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WE PRESENT A GENERAL APPROXIMATION SCHEME FOR AUTONOMOUSFDE's WITH LIPSCHITZEAN RIGHT-HAND SIDE. OUR APPROACH IS BASED ONAPPROXIMATION RESULTS IN SEMIGROUP THEORY. THE RESULTS AREAPPLIED To NONLINEAR AUTONOMOUS FDE's IN THE STATE SPACE ]Rnl

SAD O LER JUTONOMOIS FDE's OF NEUTRAL TYPE IN THESTATE

LjI. SPACES I x L' and W1#2. KD I JAN 73 1473 EDITION OW I NOV 68 IS OBSOLETE CSIIE

AFSR.-Th. 8 0 -0 92 4

AN APPROXIMATION SCHEME FOR DELAY EQUATIONS +

by

F. Kappel*

Mathematisches InstitutUniversit~t Graz

Elisabethstrasse 11A 8010 Graz (Austria)

and

Lefschetz Center for Dynamical Systems- Division of Applied Mathematics

/Brown UniversityPr idence, R. I. 02912

June 16, 1980 DIS7 oBU1ON STA IMWApp'ovd for publo nlnMu

Distulbutio Uniniltd

This research was supported in part by the Air Force Office of ScientificResearch under contract 4-AFOSR 76-30920, and in part by the UnitedStates Army Research Office under contract #ARO-DAAC29-79-C-0161.

+ Invited lecture, International Conference on Nonlinear Phenomena inMathematical Sciences, Univ. Texas, Arlington, Texas, June 16-20, 1980.

Appw@Y@ 809 22 106IC

AN APPROXIMATION SCHEME FOR DELAY EQUATIONS

by

ABSTRACT

We present a general approximation scheme for autonomous FDE's

with Lipschitzean right-hand side. Our approach is based on approximation

results in semigroup theory. The results are applied to nonlinear

autonomous FDE's in the state space en x L 2 and to linear autonomous

FDE's of neutral type in the state spaces 10'x L and W ' 2

4R FURCE UFJ'C OF SCZENTIFIC RESE kCd (ASc)WJOTICE OF TRANSMITTAL TO DDC'TLS technical report has been revieweamW'fved for publ.ic release IAM 19.. and 1s

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1. Introduction.

In recent years one can see considerable interest in approximation of

delay systems by ordinary differential equations. Results of this type

proved to be very useful for the numerical solution of optimal control

problems and identification problems (see [21,[5], [63). A very successful

approach uses abstract approximation results in semigroup theory and applies

these to the semigroups associated with autonomous delay systems. This

approach goes back to [4] (see also [5]) where control problems involving

linear autonomous functional-differential equations and approximation of the

state by step functions are considered. A considerable improvement over the

scheme developed in [53 was obtained in [7] where for the same class of delay

systems a scheme is developed which allows approximation of the state by spline

functions. In this paper we give an abstract formulation of the scheme

developed in [71 which can be applied to autonomous delay systems with

globally Lipschitzean right-hand side. We also indicate the application of

the linear version of the scheme to autonomous neutral delay systems. Since

by lack of space we cannot give a detailed discussion of the relevant literature,

we include a rather complete list of papers in the references ([1,3,5,7,13,14,15,

16,17,18,19,20,22,23,24]) and refer to the discussions given there (see es-

pecially Section 5 of [5]).

2. An Approximation Scheme for Semigroups of Nonlinear Transformations.

Throughout this section X will be a Hilbert space with inner product

(.') and norm 11. A-family T(t), t > 0, of globally Lipschitzean

operators X - X is called a semigroup of type w, w EIR, if the following

properties are satisfied:

2.

(i) T.(O) I.

(ii) T(t+s) - T(t)T(s) for all t,s > 0.

(iii) For any x E X the map defined by t * T(t)x is

continuous on [0,-).

(iv) IT(t)x - T(t)yI < e'tlx-yl for all x,y E X and t > 0.

