Optimal control of dynamic systems: Application to spline approximations

A~,~ED MATH EMATW.5 AND

O3MPUTATK~N ELSEVIER

Applied Mathematics and Computation 97 (1998) 99-138

Optimal control of dynamic systems: Application to spline approximations Nwojo N. Agwu *, Clyde F. Martin 1 Department of Mathematics, Texas Tech University, Lubbock, TX 79409-1042, USA

Abstract

Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant. © 1998 Elsevier Science Inc. All rights reserved.

1. I n t r o d u c t i o n

I n this w o r k , we s t u d y the c o n t r o l p r o b l e m

~x(t)d"

= A~(t) + B~(t),

y(t) = C~(t)

(1.1)

with the cost f u n c t i o n

J(u) =

fl2[u(k)J2dt .

(1.2)

f2

*Corresponding author. Partially supported by NASA Grants NAG 2-902 and NAG 2-899. i Patially supported by NASA Grants NAG 2-902 and NAG 2-899. 0096-3003/98/$ - see front matter © 1998 Elsevier Science Inc. All rights reserved. PII: S0096-3003(97) 10101-1

100

N.N. Agwu, C F Martin / Appl. Math. Comput. 97 (1998) 99-138

Here, ~ c ~ m , y e l p , fi'EL2(f2,~t), t E f 2 C ~ , and A C ~ ( ~ m ~m), B E ~e(R/, ~m) and C C ~ ( ~ " , RP). The vectors ~ and ff are the state and control vectors of the system, respectively. A is called the state matrix, B the control matrix and C the observation matrix. Our goal is to find the control ff that will drive the system from one point to the other in state space ~m and at the same time minimizes the cost function J(u). We will also establish controllability conditions of the system (given the cost function J(u)) and then apply the results to spline approximation problems. It has been shown in Ref. [14] that system (1.1) is controllable if and only if the matrix Z = (BAB ... Am-IB) (1.3) has rank m and that M = {(A,B): £ = A£ + Bff is controllable}

(1.4)

is a manifold in ~m(,,+t/ [15]. Anderson and Moore [3], Luenberger [24], and Sage [29] have dealt extensively with the optimal control problem when the cost function is J(u)= fouZdt (the minimum energy problem). Conditions for controllability of the system are also given [24]. Spline interpolation constitutes a class of piecewise polynomial approximation that is commonly used when approximating many of the functions that arise in actual physical processes. Spline approximations of functions are preferred to most approximation and interpolation methods because of their inherent smoothness properties [11]. In Ref. [22], the minimal property associated with spline approximation is shown. A great deal of work has been done on polynomial splines, particularly cubic splines [12]. In Ref. [7] the convergence properties of a special class of quintic splines are discussed. There is a small amount of literature on exponential splines [8,16,25,27]. McCartin [25] has given an excellent theortical discussion of exponential spines and also studied its convergence rates and extremal properties. Pruess [27,28] asserts that exponential splines can produce co-convex and co-monotone interpolants. In their paper Zhang et al. [31] show the relationship between control theory and spline approximation by studying the minimum energy problem, namely: rain

J(u)=

uZ(s)ds

subject to

x(t)d. = A (t) +

t

[0, r],

or equivalently Y(t) = era,7(0) +

/oeA(t-~)bu(s)ds.

N.N. Agwu, CF Martin / Appl. Math. Comput. 97 (1998) 99-138

101

In this case, they obtained the optimal control law by observing that the operator K: L2 [0, T] ~ ~m defined by Ku =

eAfr-S)bu(s) ds

has an adjoint given by K*~' = bTeAT/T-s)Z for any ~"E Em. Thus, the optimal control can be written as u = K*(KK*) l ( E ( T ) - eArE(0)). (The interested reader should see Ref. [24] or any other standard text on functional analysis for more details.) By imposing certain smoothness requirements on u(t), they were able to obtain the spline functions. However, this approach does not eliminate the undulations associated with classical polynomial splines. In an attempt to overcome this problem, we introduce into the cost function, J(u), derivatives of the control law u(t) and hence, formulate our problem in the space H"(~2). In this work, we intend to use optimal control theory to develop methodology for spline approximations. As an illustration, suppose that the writehead of a computer is required to move from a certain position, ~(t0), to another position, E(T), then some control u(t) is needed to drive the write-head from the initial state, ~(0), to the final state, ~(T). Thus, the write-head (the system) must go through a certain set of points, namely, (t0,E(t0)), (tl,.~(tl)),..., (t,-l,~(t,-l)), O=to
(t,,~(tn)),


at given times. A spline curve can be fitted through these data points and then a control that takes the system through this trajectory is determined. We hope to find the set of controls ut(t) . . . u,(t): E(t,-1) --~ x(ti)

that achieves this while minimizing the functional J(u). Then, by applying the appropriate smoothness requirements of u(t) at the endpoints of each subinterval, [ti_~,ti], we will obtain and characterize the class of spline approximations.

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N.N. Agwu, C.F. Martin I Appl. Math. Comput. 97 (1998) 99-138

2. Formulation of the problem We consider the special case of the linear system (1.1) where the control law is u E L2:

-~x(t)d ~ =A~(t) +bu(t),

t E [0, T],

(2.1)

with

{.... 10 0/ If) 0

A=

al

1

...

Xl(t)

0

=

x2(t) I

0

0

...

1

xo_,(,)j

a2

a3

...

am

x.(t) / (2.2)

and the observation function

y(t)

= Er~(t),

ET = ( 1 , 0 , . . . , 0 ) .

(2.3)

Let the interval [0, T] be partitioned into p subintervals P:0=t0
<'"

and set hi = t i - ti-l. Our objective is to determine the control element u(t) E C"-2[0, T] that drives the system (2.1) from ~(0) to ~(T) such that the observed function y(t) satisfies the interpolation conditions

y(t,)

= cti,

i = 0, 1 , . . . , p -

Moreover, we require that

J(u)=

1,p.

(2.4)

u(t) minimize the cost function

u(*)) 2 dt.

(2.5)

This is the case where flk = 1, k = 0, 1,... ,n, in Eq. (1.2). A control that achieves this objective is called optimal. Here, 3, b E R", u(t) E L2[0, T] and A is an m x m matrix. We want to find the control law u(t) that drives the system (2.1) from ~(0) = ~ to ~(T) = x-'r and minimizes the functional J(u). Before we go any further, the following definition and theorem are in order: Definition 2.1. The linear system (2.1) is said to be controllable if for every pair of vectors (x"°, ~T) E Rm, there exists a finite time T and a control u(t) such that, T

=

+

fO

ds.

N.N. Agwu, CF. Martin I Appl. Math. Comput. 97 (1998) 99-138

103

Theorem 2.1. The given linear system (2.1) is controllable if and only if the controllability matrix M = (1) A1) ... Am-'/~)

(2.6)

has rank m. We then say that the pair (.4, b) is controllable.

The system (2.1), with A, b as in Eq. (2.2), is controllable. This is a direct consequence of Theorem 2.1 since, in this case, the matrix M = (b Ab ... Ar"-lb) has full rank. Now, our problem is to minimize the functional J(u) subject to the constraint d,

-~tx(t) = AY(t) + bu(t),

t E [0, T].

We may replace this equation by the equivalent constraint .~(t) = eAt~0 q-

(2.7)

d(t-S)bu(s) (is.

Then the control problem may be formulated in the space H#(f2)={uEL2(f2)IDiuEL2(f2),

I2C R, j E T ? , O<.j<~n}.

(2.8)

H"(f2) is the Sobolev space of order n on f2 C R with inner product defined

by (U,V)H.(O) =

(2.9)

u(k)v (k) dt

./o \k=o and the corresponding norm IlulIH. =

/

U(k) 12dt = -

IluCk>IIL=/O)" 2

(2.10)

k=0

The following theorem guarantees that the control law for the system (2.1) and (2.5) is bounded. Theorem 2.2. Let I2 c g~m be a bounded domain with ~I2 E C 1. ( a ) I f 2n > m, then H" ( I2 ) is embedded in C ( f2 ) and the embedding is bounded; that is, there exists a constant C such that Ilullc(~ ~< CllullH.(~for all u E H"(O). Furthermore, the embedding is compact. (b) I f n - m / 2 > r, then H"(f2) C cr(f2) and the embedding is compact. Proof. The proof of this theorem is found in Refs. [1,17]. To determine the control law, we convert the system (2.1) and (2.5) to the following boundary-value problem (see Ref. [2] for details):

104

N.N. Agwu, C.F. Martin I AppL Math. Comput. 97 (1998) 99-138

~" (t) = FTt(t) + if(t),

(2.11)

B~v(0) = 0,

(2.12)

7 ' ( r ) = 0,

(2.13)

where 7s(t) = (u(t), u ' ( t ) , . . . , u Z " ( t ) j r,

6(t) = ( - 1)"breaV'2~2, and ~'2, is the standard unit vector with 1 at the 2nth position and zero elsewhere. Also 0

1

0

0

0~

0

0

1

0

0 (2.14)

F= 0

0

0

( - - 1 ) n+l

0

( - - 1 ) "+2

0 1 ( - 1 ) 2" 0

and 0

1 0

0 0

1

-1

0

.........

