformable functions in L~oc(0 , oo), lira 1 ft t~o+ t f(~-)dT- = c implies ~ u

lira s f (s) = c, (Abelian result). See.g. [9, Theorem 33.4, p. 226] and [28, Propositions s-~,aOTsE~


5.1 and 5.4]. Moreover the converse holds under mild additional assumptions (Tauberian result), in particular

L e m m a 2.1. [28, Theorem 5.2, Corollary 5.5]. Let f E L~oc(0, oc) be such that e-~(')f E L2(0, oc) for some A E N and

fo' s u p If(7-)12d~ - < t>0 t

Then 1 t

lira - f f(~r)d~-=c r lira s f ( s ) = e . t-~O+ ~ Jo s--~,sER

A simple proof of Lemma 2.1 is given in Appendix A. As a consequence for any Laplace transformable function f in L~oc(0 , oc) one must

distinguish its Lebesgue value at 0+, i.e. f(0+) = lira 1 - ] r f(T)d% from another possible r--+0+ r J0

�9 ^

value at 0§ viz. s l imc~ sf(s), which (because of the Abelian result mentioned above) will

be called henceforth the Abel value of f at 0§ We note that the existence of the Lebesgue value implies existence of an equal Abel value and vice-versa under mild conditions.

Consider now the extended observability map P described by Theorem 1.1. Since P--xo E L2(0, oo) for x0 E H, we have Px0 E L~oc(0, oc) with Lebesgue value at 0§ given by

(2.1) (Px0) (0+):= lira 1_ f t ( ~ x o ) (~-)d'r . t-~o+ t Jo

1 t lim - f (Pxo)(~-)d~- = c. This case will be denoted Note that t-~o+lim (Pxo) (t) = c implies t-+o+ t Jo

(Pxo) (0+) := ]im (Pxo) (t). The Abel value of Pxo at 0+ is given by lira s(Pxo)(s). t-+O+ s-~oo,s6~

Consider now the unbounded linear functionals

C#LXO = lim -~c # S(@xoda, D(C#L)= xo E H: 3 lim h-~o+ h~o+ ~c S(~)zodo (2.2)

and

(2.3) c#x0 = lim c#;~(AI- A)-lXo, D(c#A) = {Xo E H: 3 lim e#A(AI - A)-lx0}

which are called, respectively the Lebesgue extension and the Lambda extension of C#ID(A ) = h'A, see [27, Section 4], [28, Sections 2 and 5] and [29, Section 5] (where the "Lambda" notation appears).

L e m m a 2.2. Let c # be admissible. Then

(i) c # is an extension of C#L, such that C#1D(A ) = h*A C C#L C c#A. (ii) For all x0 E D(A), C#Xo = lim (Pxo)(t).

t-~0+

/o (iii) xo E D(c~) ~ 3 t~o+lim ~ (Pzo) (T)d~- and c~xo = lim _1 t (Pzo) (T)d~-. t--+0+


A lira s(Pxo) s). (iv) Xo e D(cea) r 3 lim s(-fiZo)(S) and C#AZO = s-~,sea " "(

Proof. (i) follows by [28, Remark 5.7]. (ii) holds because by Theorem 1.1 for Xo ~ D(A), (Pxo)(t) = h*S(t)Axo is continuous at 0+. For iii) observe that by (2.2) and (1.2)

lira -h*AflS( )xodT = l i r a h*S(t)Xo-h* o CL~Z0 t-+o+ t Jo t-40+ t

and the result ibllows by Theorem 1.1. For (iv) see (2.3) and Theorem 1.1. D

L e m m a 2.3. Let c # be admissible.

(i) (Pz)(t) = C#LS(t)x for almost every t > 0. (ii) Let w < 0 be the growth constant of the E X S C0-semigroup {S(t)}t>_o on H. Then tbr

any (z E (co, 0]

(2.4) e -at (-PXo) (t) E LI(0, oc)v/L2(o, oc) for any xo E H .

Proof. (i) holds by Theorem 1.1 and (2.2). For (ii) observe that by Theorem 1.1 (Pxo)(s) = c # ( s I - A)-~Xo . (2.4) holds then by the proofs of [6, Lemma 3.5, Lemma 3.7] where V := H, W := D(A) equipped with its graph norm (here equivalent to ][Ax[[ H tbr any x E D(A)) and CS( t )B := c#r S(t)xo = (-fiZo) (t). []

3. THE TRANSFER FUNCTION, STEP AND IMPULSE RESPONSES

Throughout this section we assume that c # is an admissible observation functional. We shall not assume that d is an admissible factor control vector as is true in some examples.

Now, multiplying the first equation of the system (1.t) by A -~ we get

(3.1) dsd (A_lx(t)) = A_l~c(t) = x(t) + u(t)d = A(A- lx ( t ) ) + u(t)d

If u E D(R) then (3.1) has a strong solution for x(0) = 0, given by

(3.2) z(t) = A S(t - "r)du(-r)d~" ,

coinciding with the weak solution of (1.10). Indeed, by Fubini's theorem applied to the integral on the right-hand side and by [24, Corollary 2.9, p. 109]

(3.3) S(t - 7)di~(7)dT = A S(t - 7)du(7)dv + du(t) = -~ S(t - 7)du(7)d7 ,

where the last equality holds almost everywhere. Applying the Laplace transform to (3.2) we get

So(s) = A(s I - A)-~dg(s) = [s(sI - A) - ld - dig(s) .

If the compatibility condition

(3.4) d e D(c #)

holds then :~(s) ~ D(c #) and thus

= : =


is well defined, where

~(~) = ~ # ( ~ - A ) - l d - c # d = ~ # A ( s I - A ) - I d =

(3.5) = sh*A(sI - A)- ld - c#d = s(Pd)(s) - c#d

will turn out to be the transfer function of system (1.1). We use now distribution theory, see e.g. [31]. Extending P d onto ~ by setting it zero for t < 0 and interpreting P d as a regular distribution with support in [0, oc) (dearly P d E L~o~(0, cx~) we find

= V(~d~) - ~ # d ~ =

where :D denotes distributional derivative, 6 stands for the Dirac distribution at zero,

(3.6) h~ := P d - c#d~ ,

and the Laplace transform is meant in the distributional sense. Here 11 C L~o~(0, oo) is the unit step function. The function h~ and its distributional derivative

(3.7) 9 := ~h~ = ~ ( P d ) - c#d6

will be called respectively the unit step response and the impulse response of the system (1.1).

R e m a r k 3.1. It is important to see that on t > 0, h~ is Pd vertically shifted by - c#d and that 9 is the distributional derivative of h~ and on t > 0 of Pd.

Our next result is motivated by the fact that in many examples (see Section 5) the inclusion (3.8) below is fulfilled.

L e i n m a 3.1. Assume that c # is admissible, (3.4) holds and

(3.8) ~# c c~ ,

giving by Lemma 2.2 the consistent chain of extensions

(39) c#l~(~ ) = h*A c e# c ~ c ~,~

Consider moreover the functions .#(s) and h~(t) given by (3.5) and (3.6). Then

(3.1o)

and

(3.11)

c # d = ( P d ) ( O + ) : = lira (-fid)(7)d~-=C#Ld= lira s(Pd)(s)=c#Ad t--+O+ b s--+c~,sER

0 = h s (0+ ) := lim 1 foths(T)d7 2~meR~(s ) =: ~ ( + ~ ) t---~ 0+ t s

Proof. (3.10) follows by (3.4) and (3.9), upon using (2.1)+(2.3) with x0 = d. (3.11) follows by subtracting c#d from (3.10), using (3.6) and (3.5). []

Observe that since Pd, u E L2(0, oo) the convolution P d * u is a uniformly continuous bounded function which tends to 0 as t tends to zero or infinity, and

( ~ d . ~)(s) = (~d)(s)~(s)


Thanks to this we have

(3.12) [ ~ ( ~ d . ~ ) ] - ( s ) = s ( ~ d ) ( ~ ) ~(~), e#d ~ = c#d ~ = c#d ~ . ~

Hence by (3.5), (3.12) and (3.6)

~(s) = ~(s)~t(s) = s(-fid)(s) ~(s) - c#d~t(s) =

= [ ~ ( ~ d . ~ ) ] - (~) - [ D ( c # d ~ . ~)] ~ (s) = [ ~ ( h ~ . ~)] ~ (~) ,

whence upon noting that :D [(Pd) * u] = [T~(Pd)], u (see [31, p. 132]) we obtain

~(~) = [ ~ ( ~ d ) . ~] A (s) _ [ z ~ ( c # d ~ ) . ~] A (~) = [ ( ~ h ~ ) . ~)] A (~) = (g .1--)(~)

Consequently we have

(3.13) y = g * u = (~9h~) * u = ~(-f id * u) - c# du .

In what follows H~(II +) stands for the Hardy space of functions ~ analytic and bounded on the complex right half-plane II + = {s E C : Re s > 0} equipped with the norm ][~][H~(r~+) := sup ]~(s)l. H~(II +) is a multiplication Banach algebra and by the maximum

sCII+

modulus principle, e.g. [4, p. 129], IIpl]ii~(r~+) = ess sup ]p(jw)], ~ ( j w ) := lira ~(a + j w )

for almost all ~ E R. See also [18, Section 19.2]. Next, let H2(H +) be the Hardy space of functions ~ analytic on II + such that

sup I ~ ( ~ + j ~ ) l ~ d ~ < o 0 , or>0 J - o o

equipped with the norm

[l~ll~<~+) := kc(Jw)[ 2 dw ,

where qo(jw) := lim+ ~p(a + jw) almost everywhere. By the Paley-Wiener theory, e.g. [19,

p. 131], L2(0, oe) and H2(II +) are isomorphic Hilbert spaces through the Laplace transform 2 using llfll~,<~+) = 2~ IlfllL,(o,oo).