If for a not necessarily single-valued operator A on X, the operator

_ is single-valued and defined on X for X sufficiently small-, we

say that T(.) is generated by A if for any x E X

t -

T(t)x - lim (I- t A) -nx, t > O,n

uniformly with respect to t in bounded intervals. Of fundamental importance

is the generation theorem by Crandall-Liggett ([121):

Let A be a densely defined operator on X. If A - wl is dissipative

for some w EIR and if range (I-XA) - X for sufficiently small X,

then A generates a semigroup of type w on X.

If A is single-valued then A - wl is dissipative if and only if

(Ax-Ay,x-y) <Wlx-y1 2 for all x,y C dom A. But note, that the Crandall-

Liggett Theorem is valid in general Banach spaces. Fundamental for our approach

is the following approximation theorem for semigroups of type W:

Theorem 1 ([10]). Let AN, N - 1,2,..., and A be single-valued operators

on X such that dom AN dom A for all N and dom A X. Assume that the

.........

3.

following conditions are satisfied:

(i) There exists a XO > 0 such that

range (I-XA) - range (I-XAA) - X

for N - 1,2,... and all A E (O,X0).

(ii) There exist real constants WN, N - 1,2,..., and w such

that the sequence (WN) is bounded above and AN - WN'

and A - wi are dissipative.

. (Aiii) There exists_a subset G of dom A such that (I-TA)G X K

for X sufficiently small and

x - Ax as N+W

for all x E G.

Then AN and A generate semigroups TN(-) and T(.) of type wN

and W, respectively, and for all x E X

lim T,(t)x - T(t)xN-w

uniformly on bounded t-intervals.

This theorem follows immediately from Theorems 4.1 and 3.1 in [101 and

is also true in general Banach spaces.

Definition 1. Assume that A is an operator on X. A sequence {XN,PN,AN},

N - 1,2,..., is called an approximation scheme related to A if

(i) are finite dimensional linear subspaces of dom A, N = 1,2,...

(ii) N: X 9X N is the orthogonal proj%%etion onto X N - 1,2,...

(iii) A; - PN N, N " 1,2,...

4.

Condition (i) in this definition is the most restrictive one. It could

mean that only the trivial approximation scheme (0,0,0) for all N is

related to A. If A generates a semigroup of some type W it is of interest

to know when the operators AN generate semigroups TN(.) converging to T(.).

Theorem 2. let A be a densely defined single-valued operator on X such

that A - wl is dissipative for some w EX and range (I-)A) = X for X

suff iciently small. Furthermore, let {XNPN,AN} be an approximation scheme

related to A. Assume that the following conditions are satisfied:

(i) him PNx - x for all x E X.

(ii) There exists a set G C doam A such that (I-XA)G = X

for X sufficiently small and

lm APNx = Ax for all x E G.. N-' m

(iii) For each N there exists a AN > 0 such that

range (I-AX) W X

for all X E (o,AN).

Then the following is true:

a) AN is continuous on X and generates a semigroup of type w

on X, N m 1,2,...

b) TN(t)XN c XN for t > 0, N - 1,2,...

c) lim TN(t)PNx - T(t)x for all x C X uniformly on bounded

t-intervals, where T(.) is the semigroup generated by A.

i 5.

Proof. Condition (ii) of Theorem 1 is satisfied with wN - w. This follows

from

(ANx-1y,x-y) - (PN [APNx-APNY] ,x-y) =

(APNX-APNY,PNX-PNy) < WIPNX-PNy 2 2wx-yl

for all x,y in X. Here we have used PN P For x E G we have

JA ANx-A_ _ !. I NPN x-PNAx1 + IPN - i

< I X I + IP Ax -Ax.

By conditions (ii) and (i) both terms on the right-hand side vanish as N .

Thus, also condition (iii) of Theorem 1 is satisfied. It only remains to

prove that range (I-XA) = X for N - 1,2,... and all X E (O,X0 ), I0 a

positive number not dependent on N, and that A is continuous on X.

We have range (I-X(AN-wI)) - range (I- X AM) - X for all X with

0 < X(l+xw)- < X N Since A - wI is dissipative, we immediately get

(cf. [8; p.73] or [9; p.23]) range (I-)(AN-wI)) = X for all X > 0, i.e.,

range (I-A,) - X for X E 1 This proves condition (i) of Theorem 1

with Xo = " Since A. - WI 'is m-dissipative and defined on all of X, it

is also demicontinuous, i.e., continuous from X with the norm topology into

X with the weak topology. But AX c N and do. N < - imply that AN

is continuous. Finally, conclusion b) follows from AN C XN.