(-1) "-2

0

( - 1 ) n-I

0

-1

.........

0

(-1) "-2

0 []

B= . . . . 0

0

0

: 0

0

...

1

...

-

:

•

0

0

0

(2.15) Theorem 2.3 (See Ref. [2]). For the system (2.1) and (2.5), the following are

equivalent: 1. There exists a control u(t, 2) such that x(T) = eArx(o) +

[

eA(r-%u(s, 2) ds

is satisfied for all x(T), x(O) and all r > 0. 2. The BVP d~ dt ~k(t) = F ~ + e~z,a,

B~(O) = O,

B~(T) = O,

has a unique solution. Here, ~ = (u, u(1),..., u(Zn))T, a = (--1)"~re-AT'2, and V and B are as in Eqs. (2.14) and (2.15).

N.N. Agwu, C,E Martin I Appl. Math. Comput. 97 (1998) 99-138

105

3. The matrix 2 C = C(h) + I((h)q(h) is nonsingular. Theorem 2.4 (See Ref. [2]). When the system (2.1) is controllable, the control that drives the system from Y(to) to .~(tf) and minimizes the cost function J(u) = oto ftr X-'" z..~k=oruk~2 ~ J ds is given by

u(t,2) = e~ eF('-'°IH-1F +

ee{t-'}6(s)ds

(2.16)

,

d tO

where t E [to,tf], ~(s) = (--1)"~2,~re-A'b, f is given in Eq. (2.I4) and H, and F are given in (2.19) and (2.20).

Here G(h)

(-1) n

e-ASb

dr'

(Is',

Jo

(2.17)

h K(h) = [Jo e-A"b(YTe \ , Fs'~ / ds',

/4=

(2.18)

[ B ~ (BT ) ] '

(2.19)

~(t) = eFt,

r , = - B e ~'f

['f

e-PS~(s) d~ = 8 a ( h ) e - A " o i ,

(2.21)

-J 1o

2 From Eqs. (2.17), (2.18) and (2.23), ~¢~I + ~i~l.I~l:

I-~l~{ e - ~ , ~ •I t o

{ e-~'~e .Ito

a~- ~

'~e~-"~.~e -~'a~ ds. ,/t o

/

106

N.N. Agwu, CF. Martin / Appl. Math. Comput. 97 (1998) 99~138

where

O(h) = ( - 1 ) "+l

f

eF(h-"/E2,bTe~T' ds', a0 is a 2n × m constant matrix.

(2.22)

q(h) = W2BO(h),

(2.23)

where W2 is the 2n × n submatrix of H -1 given by [Wl • W2] = H -l . The vector is given by

~=(e-~toCe-ATto)-l(e-A'f~(tf)-e-At°~(to)).

(2.24)

3. Application of control theory to spline approximation In this section, we describe a procedure for constructing spline functions from control principles. The optimal control law for the system (2.1) and (2.5) is given by Eq. (2.16). Since a control law for the system exists, there exists a set o f points 31,... ,~-1 with xi~ = a~, i = 0, 1 , . . . ,p such that the solution o f the system (2.1) satisfies ~ ( t l ) = ~ , i=O, 1,...,p-l,p. By T h e o r e m 2.4, the control element that drives the system (2.1) from ~ - i to ~, i = 1 , . . . ,p, and satisfies Eq. (2.3) is given by

u(t)][,,_~,t,] = u,(t),

(3.1)

where ui(t) is the restriction o f u(t) on [ti-1, ti]. Now, we need to determine the unknowns ~, i = 0, 1 , . . . ,p. However, since x~ = ~i, i = 0, 1 , . . . ,p, we only have ( m - 1 ) ( p + 1) unknowns to determine. This is realized from the (m - 1 ) ( p - 1) continuity conditions on the control u(t), namely,

ulr)(ti) ~-- Ui+(r)1(ti), .

r = 0,. . ,m . --. 2,. i . = 1,

,p-- 1

(3.2)

and 2(m - 1) conditions at t = 0 and at t = T. Now, from Eq. (2.16),

/

Y

u(t) = e~ eF(t-t°)H-1I'+e Ft e-FS6(s)ds

u(r)(t) = ~T

,,t o

eF(t-to)FrH-IF +

enF r

)

,

(3.3)

e-FS~(s)ds + ZFJ~(r-l-J)(t) ./to

,

j=0

r = O,...,m-

2.

(3.4)

107

N.N. Agwu, CF. Mart& I Appl. Math. Comput. 97 (1998) 99-138

Thus,

,,,-(~)(t)= ~

(

st

eF(t-"')FrH,TlV,+e~F r

e-F%(s)ds+

ti_ I

F:~r~ ' :)(t j=0

r=O,...,m-2,

,) (3.5)

tE[ti_l,ti].

Similarly,

/~l ~( eF(t-ti)FrH~+l1Fi+l + e n F /

/~t

r-I

"~

e-Fs~i+l(s)ds + ZFJ~}+I'-Y)(t)/,

Ui+I (t) =

d ti

j=0

]

r=0,...,m-2,

tE[t~,t;+l],

(3.6)

where, from Eq. (2.11) ffi(t) :

(3.7)

(--1)n'e2n-'~Te-Atb.

Now, (3.8)

[B~(ti_,)] H, =

L

~¢~(t,)

J

and ~(t) = e F(H'-'). Thus,

u}~)(t~)

( ~ e~'UHT1F(t~)+F f r

r

eF("-')~i(s)ds+~7_~ ti- I

~

(i)

j=0

:

(3.9)

(

ui+,(ti) = e~T FrHE+',F(ti+,) + (~)

F r-(r-l-j) o'i+1

(ti)

j=0

)

(3.10)

•

Substituting Eqs. (3.9) and (3.10) in Eq. (3.2), we have ti eF(t'-')6,(S)

eT eYh'UH71F(ti) + F r

ds + ~

ti- 1 ~T = eI

r

ti+l

r~(r-l-j)

o'i+ 1

-~-

F r ff i( r - l - j )

j:0

t (~)

/

ti

.=

r = 0,...,m-

2, i = 1 , . . . , p -

1.

(3.11)

The first term on the right-hand side of Eq. (3.9) is: . 1( ~Te~'FrHTIF(ti) = "~eFh'FrHTlI'(ti)= e~TI e Fhi.r . . .1-1i \ I F 16(ti) ~ "

(3.12)

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N.N. Agwu, CF. Martin / Appl. Math. Comput. 97 (1998) 99-138

If we set t tl (ni : H j) where/-/,!, H" are 2n × n submatrices of/t,71, we obtain HS'

=

~TeF~,F,Hi IF(ti)

~T Fh,~,~,,, ,,,,, ( = e le t~ [rti:n i )

BO(ti-l,ti)2(ti)

0

= ETeFh'FrH['Bf2(hi)e

ATtil ~(/i-1,

(3.131

)

ti).

(3.141

The second term in Eq. (3.9) is:

~Fre ~

e-Fsgi(S) ds = g~FreFti fi I

e-F~i(S ) ds fi- I

- YTUe ~'~['~ e-F~E2n(-- 1)nbre-ATL~(ti-1, ti) ds l

--

Jr1

I

[" e-FsE2,bre ATsds

= ( - l)"Y~Fre Ft'

~

dti

~(ti-I, ti)

I

=(_l)nEVlFreFh(fhe-Fs'y2n~Te AT"ds' × e

'

= -~TFrf2(hi)e

,, t,) Art" ~(ti-l,

ti).

(3.15)

Finally, the third term in Eq. (3.9) is r-I

r-1

j=0

j=o

eTEFJ~I~-'-J)(t,) = -~T, ~-"~F ~ J ~(ri l - j) (ti) r-I = (-1)"+r-'YT~--~(-1)JFJ~'2,bT(A "-x 0T2(ti). j-0

(3.16)

On substituting Eqs. (3.12), (3.15), (3.16) into Eq. (3.11) we obtain ~T [eFh,Fr H[,BiO( hi )e- AVti , _ Fr f2( hi )e-AT te_l r-I -f- ( - l ) n + r - l Z (-1)JFJ~2nbT (Ar-l-J) Y]~i j-o r--I . . . ATt, 4- ( _ _ l ) n + r - l E ( _ _ l ) j F J ~ 2 n ~ T ( A r = e-'_T I [F. n'i+lBi+lf2(hi+l)e j-O

l j) T]~i+'

(3.17)

N.N. Agwu, C.F. Martin / Appl. Math. Comput. 97 (1998) 99-138

109

From Eq. (2.24),

~i = eAV', i c - l ( hi) [e-ah,~ _ ~i-1] . Thus, Eq. (3.17) simplifies to _ ~TMi(hi)eAXt, l C - I (hi)yi-l + yT [Mi(hi)eATti_l C-1 (hi)e-Ah, -- Mi.1 ( hi+l )eATt'c -1 (hi+l)Ix/ -- eT Mi+l (hi+l )eATtic -1 (hi+l )e-Ah~+~X i+1 = 0,

(3.18) where hi = ti -- ti l ~

Mi = eFhiFrHi"Bif2(hi)e -ATti-~ -

Frf2(hi)e

ATt'-'

r-l

+ (--1)"+r-IZ(--1)JFJY2,~(A'-I-J)T

(3.19)

j=0

and

r-I

Mi+, = Frt-I~!+lBi+lf2(hi+l)e AT,` + (-1)n+r-'Z(-1)JFJ~2n-br(Ar-'-J)T, j-O

i= 1,...,p-I,

r=0,1,...,m-2.