The input-output map is now represented in the frequency-domain by a multiplication operator

H2(YI +) ~ ~, > ?)(-), ~)(s) = ~(s)~t(s)

or in the time-domain by a convolution of distributions (3.13). If

(3.14) ~0 C H~(II +)

then the above multiplication operator is everywhere defined and bounded on H~(H+), be- c a u s e

II~tlH~(~§ < 11~11~<~§247 w e He(E +) .

In Section 4, this fact will be exploited to find a time-domain version of the input-output map within the space L2(0, oo), i.e., a nondistributional meaning will be given to (3.13).

We continue this section by giving some sufficient conditions for ~ E HC~(II+). In the sequel we shall denote by BV[0, c~) the class of functions of bounded variation on [0, ~ ) . Recall that a scalar valued function f belongs to BV[0, c~) iff its total variation function T]


is bounded on [0, oc). T I is defined by Ti( t ) := sup ) s I f ( t j ) - f ( t j -1) l where the supremum j = l

is taken over all n and over all choices of {tj}j~__l such that 0 _< to < tl < . . . < t~ = t. Tf is oo

nondecreasing and the total variation norm of f given by V a r f := t---.oolim Tf( t ) exists and is

finite. For more details on the class BV[0, oo) see e.g. [25, Chapter 81. It is important that BV[0, oc) equipped with the total variation norm is a Banach space and that its thnetions are generators of finite scalar-valued Borel measures on t >_ 0 (related to Stieltjes integrals).

Fac t 3.1. Assume that (3.4) holds and that Pd c BV[0, oo), then (3.14) holds.

Proof. Since Pd E BV[0, oc), Pd has one-sided limits at every t E [0, oc), whence the limit (Pd) (0+) := lim (Pd) ( t ) exists. Moreover, (Pd) (oc) := l i ~ (Pd) (t) exists too. Hence

t - + 0 +

for Re s > 0, integration by parts yields A

[ ( p d ) (t)] = - ( 0 + ) +

Next using (3.5), we get for Re s > 0,

= e- 'e + +

Consequently, there holds: II~]]H~(,+) _< V~rPd + I(Pd) (0+) - c#d I < oc. []

Recall [2, pp. 652 - 653], [3, pp. 81 - 84], [8, pp. 337 - 338] that a scalar- valued Laplace transformable distribution f with support on [0, oc) is in the Wiener class A(~r) for some a E ]R if f ( t ) = f~(t) + f~ ( t ) for t _> 0 with e-~(')f~(.) E L~(0, oo) and

c~

f~ ( t ) = ~-~f id( t - ti), where 5 denotes the Dirae delta distribution and to = 0 and ti > 0 i = 0

oo

for i > 0 are such that ~ e -~t~ [fil < oc. Such distribution is in the Callier-Desoer class i = 0

A_(0) if it is in A(a) for some a < 0. A(~r) and A_(0) denote the classes of Laplace transforms of such distributions. A(cr) is a convolution Banach algebra with norm

[Ifll~(~) := + l id i=0

For cr _< 0, [2, p. 652], [3, Fact 2], J~(cr) has elements ] that are analytic in Re s > cr and

bounded and uniformly continuous in Res > a, with sup f ( s ) <_ IIfNa(~)- Hence tbr R e s > ~ ,

a < 0, ~(cr) is a subalgebra of Ha(H+). For more information see [3] or [8].

Fac t 3.2. Assume that (3.4) holds with ~ given by (3.5), then A

(i) Pd E A_(0). More precisely, let w < 0 be the growth constant of the E X S Co- semigroup {S(t)}t>_o on H, then

(3.15) e -~t (Pd) (t) E LI(0, oo) N L2(0, oo) Va e (w, 0]


whence

(3.16) ~ d e d ( ~ ) c H ~ ( n +) Vo e (~, o] .

(ii) g C ,4(0) implies Pd e BV[0, co), (iii) ~ e A(0) implies (3.14).

Proof. (i) follows by Lemma 2.3(ii) with xo = d. Using (3.5) we have ~(0) = - c # d and

(3.17) (~'-d) (s) - 0(s) + c#~ _ ~(s) - 0(0) S S

oo

If g e ~4(0) this gives VarPd _ Ilgll~(0) + Ic#dl and (ii) follows. [] 0

Remark 3.2. Assumption (3.14) is plausible because d E D ( A ) would imply that D(s) = c#(s I - A ) - ~ A d E H~176 +) because then by Theorem 1.1 and Lemma 2.3 e-~tg(t ) = e -~ tP(Ad) E L~(0, co)M L2(0, co) for any cr E (w, 0], where w < 0 is the growth constant of the EXS C0-semigroup {S(t)}t>>_o on It.

Lemma 3.2. Let c # be admissible and let (3.4) hold. Then Pc/ E H~(II+), whence the convolution operator K, K u := (Pal) * u belongs to L(L2(O, co)). Moreover, K ( D ( R ) ) C D(R).

The main results of this section can be summarized as tbllows.

Theorem 3.1. Given system (1.1) with an admissible observation functional c #. Assume that assumptions (3.8) and (3.4) hold. Then the function ~(s) given by (3.5) is well defined with Pd E L2(0, co) such that (3.15), (3.16) hold. It is the Laplace transform of the distribution g described by (3.7), (3.6). Moreover in (3.5) c# may be replaced by cL e or c~. ghrthermore lim t)(s) = 0 and the function hs(t) given by (3.6) satisfies hs(0+) = 0.

s--+c,o,sEN:

Proof. The claim is straightforward consequence of the assumptions, Lemma 3.1 and Fact 3.2. []

4. THE INPUT-OUTPUT MAP

We continue to assume that c # is an admissible linear functional.

L e m m a 4.1. Assume that c # is admissible and that (3.4) and (3.14) hold. Then the input- output map F is a densely defined bounded linear operator from L2(0, co) into L2(0, oc), given by

(4.1) y(t) = (Fu) ( t ) = h*S(t - v)dii(T)dT - h'dis(t) - c#du(t ) , D ( F ) = D ( R 2) .

Hence F is closable and its closure T belongs to L(L2(0, co)).

Proof. If u C D ( R ) , then by (3.3) we have

x(t) fo fo = A S ( t - 7)du(T)d'r = S ( t - T)dis(~-)dT -- du(t) =

dfo' = d~ S ( t - 7)du(T)dT - du(t) .


If u e D ( R 2) then applying Fubini's theorem and [24, Corollary 2.9, p. 109] we get

/o /o where the last equality holds almost everywhere. Hence for almost every t >_ O, x( t ) E D(c #) and

y(t) = c#x( t ) = h*S(t - 7)d~(7)d7 - h*dit(t) - c#du( t ) .

The first term is in L2(0, oo) as the convolution of an exponentially decaying function h*S( . )d which belongs also to D(L), and /~ e L2(0, cx~). The remaining terms clearly belong to L2(0, oo). This means that the input-output map given by (4.1) is densely defined. For the last assertion observe that by the Laplace transform,

9 ( s ) = = =

where (3.14) holds. []

We wish to find the exact form of F. To do this we need two auxiliary results. The first one is a consequence of [26, Theorem 4.14, p. 68, Theorem 5.2, p. 89 and Theorem 5.3, p. 90], while the second one follows from the proof of [26, Theorem 4.19, p. 72].

L e m m a 4.2. Let the operator T : (D(T) C H) ~ H be densely defined. Then T is bounded iff T* e L(H). Moreover, under one of these equivalent conditions the unique continuous extension of T to the whole space H is T = T**.

L e m m a 4.3. Assume that T, : (D(T1) C H) > H is a densely defined linear operator. Assume also that T2 E L(H). Then f E D ((T2Ti)*) = ~ T~f E D(TT) and (T2TI)* = T~T~.

T h e o r e m 4.1. Assume that c # is admissible and that (3.4) and (3.14) hold. Then the eztended input-output map, i.e. the closure of F, is given by

(4.2) R ( K ) C D ( R ) , F = - R K - c#d I ,

whence for u E L2(0, c~)

(4.3) (-Fu)(t) = ~ (-rid) ( t - T)U(T)dT -- c#du( t ) for almost all t > 0 .

Moreover, ~(s) given by (3.5) is the transfer function of system (1.1).

Proof. Integration by parts, which is possible because h*S( . )d E D(L), together with (1.5) yields an extension of F onto D(/~),

(4.4) (Fu) ( t ) = (-rid) (t - 7)it(~-)dT - c#du( t ) .

Observe, that (4.4) can be rewritten as

- F u = K R u + c#du, u E D ( R ) ,


where F and hence K R are densely defined and bounded. Set H = L~(0, oo) and T = F, T1 = R and T2 = K in Lemmas 4.2 and 4.3. By Lemmas 4.2 and 3.2, F*, (KR)*, and K are in L(H). Hence by Lemma 4.3

g * f E D(R*) = D(L) Vf e L2(0, oc), - F * = R ' K * + c#dI = LK* + c#dI .

Using ~bini ' s theorem we easily find the form of K*,

f- ( K * f ) ( t ) = (-Pd)(r - t ) f ( r ) d r , f ~ L2(0, oo) .