Corollary 1. If A is linear, then the conclusion of Theorem 2 holds without

assuming condition (iii) explicitly. Moreover, TN(t) is given by

TN(t) - exp ANt.

N .,.. . .. . .. .. . w iii i I Ili

6.

Proof. The conditions on A imply (also in the nonlinear case) that A is

closed (cf. [8; p.75]). Then also APN is closed and defined on all of X.

Therefore, by the Closed-Graph Theorem APN is a bounded linear operator.

The rest of the proof is clear.

In the linear case we also can replace the condition "range (I-XA) - X

for A sufficiently small" by "A is the infinitesimal generator of a. C-semi-

group,.- This is a-consequence-of the Tmer-Phillips Theorem (see for ins-tance-

[21; p.161).

The following corollary covers a situation taylored for delay systems:

Corollary 2. let AO be a closed linear operator with dom A0 X and A,

be an operator with do% A, dom Ao such that

lx-.lyl _ L 11lx-yl

for all x,y 6 dor%, where L > 0 and 11.11 is a norm on dom %. Then

condition (iii) of Theorem 2 holds for any approximation scheme {UNPN,-A}

related to A.- A0 + A.

Proof. The conditions on', AO '=ply that A.,0 - PNAoPN is a bounded linear

operator, A - l-,ol. We have to prove that (I-A.Vy - x has a solution

*y for any x E X provided A is sufficiently small. This equation is

equivalent to

y "X-(I-A N 0 'P;Pjy + (I-XMO) "1 : h(y)

if X C (O, 0 1 ). Then for all y,y 2 C X

7.

Ih(y1)-h(y2) I AX(l-Y)-IAlPNyl-AiPNY2I

< AL(1V)-c II PNY-PNY2 < 0NL(l X)-1 yl-y2I.

Here we have used the fact that two norms on a finite dimensional space are

equivalent. Note, that PNy1 and PNy2 are in XN. The estimate shows

that h is a contraction for X sufficiently small.

Using, for instance, Theorem II of [12] and the fact that the Peano-

Existence Theorem is valid in finite dimensional spaces, we see that

u(t) TN(t)PNx for any x E X is the solution of

u(t) = ANu(t), t > 0,

u(0) - PNx,

which is an initial value problem for an ordinary differential equation on

X. For calculations one has to choose a basis for XN and to represent A

and PNx with respect to this basis (see [7] and [15] for more details).

3. Applications to Functional Differential Equations.

a) Nonlinear Equations in Rn x L

We consider autonomous equations

x(t) - f(xt), t > 0, (3.1)

with initial data

x(O) - A Cln, x(s) * *(s) a.e. on [-r,0], (3.2)

2,4

8.

r > 0, where E L (-r,OR). For a function x: [-rc) ,JRn, a > 0, x t

is defined by xt(s) - x(t+s), s E [-r,0]. A solution of (3.1), (3.2) is a

function x on an interval [-r,a), a > 0, such that (3.2) holds and x(t) =

n + ds for t E [O,a). We assume that f is a map .Y2(-r,O 1R) +IR

(_V2 is the linear space of square integrable functions) satisfying

(hl) For any function. x in L2(-r,a$R n), a > 0, which is continuous on

-... . . [0 ) t -If(x )- uniquelydefines a map in Ll(OOa'n).

(h2) There exist constants L > 0 and rj, J - 0,...,m, 0= r0 < r1 < ... < rm r,

such that for any 0,* in .V2(-r,oJRn)

f f(*)j < L( I 10(-r )-*(-r )I + 10-0 2).j0J L

Under conditions (hl), (h2) problem (3.1), (3.2) has a unique solution

n 2 nx(t;n,O) on [-r,o) for any (n,O) E X =IR x L (-r,0 In). The family of

operators T(.) defined by T(t)(n,o) = (x(t;n, ),xt(0,)), t > 0, (n,O) E X,

is a strongly continuous semigroup of globally Lipschitzean operators on X

(cf. (17]). X is a Hilbert space with norm I~l,*)Ig = (l 2 + Jl(s)i2g(s)ds)1/2-r

and corresponding inner product, where I- denotes the Euclidean norm in ,n

and the weighting function g is defined by

g(s) - m-j+l for s E (-r, J - ,...,m

Define the operator A by

doam A - {(,(0),,)I, E Wl 2 (1r,0Rn)},

A- (f(O)A), * E

9.