(3.20)

Therefore, we can obtain the unknowns i l , . . . , ~ - i by solving the linear system (3.18). This enables us to define the control u(t) piecewise on the interval [0, T]. Further, the solution of the system (2.1) is given by

£(t) = eA'x-° +

[

eA(t-')bu(s) ds.

(3.21)

Now, from the structure of the state matrix A, we observe that xi(t ) = xi+l (t), i = l . . . . , m - 1. Thus, the continuity of~(t) implies the continuity ofx/+l (t) for i = l, , m 1 Also, the continuity of u (r/(t) implies the continuity o f x l re+r) (t), r = 0 , . . . , m - 2. This leads to the conclusion that the observation function y(t) = xl (t) is an element of c2m-:[0, T] and satisfies the boundary conditions y(r)(o ) =xr+l, 0

T y(r)(T) ~- Xr+ 1~ r----0, • . . m - 1

(3.22)

and

y(ti)=xl(t,),

i= 1,...,p-1.

(3.23)

Thus, y(t) is a spline function and the above discussion proves the following: Theorem 3.1. Let Y(t) be the solution o f the system (2.1), (2.2), and (2.5) with xl(ti) = gi, i = 0 , . . . ,p. Then there exists a unique function y(t) E Cm-2[0, T] that satisfies Eqs. (3.22) and (3.23).

110

N.N. Agwu, C.F. Martin/Appl. Math. Comput. 97 (1998) 99-138

4. Classification of splines

The type of spline is determined by a set of basis functions. By control principles, we can construct these basis functions. Now, suppose the interval [0, T] is subdivided into p subintervals. In this analysis, it suffices to consider just one subinterval to determine the kind of interpolation functions of the state ~(t). Thus, on the subinterval [to, h], Y(t) = e'4(t-/°)£(t0) -[-

f'dl0eA(t-')bu (s,) 2 ds.

(4.1)

The control law is given by Theorem 2.4:

/

u(t,~.) =e -'-T eF(t-t°)H-lF +

ft eF(t-s)5(s,~.)ds / . d to

-

(4.2)

-

From Eqs. (2.17) and (2.18)

G(t) = (-1)"

e-AS'b

EiTeF(S'-/)E2,bre-At/ dr'

)

ds',

(4.3)

t-to

K(t) = foo

(4.4)

e-Aeb@~eF")ds'

and

(4.5)

C(t) = K(t)l(h) + G(t). If we substitute Eq. (4.2) into Eq. (4.1), we obtain

.~(t) = eA(t-t°).~(to) +

e'4(t-s)[) ~

eF(t-t°)H-1F +

eF('-~)~(r)dr

ds

= e~('-'°)[: + ~(~) ~.(h)e -~'oz + G(~)e-~'o~]

__~,1-.o,? + (~(,/.<~)+ ~('//e-~'~"°~-'~'° (e-'"~'- e-~'o:)] _- ~<,-,o>[: + ~<,/~-' <~)e~'°(e-'"~'- e-~'o:)] : e A(t-tO' Ix-4) -~- C ( , ) c - l ( ' ) ( e - A h '

1-

x-4))].

(4.6)

N.N. Agwu, C.F. Martin I Appl. Math. Comput. 97 (1998) 99-138

111

Now, since the matrix F has linearly independent eigenvectors (Pl ,Pz,... ,p2,), then F is similar to a diagonal matrix D = diag(fll,... , f12,) [19]; that is, there exists a nonsingular matrix P such that

F = PDP-'. Let

(4.7)

f11. 12. . ./ . / ...... / \P2n,l

P2n,2 . . .

P2n,2n

\Pl2n,1

t:]21,2n

Here, the columns of P are the eigenvectors pk, k -= 1 , 2 , . . . , 2n, of the matrix F. Let E be the Jordan canonical form of the matrix A. Then A = QEQ -l, where Q is the matrix such that AQ = QE. Thus, further analysis is simplified by replacing the matrix F with PDP -1 and the state matrix A with its Jordan matrix E. Hence, we may write

x(t) mQeE(t-t°)Q-l[ X~O+C(t)C-l(h)(Qe-EhQ-lY l - x-'°)].

(4.8)

Let the first row of the matrix QeEIt-to)Q-I(I - C(t)C-l(h)) be (4h(t),..., q~m(t)), and the first row of the matrix Qee(t-to)Q-IC(t)C-l(h)Qe-ehQ -1 be (q;l(t),... ,~km(t)). Now, the system (2.1) is controllable since rank (b Ab... Am-I b) = m. The output of the system is

y(t) = ETa(t) = e~l( eA(t-t°)x-'0 Jr-

~A(t-s)~)U(S) as

=Qee(t-tO)Q-l[~ +C(t)C-l(h)(Qe-EhQ-'~ ' -x"°)] = (~bl(t),..., Om(t))x-~ + ( ~ , ( t ) , . . . , ~bm(t)).Y' m

m

= Zx°~bk(')+ Z x ~ , ( t ) . k=l

(4.9)

1=1

By choosing f i = ~k and ~t = ~, we obtain q~') = Thus, for r = 0 , . . . , m - 1, ~ ) ( 0 ) =y(~)(O)

k (h)=y(~)(h)

~---

y(r)(t),

xl~)(O)= Xr+,(O) = X r + 0

1

=6k,~+,,

x (h)

l

=0,

=

x~+,(h)

=Xr+

k=l,

r

•

=

0 , . . . , m --

.. m.

1.

N.N. Agwu, CF Mart&/Appl. Math. Comput. 97 (1998) 99 138

112

In a similar manner, if we choose ~ = 0 and £1 = O'j, then we obtain

~l~r)(t) = y(r)(t),

r = 1,...,m.

Hence, ~r)(0 ) =y(r)=0,

~l~r)(h)=y(r)(h)= 6j,r+l,

j=

1,...,m.

The above discussion proves the following theorem. Theorem 4.1. Let A be given by Eq. (2.2). Furthermore, let the first row of the matrix Qee(t-t°)Q-l (I - C(t)C-l (h ) ) be ( dpl ( t), . . . , ~bm(t)), and the first row of the matrix Qe E(t t°)Q-tC(t)C-I(h)Qe-EhQ 1 be (~/l(t),...,~m(t)). Then, for r = O,...,m,

~)~r)(h)= O,

q~r)(O) ~- ~k,r+,,

~lllr)(t) = O,

~Ir)(h) = ~l,r+l,

k = 1,... ,m, l =- l ~ . . . ,m.

Therefore, ~b~,~'k, k = 1 , . . . , m are basis functions fory(t). These basis functions depend on the entries of the matrices eEt, K(t), and G(t). Hence, we can determine the type of spline function by carefully examining the entries of these matrices.

Proposition 4.1. Let the state matrix A be nilpotent of order m and the cost function J(u) = f f ~,~=0 (u(k) (s))2 ds. Then y(t) is a polynomial spline if and only Ifn = 0 . Proof. To prove this proposition, we observe that the spline function y(t) is expressed in terms of the basis functions ~bk and @k, k = 1 , . . . , m , Eq. (4.8). However, the basis functions themselves are dependent on the entries of the matrices e £t and e Dr, where E and D are the Jordan forms of A and F, respectively. Now, if A is nilpotent, as in the proposition, then A is already a Jordan matrix and hence the entries of eEt are all powers of t. It then follows that the spline function y(t), which is a linear combination of the entries of e& and eDt is a polynomial if and only if the entries of eDt are all powers of t. This is the case only if n = 0. [] From Theorem 4.1, we see that the basis functions q~i, and ffJi are determined by the matrices e e', and eEtQ-1C(t). But these matrices are themselves characterized by the spectrum of the state matrix A, and the matrix F. Therefore, we will classify the spline functions obtained by control principles by examining the entries of A, and F. In this classification, we consider the case where the state matrix has dimension 2 and the cost function J ( u ) = f~((u(s)) 2 + (u'(s))2)ds. For higher dimensions and derivatives of higher orders, the procedure is similar but more involved. So let A = (0

~l

,) , 2~2

~1,~2 C O, ,

~3~- (0 l) T.