Hence for almost all t > 0 and v E L2(0, oo)

dr. -(F*v)(t) = ~ (-Pd)(7 - t )v(r)d7 + c#dv(t) =

d -Ji" ~ (-~d, T*(t)v}L~(o,~ 1 + = ~ (~d)(~)v(~ + t)d~ + ~#dv(t) = 7~ c#dv (t)

where T(t) stands for the semigroup of right-shifts operator (the adjoint semigroup operator T*(t) corresponds to left-shifts). Now, observe that for v E D(L)

/7 -- * c#dv(t) (-fid)(r)v'(r + t )dr + c#dv(t) - (F*v ) ( t ) = (Pd, T (t)Lv}L~(o,~) + = =

= f~ - t )v ' (r )dr + c#dv(t)

i.e. - F * v = K*Lv + c#dv for v E D(L). Applying Lemmas 4.2 and 4.3 once more, now with T = F*ID(L), T~ = L, T2 = K*, we get the general form of F given by

R ( K ) C D(L*) = D(R), - F = L*K + c#dI = I~K + c#dI .

This yields (4.2) and in particular (4.3). This means that the derivative in (4.4) can be extracted outside the integral giving the extended input-output map. Applying the Laplace transform to (4.3) gives

~ ( s ) = ~(s)~(s), ~ e H~(H +)

and ~ is given by (3.5). Hence the transfer function of system (1.1) is given by (3.5). []

To find a link between the representations (3.3) and (3.13) we consider the Frdchet space L~o~(0 , oo) c L~o~(0, oc) with topology induced by the countable family of semi-norms {NP~fllLa(0,oo)},,eN and observe that by (4.3) F is causal, i.e. for all r > 0: P~-fiP~ = -FP~.

Hence F e L(L2(0, oc)) extends to T e L(L~or oo)) by the formula

(4.5) Fu = lira Pr-FP~u, u E L~oc(0, oo) . T--+.

As a consequence, the system unit step response h, is well-defined as h~ := F11 ~ L~o~(O, oo). From (4.5) and (4.3) one obtains (3.6). We use now again distribution theory. Consider the input-output map (4.3) under the conditions of Theorem 4.1. Extend all functions appearing in (4.3) onto 1R by setting them zero on t < 0. Finally, consider convolution in the sense of distributions. Then the functional equation in (4.2) can be made to read

F% = ~ [(Pd)* q - (c#d)u = ~ [(Pd)* u] - (c#d)(5-u) ,


where " , " denotes the convolution and all functions belong to L~oc(0 , oo), whence they can be seen as regular distributions. Hence upon noting that 7:) [(-rd) * u] -- [D(-rd)] * u (see [31, p. 132]) and 5 = :Dll, we get

C o r o l l a r y 4.1. Let the assumptions of Theorem 4.1 hold. Then the input-output map, given by (4.3), is a convolution operator in the sense of distributions, i.e.,

(4.6) F u = g * u, u E L 2 (0, oo) ,

where g is the impulse response given by (3.7) with the Laplace transform being the transfer function ~ given by (3.5). Moreover, 9 = Dhs and h~ := Fll is the unit step response of system (1.1)coinciding with (3.6).

The main results of this section can be summarized as follows.

T h e o r e m 4.2. Given system (1.1) with an admissible observation fhnctional c #. Assume that assumptions (3.8), (3.4) and (3.14) hold. Then the input-output map is given by

(4.3) and the transfer function is given by (3.5), with both Pd(s ) and sPd( s ) in H~176 Moreover the input-output map is a convolution operator in the sense of distributions given by (4.6) with impulse response 9 given by (3.7), and unit step response h, given by (3.6). Finally the impulse response, transfer function and unit step response satisfy:

(4.7) g = 7)(-rd) - [ ( - rd)(0+)] J and ~(s) = s ( - rd) (s ) - (-rid) (0+) ,

with

(4.8) 9 -- ~ h s where hs = - r d - [(-rd) (0+)] ]1 ,

where c#d = (Pd) (0+) is the Lebesgue value given by (2.1).

Proof. The statements follow by Theorem 4.1 and Corollary 4.1. []

We emphasize that we did not assume admissibility of a factor control vector d. In Subsection 5.2 we shall present an example of a physical system satisfying all assumptions of Theorem 4.2 and therefore with H ~~ (II+)-transfer function, and yet with a nonadmissible factor control vector d.

R e m a r k 4.1. The results above are related to those of [10, Proof of Theorem 6-5, pp. 298 - 303]. Therein it is assumed that the LTD+-distribution g is singular and the density of the Borel measure of -rd E BV[0, oo) giving with (-rd) (0§ = c#d,

/0 (-Fu)(t) = [-rd(T)]u(t - "r), ~(s) = d[-rd(t)]e -st . +

oo

For the latter compare with (3.15), (3.16). For g E A(O), g -- 9~ + E g i h ( . - ti) with t~ > 0, i=1

such that

fj (Fu) ( t ) = g~(t - r)u(T)dT- + ~_~gdl(t - t~)u(t - ti) , i=1

fo ~(s) = g~(t)e-Stdt + ~_, 9ie -~t~ i=1


5. EXAMPLES In this section we discuss two examples of electrical transmission lines depicted in

Figures 5.1 and 5.2 illustrating the results of previous sections.

5.1. R L C G - t r a n s m i s s i o n line. Consider a distortionless RLCG transmission line, i.e. c~ := R/I_ = G/C loaded by a resistance Ro and depicted in Figure 5.1.

10, ]! ,~ 0 I 0 [1 ~ 0

l RLCG i(1,t) TRANSMISSION ~ ~ O

LINE l ~ RO ~(o,t) v(1,t)

. O

O RC

TRANSMISSION LINE

| l l l l I I

~(0, t) [ v(1, t)

i(1, t) = 0

O FIGURE 5.1. Loaded distortionless RLCG - transmission line

FIGURE 5.2. Unloaded R C - transmission line (W. Thorn- son's cable of a finite length)

The control is an input voltage and the observation is an output voltage. We ~snme null initial conditions. The system is governed by the partial differential equations

l_it(O,t) = -vo(O,t)-Ri(O,t), t>O, 0 < 0 < 1

Cvt(O,t) = -ie(O,t)-Gv(O,t), t>_O, 0 < 0 < 1

i(1, t)Ro = v(1,t), t > 0

~(t) = v(0,t), t >_ 0 ~(t) = v(1,t), t_>o

The d'Alembert solutions of the first two equations are

i(O,t) = e -~tr162 } (5.1) 2z

v(0, t) = e - ~ r 1 7 6 - ~t) + r + ~t) 2

where r r are arbitrary sufficiently smooth functions, and v = 1 / x / i t = l / r , z = v/L-/C. Here L, denotes the velocity of wave propagation and z stands for the wave impedance of a line. Substituting (5.1) into boundary conditions, we get the system of functional equations

r = ~ r )

2~(t) = e-~[r + r / 2y(t) = e-~[r - ~t) + r + . t ) ]


where ~ = (Ro - z)/(Ro + z) is the reflection coefficient. Introducing the new variables

~ ( t ) = ~ e - ~ r ' one obtains

(5.2) { w(t) = Csw( t - r )+u( t )bo } y(t) = ~ o ~ ( t - ~)

with

i+~ where a :----

P Then

[o 1] bo=[O] [o] Cs = -b 0 ' 1 ' Co = , a

t~ e a r . > O , b : = - - [ b l < 1, p : = Let(x( t ) ) (O):=w(t+O), t>O, OE[-r,O]. - - p 2 ~

a[=(t)](o) _ a w ( t + e) _ a ~ ( t + o) _ a[=(t)](e) Ot 8t 80 00

In the Hilbert space H = L2(- r , 0) @ L2(- r , 0) with standard scalar product, the system dynamics equations (5.2) can be written in the abstract form

Y = c# x x2

where

O'X : X t 1 D(o) {x~Wl,2(-~,0) eW' ,~(- r ,0) : x~(0)--x2(-~)} '

~-x = x2(0) + bx~(-r) , D(~-) D C[-~, 0] e C[-r , 0] D D ( ~ ) ,

and ~ is a closed linear operator, and

(5.3) c% = Co%(-~), D(~#) = {x ~ H: ~o~X is right-continuous at - ~} .

To obtain the final model of boundary control in the form (1.1) we take A = ~Ikerr and find the factor control vector d E D(cr) satisfying (Td = 0, rd = - 1 [20, p. 252 and p. 258], [21, pp. 772 - 773], [22, Section 4], and [16]. The idea is then that

~(t) = ~ ( t ) + o d d ( t ) = oi l ( t ) + d~(t)]

where, with x(t) and d necessarily in D(a) , Ix(t) + du(t)] e D(A) because

~[~(t) + d~(t)] = ~ ( t ) + ~d~(t ) = ~ ( t ) - ~( t ) = 0 .