Theorem 3. Let {XNPN'AN} be an approximation scheme related to A such

that

(i) lim PNx = x for all x E X.N-m

(ii) There exists an integer k > I such that for 4E ck(-r,0iRn)

O(0) and - 4) in L2 (-r,O"Rn)

as N - o. Here 0N E W1,2 ( -r,ORn) is defined by

PN(O(o).,O) - (N( 0),'N) E dom A.

Then, for each N, AN generates a semigroup TN(-) on X such that for all

xEX

ha TN(t)Psx = T(t)xN4)-

uniformly on bounded t-intervals.

Indication of proof -(a complete representation of the results

concerning equation (3.1) will appear elsewhere (15]).

Step 1: We have dom A - X. For the set G in condition (ii) of Theorem 2

we take G = {(k(O),)I, C Ck(-r,OeItn)). Using (h2) one shows that there exists

an W E R such that A - WI is dissipative in X. The idea to introduce a

weighting function is due to G.F. Webb [25] and is essential here. In order

to verify the range condition on A one observes that (I-XA)(O(O),O)

(n,) E X is equivalent to O(s) - eS/0(0) - e(S-)/(T)dT, s E [-r,01,Er

and *(0) - h(*(O)), where h(a) - n + Xf(e* a - . f (")/.*(T)dT) for

a E M n . Using (h2) one shows that h is a contraction on en for X

sufficiently small.

10.

Step 2: By definition of A we see that APNx A x in D is equivalent to

f-N* f($) and *N 4 in L2 (-r,0"Rn). But f<> N f(O) follows from

(ii) and (h2). Therefore, condition (ii) of Theorem 2 is satisfied.

: In order to apply Corollary 2 we write A - A + Al, where

o( (0),O) = (0,€) and A1 (0(0),O) - (f(O),O) for 0 E Wl92(-r,ORn). Then

A0 is a densely defined and closed linear operator (see [17]). (h2) implies

- that- A- satisfies the -conditton in Corollary 2 with I1((o),1) = sup 10(s)t.-r<s<0

Thus, condition (iii) of Theorem 2 is satisfied.

Step 4: The semigroup generated by A is indeed T(.) (cf. [17]).

The conclusion of Theorem 3 in the case of spline approximation was also

obtained by H.T. Banks in [3] by a different method under the additional

assumption that f is differentiable.

b) Equations of Neutral Type in IRn X L2 (-r,O$Rn)

ni nLet D and L be bounded linear functionals C(-r,0. R) IR

defined by

m 10D( ) 0 (O) - I B (-rj- B(s) b(s)ds,

I(* A A,(-r ) + A(s) (s)ds,

where 0 =r 0 < rI < ... < rm r. 'AJ,B are n x n matrices and A(.),B(')0 1 m

L2 0 nxnare in L(-r,01 ).

~-~-v

- *.. -* '

~11.

As before we take X En x L2(-r,0eRn). Following an idea given in

[11] we define the operator A by

dom A - {(D(,),)I, E Wl' 2(-r,on),

A(D(O),O) = (L(,),O), 0 E W192

X will be supplied with the norm I(i,) , where the weighting function g

now is also dependent on D:

g(s) - m-J+lcti(s+ri-l) for s E (-rj ,-rj) j =,

a m(m+2) IB 12 j =

rj I -r J - 1 , .

Theorem 4.

a) A generates a C0-semigroup T() of bounded linear operators on X.

b) Let {NPNAN) be an approximation scheme related to A satisfying:

(i) lim P x - x for all x E X.N4

(ii) There exists an integer k > 1 such that for

E ck(-ron) 0N(O) -+ 0(0) and N 0 in L (-rO'In)

as N -p , where 0N E W1,2 (-r,O"n) is defined by

PN(D(O),O) - (D(ON),ON).