N.N. Agwu, C.F. Martin / Appl. Math. Comput. 97 (1998) 99-138

113

Then the eigenvalues of A are

Thus the J o r d a n f o r m o f A and the t r a n s f o r m a t i o n matrix Q are as follows:

0 Since J(u)

=

{2'

Q=

fT((u(S))2

q-

{1

{2

'

{2

{~

-{1

l

"

then, f r o m Eq. (2.14),

(ut(s))2)ds,

(o~)

F=

1

with eigenvalues and eigenvectors

f l lz l ~ Thus

f12 = -- 1,

Pl =

1 '

(~ l) ,, ~(1 ,) o (1° Ol) eO~ (~ e0) P=

1

-1

'

=2

1

1

P2 =

- 1 '

'

Also, f r o m Eq. (2.15), B=(O

t)

and on the subinterval [ti l, ti],

O(t) = e F(t-'~ ~/ =

cosh (t - ti-1) sinh(t - ti-1)

sinh (t - ti-1) cosh(t-

ti 1) J "

Thus, B~b(t) = ½( sinh (t - ti-i ) c o s h (t - ti-1) ) and f r o m Eq. (2.19), we have

Hi = (B~9(ti_l) )

_(o sinhh~

(4.10)

1)

(4.11)

coshhi

--| /' c o s h h i /7/1 - sin-lahi !\ - sinhhi

-1"~ 0 ) '

114

N.N. Agwu, C.F. Mart&/Appt Math. Comput. 97 (1998) 99-138

where h~ = t~ - t~_l. From Eq. (2.17), G(t - ti 1) = -

)

/o" (/o (~.(t) g~2(t) e-A;-b

~eFts'-/lY2bre-A~/ dr' ds'

= Qk~21(t) g22(t))QT' where

--1 {--2(t--ti_,) e('-¢l)(t-t'l)-- l e-(l+~l>(t-t~-')--l.} gl'(t) -- 2(¢;---¢1)

--1

f---~2

+

(1 - ¢,)2

f2e(C~ ~_,)(ttit)=__ 1

(1 + ¢,)2

,

e (I-¢0(t-,i-I) --I

-~ (1 + ¢1)(1 + ¢2) J) -1

~ ' - 2 e ~2 ~l>,, ,, 1>_ 1

t~2'(t) - 2 ( ¢ ; - ¢,) [ (1 - ¢~)(¢2~-~-)

e(t-~2)(t-t~ 1) _ 1 (1 --¢1)(1 --¢2)

e-(l+¢2)('-'i ~ -- l_], -~ (1 + ¢1)(1 --~ ¢2) J '

-1 f-2(t_-_t_,._i) g22(t) - 2(¢~~ ¢,) [ 1 - ¢~ +

e (l-¢2)(t-ti-') - - 1

e -(l+¢2)(t-ti-~) - 1}

(1 - ¢2)2

(1 + ~2) 2

From Eq. (2.1 8), K ( t _ ti_,) = foot-t'-'e-As',(fileF(~'))ds' = Q ( ''t(t) ]~21(t)

L2(t) )) ^ k22(t)

where -1 ~ (e ('-¢')('-'~-') - 1 /~,,(t) - 2(¢~ ~ ¢,) I. k i-~ -1 (e (1-¢')(`-'i-') -- 1

l+~i

,))

1+~1

1

k21(t)-2(¢2- ~l) I,\ 1-¢1 I (e 0-*')0-,'-') - l ~22(t) - 2(~2

'

~ (e(,-~(t-t,_,)_ 1 e-(l+¢l)(t-ti-O- 1)} ¢,) { \

l -

~,

1+¢1

e (l+~,)(t-ti-,)- 1.) / l+~l "

"

N.N. Agwu, CF Martin I Appl. Math. Comput. 97 (1998) 99-138

115

Eq. (2.22) gives hi

eF(h'-Sl)e2bTe-ATs'd r =

~-~(hi) = f

(fDll

d0

\ (-t')21

~12 ) QT, O)22 ./

where d911 : ,12:

(f)21 =

(D22 =

¢1 sinhhi + coshhi - e -¢lh' (¢2 - ¢1)(1 - ¢~) '

¢2 sinhhi + coshhi - e -¢2h'

'

sinhhi - ¢1 coshhi + ¢le-¢~hi (¢2-¢1)(1-¢~) sinhhi - ¢2 coshhi + ¢2e-C2hi

--

(¢2- ¢1)(1-¢~)

and Eq. (2.23) gives

q(h) = H['BO(hi) = (?]11 /112 "]QT, ~21

(4.12)

/722 ]

where sinhh~ - ~1 coshhg + ~le -¢th' , /~21 = 0, (¢2 - ¢1)(1 - ¢~) sinhh, sinhhi - ¢2coshhi + ¢2e-¢2h' ?]22 : 0. ?112 = (¢2 - ¢1)(1 - ¢~) sinhh~ ~11 ~ -

Therefore,

( [ql(t)thl(hi) [ql(t)th2(hi) ) QT K(t)q(hi) = Q ~,Ic21(t)~ll (hi) k21(t )tl12(hi ) ^

and

C(t) = G(t) + K(t)q(hi) = Q(~,l,(t)

+ ]£11(t)~ll(hi) ^

\ 21(t) + k21

c22(t) J

glz(t) +/¢n (t)~lz(h,)) QT

(h,) g22(t) +/¢2, (t)th2(hi) (4.13)

116

N.N. Agwu, CF. Martin / Appl. Math. Comput. 97 (1998) 99-138

where

6"ill (/) = gll(/) -[- kll(t)Oll(hi), Cil2(t) = gl2(t) + ]fll (t)fh2(hi), ~, (t) = g2, (t) +/~21(t)l}ll (hi), 0"i22(t) = g22(t) q- ]~21( t)fll2( hi),

C l(h~) = Q-T

)22 + k21012 -

= (~ill(hi)

(

£~21 +

k210.

,

^

Q-'ICI-'

C~2(hi))

(4.14)

where

[C[ = (gll(h/)-[-kll(hi)~ll(hi))(g22(hi)-}-k21(hi)~12(hi)) - (gl2(hi)-[- kll(hi)~12(hi))(g21 (hi)-[-k21(hi)~]ll(hi)). From Eq. (4.9),

eA(t to)[l--C(t)C-'(hi)] = ( dPl~t)

(4.15)

where (t) -

-~2-

1

-- ¢1 {¢2e~'(t-t' ') -- ¢'e~('-"-')} ¢2

1

-- ~1 1

q-

1

[g,,(hi)Ou(t) + gz,(h,)O,2(t)]~ze¢'(t-,' ' lCl-' [ell (hi)c21 (t) + C2I (hi)c22(t)]~2 e~2(t-t' ') [C[ -1

[~12(hi)Oll(t) + ~22(hi)~12(t)]~le ~'tt-t' ~I}C] 1

1

~2 -- ~1 [c12(hi)c21 (t) + ~22(hi)~22(t)]~l e¢2(t-ti ~)]C1-1 and 1

q~2(1) -- ¢2 _ _-- ¢1 [~'-e~l(t ti I) q_e~2(t-ti-l))'~

-[- ~ q- ~

1 1

[Cll (hi)cll (t) -{- c21 (hi)el2 (t)]e ~l (t-ti l>lCl-' [ell (hi)c21 (t) if- c2l (hi)c22 (t)]e ¢2('-' ')1Cl-'

(4.16)

ll7

N.N. Agwu, CF. Martin / Appl. Math. Comput. 97 (1998) 99-138 1

~2 - ~ [c,2 (h~)O,, (t) + ~2 (hi)O,~ (t)]e ¢' ('-t~'/] CI-' 1

[~2(h~)b2, (t) + g22(hi)Oz2(t)]e ¢2(t t~ ,)[cl-l.

~2

(4.17)

Also

e~(t-to)C(t)c-l(hi)e Ah,= ( ~tl~t)

(4.18)

where [~ll(hi)Oll(t) 4-(21(hi)O.12(t)]¢2e~l(t_t ' ,_h~/ici a

~tl(t) _ 1 4- ~

1

[~, (h~)~2, (t) 4- ~21(h~)~22(t)]~2e (¢~/t-t' ,)-¢,h,/iCi -I

1 1

~2 -- ~ [012(hi)c21 (t) 4- g22(hi)c22(t)]~l e¢2(t-t~'-h') ICl

',

(4.19)

and

t~2(t) -

1

~2 - ~l [ell(hi)Cll(t) 4-a21(hi)e12(t)]e~It-t'-~-h')lC]-I 1

~2 - ~, [~,,(hi)~2,(t) + ~2,(hi)~22(t)]e(~2~t-t,

+~ 4- ~

1 1

,~ ~,h,~lC I '

[~12(hi)O,, (t) 4- ~22(hi)612(t)]e I~'('-t' ') 2h,tlC[-' [c12(hi)c21 (t) + c22(hi)c22(t)]e ~z(t-tj L-h,)[C[ -1"

(4.20)

Now, let us consider the various cases that arise from the various forms of the eigenvalues of the state matrix A. Case 1:~2 4- ~ > 0. In this case the matrix A has two distinct eigenvalues. Case l(i): If ~, ¢ ~2, then the spline obtained is an exponential spline with basis functions ~b~(t), ~b2(t), ~91(t) and ~2(t) given by Eqs. (4.16), (4.17), (4.19), (4.20). Here, ~b1(t), q~z(t), I/t I (t) and ~q(t) are linear combinations of e t, e -t, e ~'', te ~lt, e ~-2t, and te e2t. Case l(ii): If ~2 = 0, and ~1 > 0, then ~l = -~2 and we again obtain an exponential spline. The basis functions are linear combinations of e t, e -t, e ~t, e -~lt, te ~lt, and te -~2t. Case l(iii): If ~l = 0 , and ~2 ¢ 0 , then ~1 = 0 , and ~2 =-2ct2 if 0~2 > 0; ~1 -- -2~2, and ~2 = 0 if~2 < 0. The resulting spline function is a linear combination of the functions 1, t, e', e t, e~:~, and te ~t if ~ > 0; or 1, t, e t, e -t, e ~t, and te - ~ if ~2 < 0.