Hence k(t) = A[x(t) + du(t)]. Elementary calculations yield

(5.4) d = -t+odo, do= 1 EH ,

where 1 denotes the constant function taking the value 1 on [ - r , 0] and

(5.5) D(A) = {~ e W',~(-~ -, o) �9 W~'~(-~, 0): ~(0) = C ~ ( - ~ ) } �9


Finally, the system is described by (1.1) with A given by (5.5), d given by (5.4) and c # given by (5.3). Notice also that (1.2) holds with

h = , ~ [ b l ] a - 1 EH, ~9:= l + b

The operator A generates a Co-semigroup {S(t)} t>o on H (or even a Co-group if detCs r 0). This semigroup is EXS iff ]A(Cs)I < 1 or equivalently tb] < 1 [11, pp. 148 - 154], which is the case. Solving tile boundary-value problem

we determine the resolvent ( s I - A) -~ of A [13, p. 357],

~(0) = ( ( s I - A)-Zxo) (0) =

(5.6) { f 0 = ~o ~ 0 ( ~ ) ~ - ~ + [ ( i _ ~ - ~ c ~ ) - l _ i ] ~ ( ~ ) } , 0 E [-r,O]

/_o where zo E H and ~ denotes its finite Laplace transform xo'~-~(s) = zo(O)e-~~ Hence

r

( 5 . 7 ) ( S ( t ) x o ) ( O ) = x ( t ) ( O ) = w ( t + O ) , t>_O, 0 c [-r, 0] .

/0 /0 (s ( t )xo) (e )~-~dt = w( t + O)e-~tdt = w(~)~-~(~-~ = (5.s)

/o ~ = ~o xo(~)~-~e~ + ~'e(s) .

The unique solution of (5.2) with initial condition w(t) = xo(t) , t E [ - r , O] and null control u is

(5.9) ~ ( t ) = c ~ + % ( t - k~ - ~), t ~ (k~, (k + 1)~), k = o, 1, 2 , . . . .

Therefore, using ]A(Cs)I < 1 we obtain

~(s ) = e-~tw( t )d t = e-~tC~+lxo(t - k r - r )d t = k = 0 kr (5.10) (3O

= e - ~ e C s zoo(s ) = e - ~ ( I - e -~r~ ~ - l C ~ ' " k = 0

Substituting (5.10) into (5.8) and comparing the resulting formula with (5.6) one concludes that the Laplace transform of S( t )xo coincides with ( s I - A)-lz0 which proves that (5.7) holds.

Admissibility of the observation functional c # was implicitly discussed in [13, p. 363]. Here we give the Lyapunov proof of this fact. To be more precise, a necessary and sufficient condition for admissibility of c# is the existence of a bounded~ self adjoint, nonnegative solution of the Lyapunov operator equation [t2, Theorem 3, p. 322]

(5.11) (Ax , 7-lX)H + (x, n A x ) H : --xT(--r)CoCT X(--r) VX e D ( A ) .

Indeed,


The operator

x~( , ) '

where H C L(N2), H = H r _> 0 uniquely solves the discrete matrix Lyapunov equa{ion

(5.i2) C ~ H C s - H = -CoC~o ,

is a unique solution of (5.11) (uniqueness follows from EXS, by the results of [12, Theorem 4, p. 323]). Moreover, the pair (Cs, c T) is observable if b r 0 and in this case the unique solution of (5.12) is positive-definite. Hence 7{ is coercive which proves that the system is exactly observable [15, p. 2], provided that b # 0.

By (5.6) we have

(5.13) (Pxo ) (s ) = c # ( s l - A ) - i x o = cT ( (s I -- A ) - I X o ) ( - - r ) =

= e-'rCTo [I + e - ' r ( I - e - s 'Cs ) - lCs] ~oF(S), S e H +, Xo e H .

On the other hand by (5.8)

fo A [ 4 % ( - - ~)]A(s) = co%(t - ~)e-~'dt = e -st [Co XoF(8) + Co~(s)] = ( ~ o ) ( 8 )

and therefore if Xo E H then

(-fiXo)(t) = c"~w(t - r) for almost all t _> 0 .

In particular, using (5.9) we conclude that given any z0 E H then

(5.14) (P--xo)(t) = cTxo(t-- r) for almost all t e [0,r] ,

and thus

lira (Pxo)(t) = cToxo(--r) = C#Xo Vx0 e D(c #) . t-+O+

Using Lemma 2.2 this implies that assumption (3.8) holds and thus also the chain of inclusions (3.9). Using Lemma 2.2 and (5.14) the Lebesgue extension CL ~ is described by

C#LXO = lim l f 0 t 1/0t t-+0+ t (Pxo) (~')dT = lira cTxo(Z: -- r)d~- = t-cO+ (5.15)

= l im -I " ' /-r cYxo(O) dO Vxo E D(C#L) t-+O+ t d _ r

where

D(C#L) = {xo E H : cTxo hasaLebesguevalueat O = - r } =

1 - r+0 T


It follows from the Cauchy-Schwarz inequality that lim se-2~Tz~fiF(s) = 0 tbr any :Co E H,

whence by Lemma 2.2 and (5.13) the Lambda extension cA # is described by

C~Xo = lim s (Pxo](s )= lira se-~rcT~oF(S)= (5.16) ,-,oo,,e~

= lim s [P~-Pxo]^(s) Vx0 e D(c#A)

where

D(c#A) ---- xo C H : 3 lira s(Pxo)(s) = :co E H : 3 lira se-~T cT~or(S .

Evidently d given by (5.4) is not in D(A) but (3.4) holds, i.e. d C D(c #) with

c#d = -zg. Setting z0 = d in (5.13) one gets with ~ the finite Laplace transform of d

s(-P~-d)(s) -~)(1 . . . . 1 - be- ~r se_~cT~FF(S) 1 -- be - ~ = - ~ J i ~ - b T ~ - ijb~-~----~

By (5.15) and (5.16) d is in D(c~) and D(c#a) where

1 f t-~ ceLd= lim 1 t-~c~d(O)dO -tgt_~o+ t J-r t~0+t- r = l i m - I d O = - ~ ,

and

c#Ad= lim s (Pd) ( s )= lim se-~c~d'FF(s)=--~. s-+oo,sCN S-+o%sC~

Hence identity (3.10) of Lemma 3.1 ho]ds. From (3.5) we calculate the transfer function of the system

ae-sr (5.17) O ( s ) - t + b ~ - ~ T ~

Since [b[ < 1, [e-Sr I = e -rP~es and 1 - [b[ e -2rRes = 1 - Ibl ]e-U~ I < ]1 + be-2~ I for all s E C then

a

I~(sDI _< - - v8 e n + 1 -Iv l

�9 ^ Hence ~ C H~176 i.e. (3.14) holds, and moreover l i m e g ( s ) = 0. Hence since (3.8),(3.4), and (3.14) hold, the conclusions of Theorem 4.2 hold. In particular the input -output map given in (4.3) is well-defined, which was proved also in [13, p. 363] with the aid of another method.

The situation is even better: g E ,4_ (0). Interpreting (5.17) as the sum of geometric series we obtain

oo

(5.1s) g(t) = Z a ( -b )~ [ t - (2i + 1)r] , i = 0

and for a E ( 1 in ]bl, 0]

i = 0

a e - ~ r

1 -Ibl e-~"


By Fact 3.2 Pd satisfies (3.15), (3.16) and ~, Pd are analytic in an open right half plane strictly containing H + and bounded and continuous in its closure. Moreover Pd ~ BV[0, oc). To see this, note that by (5.17) and (3.17) with ~(0) = - c # d =

- - ( P d ) ( t ) - ~ ~ X [ o , r ) ( / ~ ) - J i - ~ Z ( - b ) i x [ ( 2 i - 1 ) r , ( 2 i + l ) r ) ( t ) ,

i = l

with (Pd) (0+) = - ~ = c#d, and for i _> 0

(Pd) [(2i + 1)r§ - (Pd) [(2i + 1 ) r - ] = a ( - b ) i .

Hence P d E L2(0, ee) is a piecewise constant function having g l -summable jumps, whence in BV[0, oc). It may have both oscillatory and monotonic type. Note tha t by Remark 3.1 P--d is the step response hs shifted vertically by c#d. The plot of hs is depicted in Figure 5.3.

0 . 8

0 . 7

0 . 6

0 . 5

0 . 4 . . . . . . . .

0 . 3 . . . . . . . .

0 . 2 . . . . . . .

0 . 1

~ o

I i

. . . . . . . I . . . . . . . . , . . . . . . . . . . i . . . . . . . . ' . . . . . . . . ' . . . . . . . . . " . . . . . . . . . " . . . . . . . . . . " . . . . . . . . . . . . . . . .

I I

i

. . . . . . . . . I . . . . . . " . . . . . . . . . . i . . . . . . . . . . " . . . . . . . . " . . . . . . . . . . " . . . . . . . . . . ' . . . . . . . . . . �9 . . . . . . . . " . . . . . i

. [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

2 3 4 5 6 7 8 9

FIGURE 5.3. Plot of the step response hs for r = 1, a = 3/4, b = 1/8 (dashed line) and for r = 1, a = 1/4, b = - 1 / 8 (solid line)

Note especially that identity (3.11) of Lemma 3.1 holds and that g as given by (5.18) is on t > 0 the distributional derivative of Pd as was to be expected from (4.7).

In accordance with Lemma 1.2, admissibility of the factor control vector d can be examined using the auxiliary dual system

[. q = d*A*pJ

where A* denotes the adjoint operator of A,

A*p = -p ' , D(A*) = {p e Wl '2 ( - r , 0) @ Wl ' 2 ( - r , 0 ) : p ( - r ) = C~'p(0)} .

The observation can be calculated explicitly. For p E D(A*) we have

q = d * A * p = - ~ 1 ' p~ I-I


Now we prove that this observation functional is admissible. From [12, Theorem 3, p. 322, Theorem 4, p. 323] we know that a necessary and sufficient condition for admissibility is the existence of a bounded, self-adjoint, nonnegative solution of the Lyapunov operator equation

(5.20) (A'p, 7-lp)H + (p, 7tA*p)H = --p~(O) Vp e D(A*) .