Then, for each N, AN is a bounded linear operator which generates the Co-semi-

group TN(t) - exp ANt, t > 0, and for all xC X

lira TN(t)PNx T(t)x

uniformly on bounded t-intervals.

12.

The proof of this theorem proceeds in similar steps as that for Theorem

3 and can be found in [141. If z - (D(f),f) with f E C(-r,ORn) then

T(t)z = (D(xt) ,xt), t > 0, where x(t) is the solution of the neutral equation

d-at D(xt) - L(xt), t > 0,

with x(s) - *(s), s E [-r,0]. Theorem 4 provides an approximation of

t + (D(xt),xt) t > 0, in X. If we could prove that TN(t)PNz + T(t)z also

with respect to the sup-norm, then we would get an approximation of x(t)

itself (see (14], Section 6 for details and preliminary results).

c) Equations of Neutral Type in W 12(-rO n )

Let the operators D and L be as under b) and put D(O) (0) -D().

The Cauchy problem

m 0O

x(t) - L(xt) + I B x(t)-r + B(s)x(t+s)ds - L(xt) + D(xt), t> ,j-l J -r

x(s) - O(s), s E [-r,0,

has for any 0 E Wl 2 (-r,O'*n) a unique solution x(t; ) on [-r,co). We take

X = W1'2(-r,o.1n) and define the family T(.) of operators by

T(t) O xt(0), t > 0, f E Wl 2 (_r,O1Rn).

T(.) is a Co-semigroup of bounded linear operators on X with infinitesimal

generator A given by

' r , °'" ... I' ll ' ° ' - -NONE...

13.

do. A - E1 W (W'2 (-r,O"eI) and 0(0) () + L(),

AO * dom A.

Note, that dom A depends on D and L. We supply X with the norm

.101g -(10()12+ ,()12g(s)ds) 1 /2 , where g is as under b) with

2m2 12

C-J r -rj- IB I j = 1,...,.

Then there exists an w EIR such that A- wlI is dissipative in X (cf.

[16]).

Theorem 5. Let {XNPNAN} be an approximation scheme related to A such

that:

(t) Iim PNO for all 0 E X.N-

(ii) There exists an integer k > 2 such that for 0 E ck(-r,0"gRn )

.. . L2(-r,0 1n)iii(o) (0) and ;N in L 2 -, as N

where N E W2 2 (-r,0"en) is defined by PN- ON'

Then each AN is a bounded linear operator which generates the C0-semigroup

TN(t) - exp ANt on X and for all E X

lim TN(t)PNO - T(t)o

uniformly on bounded t-intervals.

For a proof of this theorem see [16). Theorem 5 provides, for initial

14.

data in W1'2(-r,0JR n), an approximation of xt with respect to the sup-norm

and of xt with respect 'to the L2-norm. On the other hand, the subspaces

and projection PN depend on the right-hand side of the equation even

if D(O) - (0).

4. Spline Approximation

In this section we indicate that subspaces of spline functions can be

used in order to get concrete algorithms using the schemes presented in Section

3.

In the situation of Section 3, a) and b) we can take any sequence of

subdivisions of [-r,O] with mesh points tN, j - 1,...,k, such that

maxIt -tJ1Ii tNtil < 0 < -. In case of Section 3, a) we define XN tojJ j

be the subspace of all elements (0(0),0) such that 0 is a first order,

cubic or cubic Hermite spline with knots at the tN, respectively. We have

dim XN - (kN+l)n, (kN+3 )n or 2 (kN+l)n, respectively. Similarly in the case

of Section 3, b) XN is the space of all elements (D(O),O) with 0 as

above.

In the situation of Section 3, c) we have to satisfy the boundary

condition 0(0) - D($) + L(O) and to guarantee more smoothness. One has to

take subdivisions of [-r,O such that the points -rj, J - 0,...,m, are

meshpoints and restrict D and L. In the case of cubic splines we have to

assume B1 - ... - 1Bml - 0, B(s) A(s) 0. Then XN is the subspace of

all cubic splines with knots at the meshpoints tN satisfyingm

0(0) - 0Aj(-r) + B $(-r). In the case of cubic Hermite splines we assumeJ=0 3

15.