N.N. Agwu, CF. Martin I Appl. Math. Comput. 97 (1998) 99-138

118

Case 2: a~ + ~1 ~<0. This case leads to two complex eigenvalues: ~l = 0{2 + ion,

{2

----~ 0 { 2

--

ion,

where cn = V/-(a~ + a,). Case 2(i): 0{2 ~ 0, al < 0. The resulting spline has basis functions which are linear combinations of e t, e -t, e ~2tcos t, e ~2tsin t, te "2t cos t, and te ~2tsin t. Case 2(ii): 0{2 = 0, al < 0. This gives ko, ~2 = - i x / ~ 7 = -ia~, and a spline function whose basis functions are linear combinations of 1, t, e t, e-', cost and sint. Case 3:a22 + al = 0. This implies {1 = {2 = 0{2. Case 3(i): 0{2 ¢ 0. Then ~1 = i~/--(~l =

(02

1)

eEt

e~:,(l

a2

Q=(12

01),

0

t) 1

Q - ' = ( _ 1 0 { 2 °1), l--ti_ 1

G(t-ti_,)=

l -~,o

'

St

e-ACb(fi'eFele-F~'eEbre-Ar~'dY~ds' \ 1 Jo /

( ll(t) &(t))

=-Q\~2,(t)

~22(t)

(4.21)

'

where g,ll (t) = A1 (t)e -2~2(t-t'-') + A2e -~2(t-t~ ') + A3, ~12(t) = B,(t)e -2~2(t-'-') + B2(t)e -~2(`-'-') + B3, ~21(t) = Cl(t)e-2~2(t-*, 1) + C2(t)e-~2(t-t,-O + Ca, ~22(t) = D,(t)e -2~2(t-t~-') + D2(t)e ,2(t-t,_,) + D3, where (t ti_l) 2 t -- ti-I 1 A,(t) - a2(1 - a22) t- 2~2( 1 _ ~22) +4~2( 1 _ a22) -

Az(t) - - (1 _a~)2, A3(t)

1 1 + 4~3(1 _ ~2) (1 - a~)2,

1 a2(1 _a~)2,

119

N.N. Agwu, C F Martin/Appl. Math. Comput. 97 (1998) 99-138

t - ti-1 2~2(1 - ~2)

Bl(t) -

1 4~22(1 _~22)'

t-ti_~ 1 B2(t) - ~2(1 - ~2) k ~z22(1 _ ~22),

1

1

B3(t) = 4a~(1 - ~ )

~- ~ ( 1 - ~2),

t--ti-I

Cl(t) =

2~2(1 _ ~2 )

1 1 - ~22) ~ (1 - a2)2'

4~(1

2 (1 - ~2) 2'

C2(t)

1 1 + 4~2( 1 - ~2) (1 - ~22)2'

C3(t)

1

1

D , ( t ) - 2a2(1 - ~2), =

K(t-ti_l)

f

t-ti-,

Jo

e

D2(t) -As"I'*[--TFs"

o(ele

1

c~2( 1 _ a~),

)ds'=

D3(t) - 2a2(1 - ~2),

Q (]~ll (t) ]~12(t)) , \/~21(t)/~22(t)

where

L,(t) L (t)

= E1 (t)e (l-"2)('-t'-~l + E2(t)e -0+~2)It-t'-~l + E3(t), = FI (t)e 0-'2)(t-t/-~) + F2(t)e -11+~2)(H~ ~) + F3(t), = Gl(t)e O-"2)O-t' ') + G2(t)e -O+~2)tt-`~-') + G3(t), = H1 (t)e (l-~z)(t-ti 1) + H2(t)e-O+~2)(t-t~ ,) + H3(t), t - ti-l

E,(t) E2(t)

E3 ~

-

-

1

2(1-~2)

2(1-~2)

2'

t-ti ~ 2(1+~2)

t-ti 2(1+~2)

2'

2~2 -

Fl(t)--

(1 -

t - t~_~ 1 2(1 - ~2) I- 2(1 - ~2)2'

t - ti-~ F2(t)--2(1+c~2)

1 ~ 2(1+a2)

2'

F3(t)--

1 +~ (1 - ~2)2'

(4.22)

120

N.N. Agwu, C.F. Martin I Appl. Math. Comput. 97 (1998) 99-138 1

1

Gl(t) 7----2(1 - ~2)'

G2(t) - 2(1

1

~2)'

+

1

HI(t)

-

2(1

-

f2(hi) =

-

(1 - ~22~'

1

~2)'

H2(t) -

1

2(1 + ~2)'

fh, d0

G3(t)

H3(t) -

(1

(~,,, ~,,2)

eF(h'-'¢)g2bVe aT's'ds' = Q (D21 (3)22 '

- ~2), (4.23)

where

7(7 7~?

) e(I

1 (e (1-=2)hi - 1 0)12 =

--2 k f T g

2(1 -i- =2)2 -)

°~2)hi

e -0+~2)h~ -- 1"]

f77~2- /'

I-

(h i - l_%hio~2~e(l_~2) & (_l Jr- hi(1 q-g2)'~ e_(l+a2)h~ 0)21 =

k27-1-~2)2 ]

\ 2-(i+~-77 ]

1 (e (~ =2)h,_ 1

Hi =

e -(l+~2)h' --

(.)(o BeFh'

(('1 -- 0{I)2) "~

( coshhi HT-1 - s~ffhi k, - sinhhi

'

1.~

,)

sinhhi

-1

(2(t_q-_ 0{~)"~

+ k(1-~)2]

coshhi -1) 0 '

(4.24)

Thus, 1 /4,."- hsinh ~

(;)

(4.25)

and q(hi) = H [ ' B e & Q ( h i ) =

(qll

q12),

k, q21 q22 where hi t711 ~--- " 2(1-- 0{2)

1

2(1

- ~2) 2

÷

hi cosh hi 2(1 - ~2) sinhh/

N.N. Agwu, CF Martin I Appl. Math. Comput. 97 (1998) 99-138 hicoshhi 2(1--~nhhiJ

"~

hcoshhg 2(1 +~2)sinhhi

( hi e(l-~:lh`+ \ 2(1 +~z)

121

1 2(1 +~2) 2

h~coshh~ ~ -(l+~2)h 2(1 ------:T-z-. ' + ~2) slnhhJ e

1 1 hi cosh hi 2(1 -~2) 2 ~(1 ct2)----------5 + + 2 ( 1 - ~2)2sinhh/

h~coshhi 2(1 + ~2)2 sinhhJ ' (4.26)

coshhi 1 ____ .~e ( 1- ~21h, 2(1 -- 0~2~ @ 2(1 - ~2)sinhhiJ 1 coshh~ 'x q'~'h --I e -~ * 2) 2(1 + ~z) 2(1 + ~2)sinhhiJ 1 ( coshhi~

/712 :

+

(4.27)

1 -~2 1-Ct2sinhhij, q21 = O,

?]22 = O,

Ki(t - ti-l)rl(hi) = Q (

kll (t) l"]ll (hi) [Cl2(t)rh2(hi)^

) QX.

\ k21(t)~/ll(hi) k21(t) r/12(hi)

This leads to

Ci(t) = Ki(t)q(hi) + Gi(t) Qf [ql(t)rhl(hi) - ~ll(t) !11(t)rl12(hi) - g12(t) ~ it21(t)rhl (hi) - g21(t) k21(t)rh2(hi) - ~22(t) J

QT

(4.28)

\ cizl(t) ci22(t),] ' where Cill (t) : ]¢11(t)r/ll (hi) - gll (t),

c~i2(t) : ~2Cll (t) -4- 1¢11(t)rll2(hi) - gl2(t),

c~1(t) = ~2cll (t) +/c21 (t)qll (hi) - g21(t),

ci22(t) -- ~2c21(t) + 0~2(kll (/)~]12(hi)- gl2(t)) -[- k21(t)?ll2(hi

) - g22(t),

122

N.N. Agwu. CF Martin I Appl. Math, Comput. 97 (1998) 99-138 cTl(hi)=Q_T(

=

/~21q12- g22 --(~721"11--g21)

-([q~l,2-R,Z))Q_llc[_~ kll,ll -gll

\

(4.29)

\e,,(h,) eh(h,) ','

where ]C[ -~- (kll(hi)?~ll(hi) -

gll(hi))(k21(hi)~12(hi)- g22(hi))

- (kll(hi)?~12(hi)-

gl2(hi))(k21(hi),ll(hi)- g21(hi)).