[ pl(r]) ] , where H E L(N2), H = H r > 0 uniquely solves the The operator (7/p)(7]) = H L p2(~?) j

discrete matrix Lyapunov equation

C s H C T - H = -bob T ,

is a unique solution H _> 0 of (5.20). The pair ~(CTs, bT'~o/is observable and thus H > 0, whence H is coercive. This means that the system (5.19) is exactly observable [15, p. 2].

5.2. R C - t r a nsmi s s ion line. Following [12, p. 334] consider the system consisting of an Re (resistive-capacitive) unloaded transmission line with zero initial conditions depicted in Figure 5.2. The control is an input voltage and the observation is and output voltage. The system is governed by the partial differential equations

o = - v o ( O , t ) - R i ( o , t ) , t _ 0 , 0 < 0 < 1 ]

Cvt(O,t) = -io(O,t), t >_ O, 0 < 0 < 1 i (1 , t ) = o, t >_ o

~(t) = v(O,t), t >_ o

y(t) = ~(1,t) , t > o

The substitution x(O, t) = v(O, ROt) reduces the system

(5.21)

equations to the form

{ xt(O,t) = zoo(9, t) t k O , 0 < 9 < 1 xo(1,t) = O, t_>O

~(t) = ~(0, t ) , t >__ o

y(t) = x(1 , t ) , t > 0

In the Hilbert space H = L2(0, 1) with standard scalar product, the system dynamics equations (5.21) can be written in the abstract ibrm

I X = aX TX --- U

y = C#X

where

G-X ~ X ;!

~-x = z(O),

(5.22) ~#~ = ~(1),

} DW) = {x e Hff0, i ) : x'(1) = 0} , D(T) D C[0,1]DD(a) ,

D(c #) = {x e L2(0, 1): z is left-continuous at 1} D C[0, 1] ,

and a is a closed linear operator. The factor control vector d E D(a) satisfies ad = 0, 7d = --1. Elementary calculations show that

(5.23) d = - I E L 2 ( 0 , 1 ) , 1 (0 )=1 , 0 < 9 < 1


while A = alk~T,

(5.24) Ax = x", D(A) = {x e H2(0, 1): x'(1) = 0, x(0) = 0} .

Since A = A* < 0, A generates on H an analytic self-adjoint semigroup which is EXS. Moreover, A has a system of eigenvectors { ~}==0, corresponding to its eigenvalues { ,~}n=o which constitutes an orthonorma] basis of H (see [14, Formula (21)] or [16, Lemma 3.1 with K=0]),

en(O) = v ~ s i n ( 2 + n T r ) 0 , 0 < 0 < 1 , n > 0 } (5.25) , ~ , ~

A. = - l ~ + . ~ ) , n ~ 0

Finally, the system is described by (1.1) with A given by (5.24), d given by (5.23) and e # given by (5.22). The observation functional c # has a represention (1.2) with h(O) = -0, 0 < 0 < 1 .

It was proved in [12] that c # is admissible. Since a solution of the Lyapnnov operator equation, which proves admissibility of c #, is a Hilbert-Sehmidt (HS) operator, see [12] for details, the system cannot be exactly observable.

P~eeall that (see [12, p. 325] or [14, Formula (29) with K = 0]) for z C H

1 fol{sinhv/~OcoshvG(1-~), O<r] }z(~)&?. ((sI-A)-lz)(O) - v~coshv~ sinhv~r/cosh v~(1 - 0), 0 > r/

Hence

1 fo 1 (-fiXo)(S) = ( ( s I - A)-~Xo) (1) - v~cosh V'~ sinh(v~O)xo(O)dO - ~(V'-s) (5.26)

where

(5.27) f01 . oo ~0 ~(s) sinh!sO).xo(O)dO E ( - 1 ) k e -2ks 1 - e -~(1+~ ] = = [e -s(1-~ z o ( O ) d O COSll 8 k=0

whence [1, Section 4.1, Formula (33)] and [16, Example 4.2, where a time domain derivation is also given as Comment 4.1]

1 ~ 2 e-~2~(2yv~)dv, t > o (5.28) (~x0)(t) = e-T'/4*~(~)d~ = ~

where (still with t > 0) o o

(5.u9) e(t) = ~ ( - 1 ) ~ {xo(1 - t + 2 k ) x i , ~ , ~ + i j ( t ) - x o ( t - 2 k - 1)xE=~+,,~+~j(t)} k-0

The function { is piecewise square integrable and 4k-periodic with mean value equal to zero, i.e., for any t > 0

1 ft+4k ~(t) = ~(t + 4k), ~ j~ ~(r)dr = 0


Hence its primitive (indefinite integral) is also periodic. Integrating by parts (which can be made rigorous using Fubini's theorem), we get from (5.28)

/o ) 2 oo _ _ f v ~ ~-*~(2yzT)dy = ~ Jo ~ \Jo ~(r)d~ dy =

- ~ lim (e -y r2Yvq \ 4 ~/TrtY-~~176 ]0 ~(r)dr) + ~ fo Y~e-Y2k2yx/t do /

f 2y~

The function [0, oz) ~ y, > ~(r)dr is continuous, periodic and bounded. Hence the J0

first term is zero. Now consider the second term with Xo 6 D(c #) whence the limit

c#xo = xo(1) = lim xo(O)= lim f(t) 0-+17- t-+O+

1 j~o 2yv~ exists. This and previous information guarantees that the expression 2y~ ~(r)dr

is bounded and tends to x0(1) as t tends to 0§ for any fixed y. Hence by the Lebesgue dominated convergence theorem

lim(P--xo)(t) = lim 4 /o ~ (2~-y-y-~ f2yv~ ) ,~o+ , ~ o + ~ Y~-~ J0 ~(~)dr dy = (5.30)

= Xo(1 y2e-Y2dy = xo(1) �9

By Lemma 2.2 this implies that assumption (3.8) holds and hence also the chain of inclusions (3.0).

By (5.22) and (5.23) d e D(c#), so (3.4) holds with ~#d = -1 . It follows from (5.26) and (5.27) that

-~(Pd) (s) = c #(sI - A) - ld - 1 - cosh x/~ _ ~(x/~) s C II + (531) ~ co~h 4 ~ v ~ '

1 - cosh s ~(s) -

scosh s ' A

where Pd is analytic and bounded in a neighborhood of s = 0 as the coprime fraction of two

entire functions, the denominator function being cosh x/~ with simple poles at - ~ + nTr

for n = 0 , 1 , . . . . Note that lim sPd(s) = - 1 = e#d. Then using (3.5) one gets the s--+c~,sCN

transfer function

1 (5.32) ~(s) - cosh x/~' s e n +

A

with the same poles as Pd. It is easy to see that

Orabowski, Callier 23

where the norm is attained at s = 0. Hence (3.14) holds. From (5.30) with x0 = d we easily find

(-rd)(0+) := ~ Z ( - r d ) ( t ) = - 1 = ~#d = lira 8 - r d ( s ) . #: s-+oo , S E ~

This confirms identity (3.10). Since (3.8), (3.4) and (3.14) hold, the conclusions of Theorem 4.2 apply. However the situation is much better as follows from the analysis below.

We use the method of getting the inverse Laplace transform using residue calculus, see e.g. [9, Chapter 2@ Applying this method to (5.32) one obtains by [23, Problem 4.6.14(b)] for Isl small the series expansion

(7 ) (5.33) .~(s) = 2 E ( - 1)#~ + k~r

corresponding to the t ime-domain impulse response function given for t > 0 by

- t 7r + kpr O0

g ( t ) = 2 E ( - 1 ) k ( ~ r ~ + k p c ) e

(5.34) k=o

The same method applied to (5.31) gives for t > 0+

(-rd)(t) = - 2 ~ ~ ~ = k=o ~ + k~r

(5.35) s 2o ) § 1 §

~=0 t ~ + 2npr 37r e . ~ - + 2npr

Comparing the second line of (5.35) at t ~- O+ and (5.33) at s = 0 one has using (5.32)

- ( - r d ) ( 0 + ) = 8 1 ~=0 (4n + 1)(4n + 3) = ~(0) = 1

as can be expected from identity (3.10). Standard analysis shows that

(a) for every e > O, the series on the first lines of (5.34) and (5.35) are absolutely convergent uniformly in t E [c, co),

(b) for every c > O, g(t) and -rd(t) are uniformly continuous for t C [G co), and

(c) g(T)dm = --rd(t) whence d[(-rd)(t)! = g(t) for t > 0. dt

The second line of (5.34) will be used later. Formulas (5.34) and (5.35) are called "large t formulas" in mathematical physics as they best reveal the "large" t behaviour.