B(s) A(s) E 0 and take XN to be the subspace of all cubic HermiteNm m

splines with knots at the t such that j(0) = A.O(-r) + I B0(-r)j J.i0 i -i

For details, explicit calculation of bases for ., matrix representa-

tions of AN and numerical examples, see [7], [14] and [16].

REFERENCES

[1] H.T. Banks, Approximation of nonlinear functional differential equationcontrol systems, J. Optimization Theory Appl. 29 (1979), 383-408.

(2] H.T. Banks, Spline-based approximation methods for control and identifica-tion of hereditary systems, Proc. Int. Symp. System Optimization & Analysis,Rocquencourt, France, Dec. 11-13, 1978, Lecture Notes in Control andInformation Sciences 14 (A. Bensoussan and J.L. Lions, Eds.), pp. 314-320,Springer-Verlag, 19797

[3] H.T. Banks, Identification of nonlinear delay systems using splinemethods, Proc. Int. Conference on Nonlinear Phenomena in MathematicalSciences, Univ. Texas, Arlington, Texas, June 16-20, 1980, to appear.

[4] H.T. Banks and J.A. Burns, An abstract framework for approximate solutionsto optimal control problems governed by hereditary systems, in "InternationalConference on Differential Equations" (H. Antosiewicz, Ed.), Academic Press,New York (1975), pp. 10-25.

[51 H.T.Banks and J.A. Burns, Hereditary control problems: Numerical methodsbased on averaging approximations, SIAM J. Control Optimization 16 (1978),169-208.

[61 H.T. Banks, J.A. Burns and E.M. Cliff, A comparison of numerical methodsfor identification and optimization problems involving control systems withdelay, TR 79-7, Lefschetz Center for Dynamical Systems, Brown University,Providence, R. I.

[71 H.T. Banks and F. Kappel, Spline approximations for functional differentialequations, J. Differential Eqs. 34 (1979), 496-522.

EEL

16.

[8] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach

Spaces, Noordhoff Int. Pub. 1976.

[9] H. Brezis, Operateurs Maximaux Monotones, North-Holland 1973.

[10] H. Brezis and A. Pazy, Convergence and approximation of semigroups ofnonlinear operators in Banach spaces, J. Funct. Analysis 9 (1972), 63-74.

(11] J.A. Burns, T.L. Herdman and H.W. Stech, The Cauchy problem for linearfunctional differential equations, preliminary draft presented at theonf. Integral and Functional Differential Eqs., Morgantown, West

Virginia, June 1979.

[12] M.G. Crandall and T'M.Liggett, Generation of semigroups of nonlineartransformations on general Banach spaces, Am. J. Math. 93 (1971), 265-298.

(13] A. Halanay and V1. Rasvan, Approximations of delays by ordinary differen-tial equations, INCREST - Institutul de Matematica, Preprint series inMathematics No. 22/1978.

[14] F. Kappel, Approximation of neutral functional differential equationsin the state space Rn x L2, to appear in "Colloquia MathematicaSocietatis Jinos Bolyai, Vol. 30: Qualitative Theory of DifferentialEquations", Janos Bolyai Math. Soc. and North Holland Publ. Comp.

[15] F. Kappel, Spline approximation for autonomous nonlinear functionaldifferential equations, Manuscript, June 1980.

[16] F. Kappel and K. Kunisch, Spline approximations for neutral functionaldifferential equations, Inst. fUr Math., University of Graz, PreprintNo. 5, July 1979.

[17] F. Kappel and W. Schappacher, Autonomous nonlinear functional differen-tial equations and averaging approximations,J.Nonlinear Analysis: TheoryMethods and Appl. 2 (1978), 391-422.

[18] F. Kappel and W. Schappacher, Nonlinear functional differential equationsand abstract integral equations, Proc. Royal Soc. Edinburgh 84A (1979),71-91.

[19] K. Kunisch, Neutral functional differential equations in LP-spaces andaveraging approximations,J.Nonlinear Analysis: Theory, Methods and Appl.3 (1979), 419-448.

(20] K. Kunisch, Approximation schemes for nonlinear neutral optimal controlsystems, Preprint, May 1980.

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