Now

C(t)C_~(h~ ) = (c~(t)

c~2(t )'~ (~(hi) ~2(hi)'~ c~2(t) J \ U21(hi) ~22(hi) ] = (~ ]l(h)cll(t) + U:l(hi)cl2(t) ?-~2(hi)c|l(t) + U22(hi)c12(t)) \ 6~l(h~)c2~(t ) + g~l(hi)c:z(t) ?.~2(hi)c2~(t) + U~(hi)c22(t) ' \C~l(t)

(4.30)

eA(t-ti-1)[I - C(t)c-l(hi)]= (~bl~ t) q~2~t)),

(4.31)

where q~(t) = e~2(t-t'-~){(1 - ~2(t - ti_l)) (1 - oill(hi)cll(t ) - ~l(hi)c12(t))} - e~2(t-t'-') {(t - ti_l)(~ill(hi)c2,(t) + oi2,(hi)c22(t))}, (4.32) ~b2(t) = e ~2(t-t' '){-(1 - ~2(t- ti-l))(Oi12(hi)cH(t) +?.i92(hi)c12(t))} + e~2(t-t'-') { (t - ti_l) (1 - ~12(hi)c2, (t) - ~i22(h,)c2z(t))}. Similarly,

eA(t-t,-,)C(t)C-l(h,)e-Ah,=(~kl~t)

~kz~t) ),

(4.33)

(4.34)

where

~1 (t) ~ ea2(t-ti-I-hi){(1 - ~2 (t -

ti-l) )(1 + o~2hi)(Gill (hi)Cll (t)

+ el2,(hi)c,2 (t)) } + e ~2(t-t'-~-h') { (t - ti-, )(1 + ~2hi) (?.',1(hi)c2, (t) -]- c,i2!(hi)c22(t)) } -I- e a:(t-t~-'-hi) {~2hi(1 - ~2(t -- ti-i ))(e'il2(hi)Cll (t) + ?J92(hi)cl2(t)) } 4- e~2(t-t'-~-h'){ot~hi(g.il2(hi)c21 (t) + ?.i92(hi)c22(t)) } (4.35)

123

N.N. Agwu, C F Martin I Appl. Math. Comput. 97 (1998) 99-138

and ~k2(t) = C2(t-t,_,-h,){_hi(1 - ~2 (t - ti_ 1))(~'il 1(hi) cll (t) + C~l (hi)elE(t)) } _ C2(t-t,_,-h,){hi(t - ti-i )(~'i11(hi)¢21 (t) -'[- ~'/~1(hi)c2a(t)) } + e'2(t-t'-~-a'){ (1 - 0t2hi)(1 - ot2(t - ti-,))(~i,2(hi)c,, (t)

+ ~i2z(hi)el2(t)) } + C 2(t-t'-~-hi){(1 - o~2hi)(~i12(hi)c21 (t) + oi?2(hi)c22(t)) }.

(4.36)

Thus, the resulting spline has basis functions which are linear combinations of 1, t, F, e -a2t, te -~2t, e -2a2t, te -2~2t, t2e -2=2t, e (l-=2)t, te (l-=2)t, e -(l+a2)t, te-(l+ct2)t. Case 3(ii): ~t2 = 0. In this case, the state matrix is in Jordan form and hence, G(t) and K(t) may be computed directly as follows: A=

(0 '0) 0

"

On the subinterval [ti-l, ti],

/4,.=

~ea,

=

,)

(4.37)

e,:d-~, e,,o-., ,2

1 (-coshhi H/-l -- sinhh-----~ sinhhi

10)

(4.38) '

" 1 (10) /-/I - sinhh-~ ' O(hi) _-

f0''

(4.39)

eF(hi_s,)~2~Te_ArS, ds, =

0)11 0 ) 1 2 \ O)21 0)22

(

)

(4.40)

where 0)'11 = hi - sinhhi, 0)t12 = c o s h h / - 1, i ~ --0)112~ 0)22 i sinhhi. O)21 Also .(h,) = H"B~(h,) -- ( "" \ q21 where ~ l l l = ( 1 - c o s h h i ) / s i n h h i , Eq. (2.18),

),

(4.41)

r/12=l, r/21=0, and q22~-0.

From

.,2 q22 /

N.N. Agwu, CF Martin / Appl. Math. Comput. 97 (1998) 99-138

124

K(t~ =

e Fs' dJ

f0'e -As'

)

(4.42)

= ( kl I (t-) k12(~ )

(4.43)

where kll (t-) = - t s i n h J + c o s h t k21 (t-) = sinh~,

1,

k12(t~ = i c o s h i -

sinhi,

k22(t-) = c o s h ~ - 1,

Ki(hi) = (-hisinh(hi) + cosh(hi) - 1 hi cosh (hi) - sinh (hi) sinh (hi) cosh (hi) - 1 ; Ki(t)rl(hi ) = ( ~,l(t-)

(4.44)

\ k2, (t3

where

~,, (t3 = kH (t-).,,,

k,2(t-) = k,l(t-).,~,

k2l (t-) = k~l (t3.., =

\k:,(h) L:(h)

F r o m Eq. (2.17),

H(0:

Gi(t)=(-1)/e-ASb

~

)

eF(s-r)~2bTe-ATrdr ds

0

= (g,l(t~

g21 (t~

gl2(t-)~

(4.45)

(4.46)

g22(t~/'

where gll (t-) = ]~3 + sinh~ - 7coshi, gl2 (t-) = -½i 2 - c o s h i + i s i n h i + 1, g21(t-)-- c o s h i - 1

-½~,

g22(t-) = t -

(½h~ + sinh (hi) - h/cosh (hi)

G'(hi) = \ cosh(hi) - 1 - lh~

sinh~,

-½h 2 - cosh(hi) + hisinh(hi) + l'~ hi - sinh ( h i ) )

125

N.N. Agwu, CF Martin I Appl. Math. Comput. 97 (1998) 99-138 Thus, ci(~ = x'(tO,7(hi) + oi(t3

(]~11 (t-) -~-gll (t-) ]~12(t-)-]- gl2(t-) ) = ( Cill(t-) Cil2(t~ \ k~, (t3 + g2, (t-) k22(t-) q- g22 (t~ \ c~l (t-) c22(t-)J I

^

^

i

(4.47)

where 4 , (t-) =

(gsinhg- coshg+ 1)(coshhi- 1) ~3 sinhhi + -~ - t c o s h t + sinht,

F 42(t3 = - ~ , 4~ (t3 = - 1

~2 2

~sinht (c°shhi - 1) + cosht,

42(t~ = 9, 9=t ti-~, i = 1,... ,n - 1. Hence, when t = ti, we obtain t = hi, and

(4.48)

-

Ci(hi) = gi(hi)rl(hi) + Gi(hi) = (]~11(hi) q-gll (hi) kl2(hi) q- gl2(hi) \~:2, (hi) + g~ (hi) k22(hi) + g22(hi) ^

= (C{l(hi) c{2(hi)~. ~,,ci21(hi) ci2(hi )/]

) (4.49)

Furthermore,

czl(hi) =d(Ci(hi,)-l(

c~2(hi' -ci21(hi)

-cil2(hi) ) = (Cll(hi' Cill(hi) \ c21 (hi)

Cl2(hi)), e22(h,)

where d(Ci(hi)) denotes the determinant of C,.(hi) and is given by

d(fi(hi) ) = Cll (hi)c22(hi) - Cl2(hi)c2, (hi).

(4.50)

Therefore, from Eq. (4.8), we have the following:

eA'-eA'Ci(t-)c[l(hi)=(lo

1)-(;

1)

+ ¢,2(t3<, 6'11(t-)Cl2 q- C,2(t-)e22 "~ ~k¢22 (t-)Cll -1- C22(t-)c21 C21(I-')CI2"1-¢22(t-)c22 ; ~__ ( (/)1~t-) (])2~t-)), x (c.(t3<,

(4.51)

where

q~l(t-) ~- 1 -[c,~ (t-)cll-~-c12(t-)c21-t-t-(c21 (t-)Cll q-c22(t-)c21)]

(4.52)

~2(t~ = 9- [ell (t-)el2 -~- c12(t-)e22 At-9(c21(t-)el2 -~-c22(t~e22)].

(4.53)

and

126

N.N. Agwu, C F Martin I Appl. Math. Comput. 97 (1998) 99-138

Substituting for co(t ) , these equations simplify to: ~bl(t-) -- 1 q-~'11t--e21-~71-~-'ll~--C11

[

sinhi+

(I - cosht-')(coshhi- 1)] ] sinhh; (4.54)

and (~2(t-) = (1 --~ 6",2)t -- C22"~ + ~'12"~ -- 6"12 sinhT-q

(1 - cosht-)(coshhi- 1)] sinhh~ ] (4.55)

Similarly, eAtCi( t-)C l (hi )e -Ahi 0

k c21(t-)011 + c22(t-)OZl

= ( O,Jt) g,~t)),

Cll (t-)Cl2 -~- 6.12(t-)C22 c~1(t3~,~ + c2~(t3~2) ( l0 -hi)l (4.56)

where ~1 (t-) = [CII(t~Cll ~- 6.12(t-)C21 -[- t(C21 (t-)c,I --~ 6.22(t-)c21)] -~- 1 -- ~l(t-)

and

~/2(t-) = =

(4.57)

-hi[Cll(t-~Cll--~c12(t--)c21--[-t(c21(t-)Cll --[-c22(t-)c21)] + [c. (t3~,2 + 6.12(t3~ + ~(c~,(t3~12 + 6.~2(t3e2~)] -hi(1 - ~bl (t-)) + i~b2(t).