R e m a r k 5.1. The expansions (5.35) and (5.34) coincide with the Fourier expansions of -rd E L2(0, co) and its derivative on t > 0 with respect to the orthonormal basis (5.25). To


see this, note that using (5.35) and (5.34) one gets OO OO

(Pd)(t) = E e~"t<d' e~)He~(1) = E e~"t(d' en>I-I(h, Ae~>H,

d [(~)(t)] dt

For discovering the "small t formulas" note that (5.28) and (5.29) give

(Pd)(t) = e-r = -1 4- - ~ e-=2#(2xv~)dx (5.36)

where

(5.37)

t ~ 0 , n ~ 0 n = 0

E e~t(d, Ae~)H(h, de~)H = E ea~te~(O)e~(1) = 9(t), t > 0 . ~=0 n ~ 0

t>O

oo

~(t) := ~ + ~(t) = 2 ~ {~[t - ( 4 . + 1)] - ~[t - (4~ + 3)]} . n = O

Recall here that

/0 F 2 Y 2 2 e_~dx (5.38) eft(y) := ~ e -~ dz and erfc(y):= T o y

are respectively the error- and complementary error functions such that for all y > 0 eft(y)+ erfc(y) = 1 and erf(ee) -- 1. Now by (3.6) and taking ( am ) and (5.38) into (5.36) we get the "small t formula"

(5.39) he(t) = 1 + (-fid)(t) = 2 erfc \ ~ - ~ - / - erfc \ - - ~ . ] , t > 0 + n=0 k

where

(a) he(t) is the step response with he(O+) = 0; this confirms identity (3.11), (b) one uses the complementary error function to exploit the "small f ' behaviour of 9(t)

below by term by term integration.

The plot of h~(t) is given by Figure 5.4. Finally in (5.39) h~(t) is the sum of (positive) 4 n + l

decrements of the complementary error function at scanning points 2 v ~ over intervals of

1 length ~ to the right. Thus (as this function is convex and strictly decreasing on A+ (from

one at zero to zero at infinity)) the series in (5.39) is absolutely convergent uniformly in t > 0, whence h~(t) and -Pd(t) are well-defined and continuous on t _> 0.

The "small t formula" for 9 is the inverse Laplace transform of (5.32) given by [1, formula(19), p. 257] which can be made to read

2k + ] - ( 2 k + 1)2/4t g(t)= Z ( - 1 ) ~ T ~ e , =

(5.40) k=o = ~ f 4 n + l _ (4n+l )2 /4 t 4 n + 3 e _ ( 4 n + 3 ) 2 / 4 t ~

Standard analysis using (5.39) and (5.40) shows that


(a) for every c > 0, the series on the first line of (5.40) is absolutely convergent unifbrmly in t E [0, e - l ] ,

(b) g(0) = 0 (more precisely for every integer p > 0 there is a constant Kp independent of t > 0 such that Ig(t)l < Kptl+p),

(c) for every c > 0, g(t) is uniformly continuous for t �9 [0,c-1], and

(d) g(T)d~- = hs ( t ) w h e n c e d[hs(t)] _ d[(Pd)(t)l _ g( t ) for every t > 0 + . dt dt

N o t e the complementary "small" t aspect of the properties, where (d) states what was to be expected and the behaviour near g(0) = 0 cannot be obtained from (5.34). The plot of g is given by Figure 5.5.

0 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /

0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ 0.5 . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ 0.4 . . . . . . . . ~ . . . . . , . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

/ o.~ ........ ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

o . 2 . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . .

/ OJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . .

/ i )

�9 I i T 0 5 I 1 5 2 ~15

l / ~, @ . i . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B

1 .41 [ " " j . . . . . . . i \ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L i . . . . . . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . .

12 [ i ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

t l ',

' i . . . . . . . .

i o 8 F" i . . . . . . . . . . . . . . . . . \ : . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . .

o.6 i . . . . . . . . . . . . . . . . \ . . . . . . . . . . : . . . .

0 ~ , : .4. F , . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i 01 / { i . . . . . . . . r - . . . . . = : = -

o 0.5 1 1.5 2 2.5

FIGURE 5.4. Plots of the step responses hs = . r id- c#d computed with the aid of (5.39) (dashed line) and confirmed by (5.35) (dotted line)

FIGURE 5.5. Plots of the impulse responses 9 computed with the aid of (5.40) (dashed line) and confirmed by (5.34) (dotted line)

L e m m a 5.1. The impulse response g and the functions Pd and hs have the following properties:

(i) Let {e-1/4t -t V4} (5.41) 7(t) := rain t x / ~ ,we , t > 0 .

1 Then in (5.41) the minimum is achieved at its first argument for t < - and at its second argument elsewhere and ~r

(5.42) (1 - 3e-2~)3,(t ) G 9(t) < q/(t), t > 0

where 3e -2" ~ 5.6 x 10 -3. Thus g(t) ..~ 7(t) with an error of less than one percent.


( i i )

( s .4a)

( i i i )

and with

(5.n~)

f t ~ d (Pd)(t) = - g(T)d% ~ ~d) ( t ) ] =9(t) and (Pd)(t) < 0, t > 0 .

fo t

ha(t) = (-fid)(t) + 1 = g(~-)d% t > 0 ,

I 2 erfc 1 1 }

i- 774-e-t~2/4' t E [~,oo)

ha(t) ~ ~(t) with an error of less than one percent. The function r/has a small jump of 1

- 7 x 10 . 4 at t = - . 77

1 / 1 Proof. (i). The equation - - C - - e - 1 / 4 t = ~re-tTr2/4" has on t > 0 a unique solution at t

t v ~ - 77'

with the left-hand side dominant for t _> _771 and the r ight-hand side for t c [0, 1] . This

explains the comment on (5.41). Observe now that with a > 0 and b > 0

(5.45) d--xd(-axe-bx2) = a e - b x 2 ( 2 b x 2 - 1 ) "

Fixing t and using (5.45) with a = 1 and b = t ~ in the second line of (5.34) one obtains

j ( t ) = ~ - x e - b x = ~ e - b X 2 ( 2 b ~ - 1 )d~ n = O t. .1 4n+1 n = O d 4 n + l

1 77 For t > - and x > 1 we have (2bx 2 - 1) = tTr2cc 2 - 1 > - - 1 > 0 such that the integrand

- T r - - 2

is positive. Hence, due to (5.45), g(t) is for t >_ x__ bounded above by 7r

f~ 77j~ e - b X ~ ( 2 b x ~ - 1 )dx = ~e-t~V 4

and bounded below by

3 2 f e - bx (2bz 2 - 1)dz = ve- t~2 /4(1 - 3e -2tTr2) >_ 7re-tTr2/4(1 - 3e -27r2) .

Therefore there holds

(5.46) (1 - 3e-2~)7(e-tTr2/4 < g(t) < ~re -tTr2/4, t >__ 1 . 7(


1 1 1 Similarly, fixing t e [0, ~] and using (5.45) with a = ~ and b = ~ in the second line of

(5.40) there holds

2~ e - 1 / 4 t e - 1 / 4 t 1 (5.47) (1 - 3e- ) ~ - ~ - < g(t) < t----~-' t e [0, ~] .

Hence (5.42) follows by (5.41), (5.46) and (5.47). (ii) and (iii) follow by previous observations. In particular the claim concerning ~?(t)

1 1 given by (5.44) follows by integrating (5.42) for t _< - on [0, t] and ibr t _> - on [t, oe). []

7C 7C

7r 2

Take now cr E (--~- , 0] and note that by Lemma 5.1

f0 f0 1 1 I b t h ( ~ / = e -~ ' Ig(t)l d t = e-~tg(t)dt = g(a) - cosh v/F - cos ,/-:-J

and

f0 /0 I[(Pd)(t)llA(~) = e -zt ](Pd)(t)l d t = - e-r =

_ i - cos k~)kJ

--G COS

Hence we showed explicitly that both 9 and Pd are functions in .4_(0) and therefore A

and Pd are analytic in an open right hAalf-plane strictly containing II + and bounded and

continuous in its closure: both ~ and Pd are in H~176 +) but the situation is much better. Finally since 9 6 L~oc(0, oo) and is continuous on ]l{, it follows by (4.8) that here the unit step response hs = D - l g is continuously differentiable on N and so is P d = hs + ]l(Pd)(0+) on t > 0+. Hence here (4.7) reduces to y is the derivative of P d on t > 0+ which is the case.

L e m m a 5.2. The factor control vector d given by (5.23) is not admissible.

This lemma was proved in [16, Section 3.3] using the duality theory as in Lemma 1.2 with d*Az = z'(0) for z E D(A) and a version of the spectral criterion of admissibility. For the convenience of the reader this proof is abbreviated in Appendix B.

An anonymous reviewer kindly suggested the authors to follow the procedure of Curtain and Weiss [7, Example 6.2] to check whether or not a state space exists on which both control action and observation would be admissible and the system transfer function is in H~176 An affirmative answer as well as some comments are presented in Appendix C.

6. HANKEL OPERATOR AND INPUT--OUTPUT MAP

This section prepares the next section on abstract linear systems. Throughout we

assume that c # is admissible, (3.8), (3.4) and (3.14) hold, and d is an admissible factor control vector. Recall the glossary of notions we have introduced in the previous sections.

�9 In the sequel T E L(H, L2(0, oo)) and Q C L(L2(0, oe), H) are the extended observabil- i ty- and teachability maps described respectively by Theorem 1.1 and (1.7), and (1.8).

�9 F C L(L2(0, co)) is the extended input-output map given by (4.3).

28 Grabowski, Catlier

�9 R~ C L(L2(0, oo)) is the reflection operator given by (1.9). �9 {T(t)}t>0 and {T*(t)}t>o both contained in L(L~(0, oe)) are respectively the right-shift-

and left-shift semigroups defined by the generators R and L = R* described in (1.3). They are described for all f E L2(0, oo) by (1.4).