(4.58)

Thus, the spline obtained has basis functions which are linear combinations of I, t, t 2, t 3, e t, and e -t.

5. Convergence and numerical results

Here, we will examine convergence rates for the spline approximant for case 3(ii) and then give the results of computer simulations. 5.1. Results for a nilpotent matrix Since the state matrix under consideration is 2 × 2 (that is m = 2), then it follows that r = 0, and, from Eq. (3.18), the requirement ul )(ti)= Ui+ (r)1(ti) yields the following:

N.N. Agwu, CF. Martin I Appl. Math. Comput. 97 (1998) 99-138

M,(h,) =

:H:'~(h3e

--

-:~'-' -

127

~(h,)e-:'~-'

(e~n;'8 -/)~(h)e-:"-'

(o 1

,

~Mi(hl)e :t'-~ C -~ = (di ei),

(5.2)

~TMi(hi)eATt'-~C-le -Ah' =(di - h,di + ei),

(5.3)

where di =

ei =

--(-Dill--~ ~ ( - D 2 1 j

C'~ll-Jr

[

coshhi

] [

i 6i12 + _~oli 2 + ~cos./ o)22j~,22

--(-Oill+ ~coshhi 0)21

" -Art' = ( 00 M/+I (hi+l) = H;+lBI2(h,+l)e eTmi+l (hi+l)e ATt'C -I

i ]-i

=

,

'

(5.5)

(sinhhi+')-l)12(h)e -'4"t', 0

(d/+, e,+l),

(5.6) (5.7)

eTm/+l(hi+l)eaTtic-me -Ah'+' = (di+l - hd/+l + ei+l),

(5.8)

where .i+1 =i+1 _1- - i+1 =i+l

di+l ~_-w21 Cll ~ ('022 t:21

sinhhi+l

'

(5.9)

o/i+ 1 7-.i+1 ,,..hi+1~i+ 1 21 t;12 + w22 ~22

ei+t = sinhhi+l ' ~fMi(hi)e -:h' C-'e -ah' + #'M,-+~ (hi+,)e -AThi+l C -1 = (a~ + e l - hd~ + d 2 + e 2 ) .

(5.10) (5.11)

Substituting in (3.18), with ~ = (a,- fli) a', we obtain -(di - -

e~)(;:~:)+(di+di+~ ( di+l

-hidi+e,+ei+~)(fli)

i+') -hi+ldi+l + ei+l ) \( afli+l

= O,

i= 1,...,p-

1.

(5.12)

This gives -

-

eifli-i + (-hiai + el + ei+~)fli - (-hi+tai+l + e,+,)fli+, : di(xi-i - (di + di+l)O~i + 6+l(Xi+l,

(5.13)

N.N. Agwu, C.F Martin / Appl. Math. Comput. 97 (1998) 99-138

128

(Y

0

.

.

x

Z

0

...

0

y

Z

...

0 \0

.

0 0

f12 / f13

..

x

Y

z

• ..

0

x

Y J \tip-1 I

•

DI

d2~1- (d2+ d3)~2 + ds~3 d3~ 2 - (d 3 + d4)0~3 -[- d4~ 4

(5•14)

dp-2O~p-3 - (dp-2 q- dp-l )O~p-2 if- dp-l~p-I

De where x = -e~, y = -hdi + ei + ei+l, and z = - ( - h d i + l -[- ei+l), Do = dido(d1 + d2)~1 + d2~2 + eiflo, and Dp = dp-l~p-2 - (dp_l + dp)~p_l +dp~p + (-hd~+l +ei+ l ) flp.

Lemma 5.1. The coefficient matrix of Eq. (5.14) is strictly diagonally dominant and hence nonsingular. On solving the above system of equations for fl, we then apply Eq. (4.9) to get, on each subinterval [ti_~,ti], the spline function

y(t) = ai l~b,(t) + fli_ldP2(t) + ~i~,(t) + fli~b2(t). Here, 41,

(5.15)

q~2, if/l, and ~2 are given by Theorem 4•1•

5.2• Convergence of the approximation In this analysis, we will denote by s(t) the spline approximant, obtained by control principles (given by Eq. (5.15)), of the function f(t). Error estimates for the classical cubic spline have been discussed extensively; see, for example, Refs. [12,20,21]. In this section, we will obtain the convergence rates for our approximation for case 3(ii); similar procedure will yield the error estimates for the other cases. Furthermore, we restrict our analysis to the case of uniform

N.N. Agwu, C.F Martin I Appl. Math. Comput. 97 (1998) 99-138

129

mesh with mesh width h. From Eq. (5,13), the spline approximation s(t) satisfies the equation (5.16)

Xfli-1 + Yfli + Zfli+l = dioci-1 -- (di At-di+l)O~i-4-di+lO~i+l,

where x = - e i , y = - h d i + ei + ei+l, and z = - ( - h d i + l + ei+l). Now, s(t) interpolates f ( t ) at the mesh points P:to<<.6 < . . . . < , t f; in other words, s(ti) = ~ = f(ti). Clearly, fl~ = s'(ti). Hence, Eq. (5.16) can be written as (5.17)

XSI_ 1 + YSli + ZSi+ 1 -~- dlfi-1 -- ( d l + e l ) f + elf,+,,

where s~ = s'(ti) and f. = f(ti). For h sufficiently small, we have the following: 0)11

h -

=

sinhh

~

lh3,

0)12 = coshh - 1 ,~ lh2 + l h 4 , 0)21 = - c o s h h + 1 ~ lh2 - l h 4 , (5.18)

0)22 = sinhh,

where, for each truncation, we have omitted terms that are of higher order in h than the ones retained. Also, from Eq. (4.48): C l l ( h ) -'- l h 3 + s i n h h - h - ( c o s h h

c2~(h) = -½h ~,

-

1)2/sinhh,

~,~(h)=-~h ~, (5.19)

c::(h) = h.

Thus, det C = c11c22 - c12c21 -----h4/12 + h sinhh - h 2 - h(coshh - 1)2/sinhh h6/120,

(5.20)

~1~ = c22/ det(C) ~ 120/h 5,

g12 = -c]2/ det(C) ~ 60/h 4,

~2] = - c 2 1 / d e t ( C ) ,~ 60/h 4,

c22 = c~]/det(C) ~ 30/h 3,

d, = ei =

[

coshh

]_

[

coshh

(5.21)

1

--(Dll + si-~-~(D21JCI1 -~- --0)12 + s i - ' ~ ( D 2 2 J C21 ~ --5/h2,

f

--0)11 +

si-~o2~J c,2 +

E cos ] --0)12 q- s i ~ 0 ) 2 2

6'22 ~ - 3 / 2 h ,

0)21~'11 -J-0)22G-'21 ,~ 5/h 2,

di+l =

sinhh 0)21C12 "~- 0)22~"22 ~ 7/2h. ei+l = sinhh

(5.22)

130

N.N. Agwu, CF Martin / Appl. Math. Comput. 97 (1998) 99-138

Thus, we obtain the following: - hdi + ei + ei+l = 14/2h, - hdi+l + ei+l = - 3 / 2 h .

(5.23)

Theorem 5.1. Let f ( t ) E C 3 and let 6(t) = s(t) - f ( t )

be the error that results when f ( t ) is interpolated by the spline s, defined above, on the partition P: 0 = to < ti < • • • < tp = 1. Then there exists a constant K such that

]s - f[ ~
(5.24)

To prove Theorem 5.1, we first discuss the following lemma. Lemma 5.2. Let P be any partition o f the interval [a, b]. I f f (t) E C3[a, b], then Is'(t,) - f'(t,)l ~<~4h211f(3)(4,)

II

(5.25)

f o r each node ti, i = 0, 1,... ,p and ti-i <<.¢i <~ti.

Proof. Using Eq. (5.23), Eq. (5.17) may be written as 3s,

14,

i-1 + ~ s i

3 ,

5

+~-~si+, = - ~ i - ,

5

+~i+,.

(5.26)

This simplifies to 3si_~ + 14sl + 3si+~ --

1~0 ~ - l - J~+l),

(5.27)

where we have used the interpolation data ai = f . Suppose that each term of the form c(r) J i-I-1 is expandable as: (r)

ft. ±1

----~

f/(r) -4- h f (r+l) ~ !h2£(r+2) 4- lh3f(r+3) q- lh4f/(r+4) i . . . --2

Ji

Then, it can easily be shown, using Taylor's formula, that 3h 2 ,, 3f'_ 1 + 14f' + 3f,~_l = 2 0 f / + - - ~ - ( f (4-) + f " ( ¢ + ) ) ,

(5.28)

ti-i <~~ <<.ti+t. I f f (3) is continuous on [xi-i ,x~+l], then by the Intemediate Value Theorem we may write ?