�9 For all f > 0 and t k 0, P~ and P~,r+t both in L(L2(0, oo)) are truncation operators given ibr all f E L2(0, oo) by (1.11) and (1.12). Shifts and truncations are related by

(6.1) PTT(T) = O, T*(T)T(T) = I ,

(6.2) P~.~ = By + P~.T+,, P~,~+, = T ( z ) P , T * ( ~ ) .

As in [8, Lemma 8.2.8, p. 401] we define the Hankel operator for (1.1) as H E L(L2(0, oe)) such that on t _> 0

d [h*S(t)(Qu)] (6.3) (Hu)(t) := (PQu)(t) By (1.7) and (1.8) we get

d [h*S(t)A Wu]= d [ 9s ~176 ] (6.4) (Hu)(t) = ~ ~ h*A S(t + s)du(s)ds

The following results are elementary and describe output trajectories due to states propagated by the system semigroup and properties of the input-output map.

Lemma 6.1. Assume that c # is admissible and that (3.4) and (3.14) hold. Then

(i) For all z E H and for all t > 0 and s _> 0 there holds

(~s ( t )x) ( s ) = (~x)(t + s) = (T*(t)-~z)(s) (ii) F is right-shift invariant, i.e.: T(T)F = FT(~-) for all T > 0.

(iii) F is causal, i.e.: P~-FP~ = FP~ ibr all T > 0.

The following result describes output trajectories due to states reachable at f > 0.

L e m m a 6.2. For all u E L2(0, oo) and for all T _> 0 and Mmost all t k 0 there holds

(i) ( H ( R ~ ) ) ( t ) = ~ (Pd)(t + ~ - s)~(s)ds.

(ii) H(R~-u) = T*(o-)F(P~u).

Proof. (i) By (6.4) and [24, Corollary 2.9, p. 209] we get successively for t >_ 0

f ] d [h*S(t)A S ( ' r - s)du(s)ds = (H(P~u)) (t) =

= d~ S ( T - s)du(s)ds - (S(t)d)u(T =

= d~ ~ [h*S(r- s)(S(t)d)Ju(s)ds- h*(S(t)d)u(T)


where the expression between the square braces in the integrand is differentiable by Theorem 1.1. Hence for all t > 0

d fo ~ d [h*S(~--s)(S( t )d)]u(s)ds=

_ d ~o ~ d [h*S(~--s)(S( t )d)]u(s)ds= dt d(T - s)

/o /o" " d (~d) (t + ~ - ~ )~(~)e~ , - dt (P(S(t)d)) ( 7 - s)u(s)ds =

where in the last line we used Lemma 6.1(i). (ii) For all t >_ 0 with T fixed

( ~ ) (t + ~ - ~)~(~)~ (T*(~-)F(PT.u)) (t) = (-F(PTu)) (t + 7-) =- -~

Thus (ii) follows by (i). []

7. BOUNDARY CONTROL SYSTEMS IN FACTOR FORM AS ABSTRACT LINEAR SYSTEMS

AS in Section 6 we assume that c # is an admissible observation functional, (3.8), (3.4) and (3.14) hold, and d is an admissible factor control vector. We prove that system (I.i) is an abstract linear system as in [28, Definition 2.1] which is regular. Moreover we discuss its description.

(~<>v)(t) := 7. v(t - ~), t>~

denotes the concatenation of functions u, v defined on 1~, whence

(7.1) ~ v : PT.~ + T r , 7.

where PT is the truncation operator given by (I.ii) and T(~-) is the right-shift operator given i~ (1.4).

D e f i n i t i o n 7.1. [28, Definition 2.1]. Let U, X and Y be Hilbert spaces, f~ = L2(0, oe; U) and r -- L2(0, oc; Y). An abstract linear system on f~, X and F is a quadruple E = (S, ~, ~ , F) , where

(i) S = {S(t)}t>o is strongly continuous semigroup of bounded linear operators on X, ( i i ) ~ = {~}t>0 is a family of bounded linear operators from s to X such that

(7.2) ~ + , ( ~ v ~) = s ( t ) ~ , ~ + ~,~ , 7"

for anyu, vE~andanyT, t_0.

(iii) ~ = {~t}t>0 is a family of bounded linear operators from X to F such that

(7.3) ~r = ~x ~ qgtS(T)X , T

for any x E X and any T, t ~ 0, and kO 0 = 0,


(iv) F = {Ft}t>>_o is a family of bounded linear operators from f~ to F such that

(7.4) F~+t(u ~ v) = F~u ~>(gt~ru + Ftv) , T T

for any u, v E f2 and any % t > 0, and F0 = 0.

U is the input space of E, X is the state space of E, and Y is the output space of E. The operators ~ are called restricted input-state maps. The operators ~ are called restricted state-output maps. The operators F~ are called restricted input-output maps.

For system (1.1) the appropriate spaces are

(7.5) U = Y := N, X := H, [2 = r := L2(0, oo) .

Moreover as (1.1) has an associated EXS semigroup {S(t)}t>_o, a teachability map Q, see (1.8), an extended observability map P, see (1.5), and an extended input-output if, see (4.3), it is natural (following the philosophy of [28, Section 2]) to choose the following restricted maps with ~- _> 0:

(7.6) {S(T)]~>0 := the E X S Co-semigroup on H associated with (1.1) ,

(7.7) (9~u := QP~u, u e Le(0, oo) ,

(7.8) (9 ,x ) ( t ) := (P~P--x)(t), x e H, t > 0 ,

(7.9) (FrU)(t) := (Pr-ffu)(t), u e L2(0, oo), t > 0 .

T h e o r e m 7.1. With the spaces and maps defined by (7.5)+(7.9) there holds that system (1.1) is an abstract linear system E = (S, (~, 'Is, F) on L=(0, oo), H and L2(0, oo).

Proof. As required {S(t)}t>o c L(H), {(ft}t>0 C L(L2(0, oo), H), {~t}t>0 C L(H, L2(0, oo)) and {Ft}t>o C L(L2(0, co)). It remains to prove (7.2)+(7.4).

(7.2): Note that with W given by (1.7) the family {WP~}~_>0 c L(L2(0, oo)) repre- sents the restricted input-state maps of the state differential equation 2(t) -- Az(t ) + du(t), where d E H, thus with a bounded control operator. Hence with ~- and t _> 0 and u and v in L2(0, oc),

w +t(u o = + . T

Now by (7.7) and (1.8) q~t := QRt = AWRe, whence (7.2) follows. (7.3): With 7- and t _> 0 and x in H there holds successively using (7.8) and (6.2),

next (6.2) and Lemma 6.1(i), and finally (7.8) and (7.1)

~r+tx = Pr+tPx = P~-Px + P~-,r+tPx

= PrPx + T(T)PtT*(T)-Px = P,P,-fix +T( t )P tP(S (T )X)

: P ~ z + T(t)gt(S(T)z) = 9~z<>gt(S(7)z) . T

(7.~0: With T and t _> O and u and v are in L2(O, oo) there holds successively using (7.9) and (7.1), next (6.2), and finally (6.2)and Lemma 6.1(ii)

F,+t(u+v) : P~+tF (P~u + T(t)v) = T

= + + ( T R o t =

= (ff u + + (YP u +


Hence using Lemma 6.1(iii), (6.1), Lemma 6.2(ii) and (6.1), next (6.3), and finally (7.7) +

PTffu + T('r) [PtH(R~u) + Pt.flv] =

P, + + Play) :

= P~F~u + T(T) (qgt~u + Ftv) = F,u 0 (~ t~ ,u + Ftv) . T

[]

System (1.1) as the abstract linear system E of Theorem 7.1 is henceforth syn- onymicly called abstract linear system (1.1).

Definition 7.2. [28, Definition 2.2]. Abstract linear system (1.1) is said to be regular iff its unit step response hs given by (3.6) has a Lebesgue value at t = 0+, i.e. d e := hs(0+) :=

l fat lim hs(T)dT exists, in which case d e andN ~ u J ~ d#u are the feedthrough parameter t-+0+ )- and -operator of abstract linear system (1.1).

Fact 7.1. Abstract linear system (1.1) is regular with feedthrough parameter d e = 0.

Proof. This is a straightforward consequence of Lemma 3.1. []

Theorem 7.2. Consider abstract linear system (1.1). Then

(i) abstract linear system (1.1) is described for t > 0 and u E L~oc(0, oo) by the equations

{ it(t) = A[x(t)+du(t)] , t>_O } x(O) = Xo E H y(t) = e x(t), t > 0

where 1 ~ the state differential equation has to be solved weakly in H as in Lemma 1.1 giving

/o' x(t) = S(t)xo + A S(t - ~')du(~-)dT E H ,

2 ~ the output equation is valid almost everywhere with %# the Lebesgue extension of c # given in (2.2), and

3 ~ c~ may be replaced by %# given by (2.3). (ii) The transfer function of abstract linear system (1.1) is given by (3.5).

Proof. (i) According to [28, Formula (2.4)] abstract linear system (1.1) is for x0 E H and u E L~oc(0, co) described by the equations

z(~) = S(T)xo § c~PTu, P~y = ~ z a § F~P~u, T >_ 0 .

Note that x(~-) E H and P~u as well as Pry E L2(0,c~). Moreover because ~rP~u = 'b~u and F is causal these equations are equivalent to the more usual

x(t) = S(t)xo + ~tu t >_ O, y = Pxo +-flu ,

where x(t) E H and u as well as y E L~o~(0, oo). By Lemma 1.1 the state equation is realized as mentioned above. By [28, theorem 2.3] the state equation is realized as the strong solution

(7.9) and (7.1)

Fr+t(u ~ v) = T


of an equivalent state differential equation defined on an extension of H (see [16, Section 4.5]) and the output equation is realized as mentioned above.

(ii) Observe that T is by Theorem 4.1 in L(L2(O, oo)) and by Lemma 6.1(ii) right- shift invariant. Hence the multiplier transfer function (3.5) detected in the proof of Theorem 4.1 agrees with the transfer function mentioned below [28, Theorem 3.1], a special: case of [28, Definition 3.7]. []

8. CONCLUSION

The most important results of this paper are:

�9 transfer function Theorem 3.1 and input-output map Theorem 4.2 illustrated by transmission line examples in Section 5, and

�9 abstract linear system Theorem 7.1 with .its description in Theorem 7.2

A less satisfactory feature is the need for the H ~176 assumption (3.14) in spite of the fact that A is the generator of a C0-semigroup S(t) which is EXS. Further research is needed to see if this assumption is not redundant.

As a final comment, the first example of a system with H ~176 transfer function, an admissible observation functional, and with a control vector which is not admissible has been given in [5, Example 3.4, p. 22]. However, in our similar second example H -- L 2 (0, i) is taken as the (easy) state space. Such examples can have a great significance for the development of a general approach to the lq problem. It is known [13] and [30] that boundedness of the observability map (admissibility of the observation functional) as well as the input-output map are fundamental properties for a unique solvability of the lq problem. Contrary to [30] it is not assumed in [13] that the reachability map is bounded.

APPENDIX A: A NEW PROOF OF LEMMA 2.1

Pwof. As in [28, Corollary 5.5] it is enough to prove Lemma 2.1 for A = 0. By the Cauchy- Schwarz inequality

(,/0 , , (8.1) -i y(~)d~ < ~ ly(~-)l~ d~- <_ sup z f ly(r)fdr<oo Vt>O. t>o JO

Hence for any t > 0 the directed set ~ y(r)dr is bounded and therefore it has a > 0

a directed subset { ~ f o a"t y(r)&- which converges to co as an "N 0. Now, by (8.1) ) a a > O

and the Lebesgue diminated convergence theorem,

(;) /7 I j0 1 8 ^ S t s 2 e _ s t 1 fa~t -~y = y(r)dr dt > c~ as a ~ % 0

for any s E 1-I + rh R. If the limit l imees~(s ) := c exists then for s E I-I + n R the limit S ,

- -

a'NO


} only accumulation point of the directed set ~ y(r)dr is {e}. a>0

lim -1 fot y ( T ) d T = C. t-+O+ t

Proof. If d were have

oo>

(8.2) =

This shows that

[]

APPENDIX B: PROOF OF LEMMA 5.2

admissible then by Definition 1.2, Lemma 1.2 and Definition i.I we would

k [IQfll~ : l(Qf, e~>HI 2 = Y ] < f , Q * e ~ ) L ~ ( o , ~ ) ] 2 = n = 0 n = 0

, 2 y]<f,d = ~ l < f , eh(0)~~ 2-- n = 0 n = 0

r ~ : 0 n : 0

where ] �9 H2(II +) is the Laplace transform of f , Q denotes the teachability map associated r oo with d and { ~}~=0 stands for the orthonorma] basis of eigenvectors of the operator A given

by (5.26). In particular, for f �9 L2(0, oc), f(t) = t-1/4e -t we have by [1, Formula (1), p.137] ](s) -- (s + 1)-a/4F(~) and therefore

: 4 3 / 2 > - - oo F 8-.:o ~:o [7{+(2n +1)2] - 4 ==o2n+l

which contradicts (8.2). []

APPENDIX C: FURTHER ANALYSIS OF EXAMPLE 5.2

In the sequel (-A) a for ~ ~ 0 denotes the o~-th fractional power of (-A) with domain D[(-A)a]. For ~ ~ 0, I-I a and H_ a denote respectively the restriction of H and its

completion with respect to the norms induced by the scalar products

<xl,x2>~o := <( -A)a ,1 , ( -A)a~2>u ,

and

<xl,XJH_~ := <(-A)-ax, , ( - A ) - a x J H .

Note that x E Ha iff ( -A)ax E H. As in [7, Example 6.2, p.54] the symbol A will mean an element of L(Ha+I, Ha) as the generator of a restriction or an extension of the semigroup S(t) on Ha, for a C ~. Moreover as in [7, p. 53] we use the duality pairing

(x,y)Dp := ((-A)ax, (-A)-ay)H, Vx �9 Ha, Vy �9 H-a ,

with a > 0. For o~ = 1 it is identical to the duality pairing [16, Formula (62)]. Moreover it implies that the space Ha is dual with respect to H-a, giving H-a = L(Ha, JR).

If there exists a/3 < 1 such that h �9 HZ then

c#x = (Ax, h)H = --((--A)l-/~x, (-A)Zh)H ,


and the observation functional c # extends to a functional in L(HI_~,]R) and there exists a vector C = Ah C H_t+~ such that c#x = {x, C}D P.

Observe that

h E Hf~ ~ ] _ Anl2 ~ 2 = 27r4/3_4 + n < 0o .

n=-O n=0

This holds for/3 = 3/4 - c, c > 0, whence c # extends to a bounded functional on H�88 and

the vector C determining c # belongs to H_�88 Similarly,

' v dE H~ ~# {-A~ 2~ ek(0) 2 = 2rr 4'~-2 + n < 0 o ,

holds for a = 1/4 - % ~7 > 0. The observation functional d # (which restricts to d*A on H1 = D(A)) dual to the factor control vector d C H or to the control vector B = Ad C H-l , [16, Sections 4.4, 4.5], reads

(8.3) d#z = x'(O), D(d #) = {x e L2(0, 1): z is right-differentiable at 0} D C'[0, 1] .

It extends to a bounded functional on H�88 whence the vector B defining d # by d#x = (x, B)Dp belongs to H _ a , .

L e m m a 8.1. We have:

(8.4) k V/~ '1- n~ f [(2+nrc)21 2<0o gf E H2(II+) �9

Pro@ By Bari's theorem as given in [16, Lemma 2, pp. 90 - 91], (8.4) holds iff the system

_> 0 or {/r~},,,e~u{o} C L2(O, oo), f n ( t ) : : +n~-exp - ~-+nrc equivalently iff its Gram matrix

,~,~e~u{o} induces a bounded operator on g2. The latter follows by using Schur's test [17, Theorem 5.2,

p. 22] with p(t) = q(t) = ( t + �89 []

Consider now the admissibility of the observation functional c # on Ha. By the definition of admissibility an equivalent requirement is that the observation functional c # (-A) -a is admissible on H. Hence by tlle spectral criterion for admissibility as in I14, Formula (6)],


[16, Formula (7)], the observation functional c # is admissible on Ha iff CO ^ 2

s ( - ; , o ) =

(s.5)

Therefore, by gemma 8.1, (8.5) holds if

) n~Nu{o} 1 c# which holds for c~ >_ - ~ . Hence is an admissible observation functional (or C is an

1 admissible observation vector) on H~ for ee >_ -7" We test now the admissibility of the observation functional d # (or of the control

vector B = Ad) on H~. Using the definition of admissibility of the control vector and duality, an equivalent requirement is that d#(-A) ~ is an admissible observation functional on H. Hence d # is admissible on H~ iff

l(-An) e%j = (8.7) .=o

[71" \ 4 a + l 71" ^ 2

= ~ 2[7 § nTc) f + n~ < co V / E HS(H +) .

Therefore, by Lemma 8.1, (8.5) holds if

~r ) 4 ~ + ~ ~ ,

?~T'_ J n E N U { 0 }

1 d# which holds for a _< - ~ . Hence is an admissible observation functional (or B is an i

admissible control vector) on H~ for a < - 7 . From the conditions on c~ above it follows that H_�88 is a unique candidate for a state

space upon which control and observation are admissible. Now in order that the proposed state space can be realized, the positive parameters e, 77 should be chosen such that 1) c # should be bounded on H~, which was true on H�88 and 2) d # should be bounded on H~,

which was true on H�88 Hence with e = ~7 = �89 c # E L(H~, R) (or an observation vector

C E H_�88 and d # E L(N~, R) (or a control vector B E H_S) are both admissible on H_�88 The latter therefore exists as a state space, upon which both control and observation are admissible as in [7, Example 6.2].

Further comments are in order here. Although the construction above allows one to find a state space on which both control and observation are described by admissible operators and (as was established earlier) the system transfer function is in H~(II+) , one observes that this procedure of finding such space is limited to discrete self-adjoint semigroup generators. In more complicated situations more sophisticated tools may be needed. At this point it is not clear if these tools are adressed to or recommendable to the control engineering community, especially when for solving some problems (as it is emhasized in our paper) not all abstract linear system axioms are needed and one can use a simpler Hilbert state space.


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Ins t i tu te of Automat ics Academy of Mining and Metallurgy al.Mickiewicza 30/B1, rm.314 PL-30-059 Cracow, Poland e-maih [email protected]]

Facult6s Universitaires Notre-Dame de la Paix Department of Mathematics 8, l~empart de la Vierge B-5000 Namur, Belgium e-maih frank.callier@fundp, ac.be

A M S C l a s s i f i c a t i o n . Pr imary: 93B, 47D. Secondary: 35A, 34G.

Submitted: September 21, 1999 Revised: September 20, 2000

Boundary control systems in factor form: Transfer functions and input-output maps

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