3f/_ 1 + 1 4 f ' + 3f+ 1 = 20f' + 3h2f"(~). (5.29) Let s'g - f ' = E~. Then, subtracting Eq. (5.28) from Eq. (5.27) and replacing f / w i t h (1/2h) ~÷~ - f~_~] - (h 2 / 6 ) f 3(¢), we obtain 3Ei_l + laE~ + 3Eg+l = ½h2f(3)(~).

(5.30)

That is, GE = H ,

(5.31)

131

N.N. Agwu, C.F. Martin / Appl. Math. Comput. 97 (1998) 99-138

where 14

3

0

0

...

0

3

14

3

0

...

0

0

3

14

3

...

0

0

0

...

3

14

3

0

0

...

0

3

14

G=

and/-/,. :- l h 2 f ( 3 ) ( ~ ) . If we multiply both sides of Eq. (5.31) by the matrix D = d i a g ( 1 / 1 4 , . . . , 1/14), we obtain

(5.32)

DGE = DH and 1

3/14

0

0

...

0

3/14

1

3/14

0

...

0

0

3/!4

1

3/14

...

0

DG=I+B=

(5.33)

0

0

...

3/14

1

3/14

0

0

...

0

3/14

1

where IIBII~ = 6/14 ~< 1. Thus, (DG) -I = (I + B) -1 exists, (see Ref. [32], p. 61). Hence,

IIEIIo~ = II(Oa)-~Onll~ ~< II(Day' IIo~IlOll~ Ilnll~ 1 1 h2 <~ 1 - ( 6 / 1 4 ) 1 4 3 Ilf(3)ll <<-l h2l[f(3) II, (4/27h2)[lf 13)[[ for the classical case, see Ref. [20]).

(5.34) []

Now, to prove Theorem 5.1, we observe that 6 ( 0 = s(t) - f ( t ) E C 2 and since s(ti) = f ( t i ) , i = O, 1 , . . . ,p, we obtain, by the Mean Value Theorem, ~(t) =

f'

ti_i

~'(r)dr.

(5.35)

N.N. Agwu, CF. Martin / Appl. Math. Comput. 97 (1998) 99-138

132

This gives

II~(t)ll ~<

16'(r)l dr 1

~< 113'11max I ( t - ti-l].

(5.36)

From Eq. (5.34),

I1<~'11~ ~h~llf(31 [I • Hence,

116(t) N~<~4h3 liT/3t II.

(5.37)

This completes the proof of the theorem.

5.3. Numerical examples A L G O R I T H M F O R C O N S T R U C T I N G THE C U B I C - E X P O N E N T I A L 3 SPLINES To construct the spline interpolant s(t) for the function f(t), defined at the nodes to < ti < --. < t,, satisfying s'(to) = f'(to) and s'(t,) = f'(t,): I N P U T A , b , F , B , n ; to, tl,...,tn; ~i----f(tl),".,O~n-I = f ( t , - 1 ) ; fl0 -----/'(to); fin = ft(tn). O U T P U T ilk, k = 1 , 2 , . . . , n - 1. Recall, from Eq. (5.15), s(t) = O~i lq~l(t) + fli_l(92(t) - t - ~ / ~ / l ( t ) -I- fli~12(t ) for ti-i <<.t<~ ti. Step l : F o r i = 0 , 1 , . . . , n - 1 seth~=t~-b_l. Step 2: Compute f2(h~), C(t), c~i, and ~kg, i = 1,2, from Eqs. (4.40), (4.47), (4.54), (4.55), (4.57), (4.58), respectively. Step 3: Set di =

i +~(0211e,ll

[-(011

sinh ni

+

I

i

coshhl

i ]-i

'

ei = [ --(0il 1 + ~coshhi ( 0 2 1 ] i ] c,2 -i + [ -(0,~ i + ~coshhi ( 0 ~ 2 ] i ] c22, -' i+1 -i+1

i+1 -i+1

di+l _~_(021 ell + (D22 e21 sinh hi+ l •+1-i+1

j_

i+l-i+l

(021 C12 ~ 6022 C22

ei+l ~-

sinh hi+ 1

'

where (0~; and c~; are the entries of the matrices O and C, respectively.

3 So called to reflect the fact that it contains cubic polynomials as well as exponential terms.

N.N. Agwu, CF. Martin I AppL Math. Comput. 97 (1998) 99-138

133

Step 4: Set ll = dido - (dl + d2)~l + d2ct2 + eiflo, and ln-I = dn-lO~n-2 - (dn-I + dn)~,-I + dn~n + ei~o. Step 5: Set l j - - d j ~ s _ l - (dj + dj+l)dj + dj+,~j+l, j = 2,... , n - 1. Set L = (/1,12,..., l,_2,/n-l)'. Step 6: Form the tridiagonal matrix, M, with: lower diagonal elements, -ej, diagonal elements, -hjdj + ej + ej+l, and upper diagonal elements, - ( - h f l j + l +

e;+l). Step 7. Solve the system M/~ = L. Step 8: O U T P U T fli, J = 1 , 2 , . . . , n -

1.

Example 5.1. (f(t) = sin (Trt)). Consider the test function

f(t) = sin(Tit), t E [0,1]. We set the boundary conditions:

~0 = - ~ ,

/~p=~

and then construct the spline function for h = 0.2, 0.1,0.05. Graphs of the spline function s(t) are shown in Figs. 1-3. Example 5.2. (f(t) = e-3°t3). Consider the test function

f ( t ) = e -3°t3, tE[O, 1].

cubic-exponential splJne with uniform spacing,n--.5 1.2

1

"

\\X

0.8

~o.6

x\ x\\

0.4

0.2

o'.~

o:~

o:~

o:,

o:s

o:~

o'.~

t

Fig. 1. F u n c t i o n :

f(t)

= sinQrt), n = 5.

o:,

o'.~

134

N.N. Agwu, CF. Martin I Appl. Math. Comput. 97 (1998) 99-138

cubic-exponential splino with uniform spacing,n=lO

1.2

i

1 0.8 0.6 0.4 0.2 0 -0.2

0

011

012

013

014

F i g . 2. F u n c t i o n :

1.2

.

.

015 t

016

017

018

019

f(t) = s i n ( n t ) , n = 10.

cubic-e~oonential spline with uniform spacing,n=20 . . . . .

1 0.8 0.6 0.4 0.2

-0"20

011

012

013

014

015

016

017

t

F i g . 3. F u n c t i o n :

f(t) = sin(Tit), n = 20.

018

019

N.N. Agwu, C.F. Martin / Appl. Math. Comput. 97 (1998) 99-138

135

We set the boundary conditions:

flp=-30e -l°

fl0=0,

and then construct the spline function for h = 0.2, 0.1,0.05. Graphs of the spline function s(t) and its derivative are shown in Figs. 4-6. Example 5.3. For our third example, we consider the function

f(t)=

0

if - l < t < O ,

1/2

ift=O,

1 ifO
=0,

=0

and then construct the spline function for n = 80. Graph of the spline function 7. Fig. 8 shows the graph of this same function using the classical quintic spline. Comparing these two figures, we see that there is an improvement over the classical quintic approximant. However, we must realise that the scheme used to obtain Fig. 7 contains only cubic terms. Further, only the first derivative of the control law is used in this case. Increasing the order of the derivatives of the control law used in the cost function, J(u), will result in greater smoothness of the interpolant.

s(t) is shown in Fig.

Spline approximation of f(t)=eA(-lOt^3) with uniform spacing; n=5 1.2 1

~\\\\x\~\

0.8

0.6

0.4

0.2 o

-0"20

,11

o12

o13 oi,

Fig. 4. F u n c t i o n :

t

& f(t)

016

= e -1°t3, n = 5.

o16 oi,

136

N.N. Agwu, C.F. Martin / Appl. Math. Comput. 97 (1998) 99-138 Spline approximation of f(t)=lP(-10tA3) with uniform spacing; n= 10 1.2

1

0.8

0.6

0.4

0.2

-0.2

oi,

o12 o13 oi, Fig. 5. F u n c t i o n :

ols t

f(t)

oi~

oi~

o18

oi~

= e -1°'3, n = 10.

Spline approximation of f(t)=eA(-10t.~3) with uniform spacing; n=20 1.2

1

0.8

~o.6 0.4

0.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

Fig. 6. F u n c t i o n :

f(t)

= e -1°'3, n = 20.

0.8

o19

N.N. Agwu, C.F. Martin I Appl. Math. Comput. 97 (1998) 99-138 cubic-exponential spline with uniform spacing, n---80

1.2

r

r

r

1

T

r

~

-0.2

0 t

0.2

0.4

"r-----------T--

1

0.8

0.6

0.4

0.2

0

-0.2 -1

-0.8

-0.6

-0.4

0.6

0.8

Fig. 7. I n t e r p o l a t i o n b y c o n t r o l t h e o r y a p p r o a c h .

classical quintic spline with uniform mesh n = 80 1.2

1

ik-

0.8

0.6

0.4

0.2

•

-0.2

'-

-018

-016 -01, -012

q

~

012

014

Fig. 8. C l a s s i c a l q u i n t i c i n t e r p o l a t i o n .

016

016

137

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N.N. Agwu, CF. Martin I Appl. Math. Comput. 97 (1998) 99-138

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