A method for the asymptotic solution of singularly perturbed linear terminal control problems

19 The column v in Table 2 lists the number of computations of the function at each stage of the search; rp' is the maximum of the function, Z' is the corresponding argument, s' and 'p' are the solutions obtained, N is the number of computations of the function needed, and QN are the final values of the criterion. Table 2

Qd) n(r) Cd) W(x) W=) $a)

1

a i a 1

a

1

i

030 0.55 O.&ON7 0.850077

&.a 4.9 a.63 8.83 a.49Ba3 2.498125

O.ea8sbl O.W77%8 0.552012 0.55i040 0.8497748 0.8460658

4.zede9a 4.28M88 s.&?0031 9.~ a.498888 2.447483

81

ai w, ?a 10 10

63m%lo-' 9.76%8.10-' 4.4257.W' 4.27#B.t0-0 8.874403.10-4 1.i7#271.10-~

Incidentally, the coefficient T figuring in (E), (lo), (17), (23) and (25) already becomes equal to two after the first iteration, both of the successive algorithm using formula (15) and of the block algorithm using formulae (20).

REFERENCES 1. SUKHAREV A.G., Optimal Search for Extrema, Izd. Moskov. Gos. Univer., Moscow, 1975. 2. CHERNOUS'KO F.L. and MELIKYAN A.A., Game Problems of Control and Search, Nauka, Moscow, 1978. 3. KOROTCHENKO A.G., On a search algorithm for the greatest value of one-dimensional functions. Zh. vychisl. Mat. mat. Fiz., 18, 3, 563-573, 1978.

Translated by D.L.

U.S.S.R. Comput.Maths.Math.Phys.,vol.3O,No.2,pp.i9-28,1990 Printed in Great Britain

0041-5553/90 $10.00+0.00 01991 Pergamon Press plc

A METHOD FOR THE ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED LINEAR TERMINAL CONTROL PROBLEMS* A-1. KALININ

Terminal control of a singularly perturbed linear system, with constraints on the right endpoint of the trajectory is considered. The asymptotic behaviour of the solution is analysed and on that basis an algorithm is proposed for the asymptotic allocation of optimal control switching points. A computational procedure is outlined which utilizes the asymptotic approximations to obtain an exact solution for any given value of the small parameter. 1. Statenrent of the problem. In the class of scalar piecewise-continuouscotrols u(t),t~T-[O,t.], we consider the following optimal control problem for a linear time-independentsystem: etcl'z(t.)+cr'y(t.)~max, pi-A,z+A,y+b,u, z(0)=sO,Ji'A,z+A‘y+b*u,K(o)+, In(t)

t=:T,

Il,z(t.)-g1,&&.)=gz,

(i.*a) (l.lb) (l.lc)

where H is a small positive parameter, e is an nI-vector, y is an nt-vector, g, is an m,vector, gz is an ml -vector (m,Gz,, m,CnJ, and the other elements of the problem have the appropriate dimensions. Asswnption 1. The matrix A, is stable, i.e., the real parts of all its eigenvalues are negative. Despite the fact that (1.1) is a linear optimal control problem, its numerical solution i~Zh.vychisl.Hat.mat.Fiz.,30,3,366-378,1990

20 presents serious difficulties since both the system itself and its adjoint /l/ are "stiff" /2/ for small u. In this connection the asymptotic approach comes to the fore. There is a considerable literature on singularly perturbed problems. Surveys of the main results may be found in /3, 41. Previous attempts to determine the asymptotic behaviour of solutions have usually been confined to problems with an open control domain and smooth controls. In the event that there are constraints on the values of the controls, expressed as closed inequalities, most studies have been confined to the examination of the limit problem, obtained in some topology or another when the small parameter in the perturbed problem is allowed to tend to zero. One of the few exceptions is 151, in which, under rather strong assumptions, an algorithm is worked out for the asymptotic solution of problem (1.1). In this paper we conpropose a substantial modification of that algorithm, which permits the adoption of siderably weaker assumptions than in /5/. To facilitate the analysis we will introduce a few concepts.

Definition. A piecewise-continuouscontrol u(t,u),tC7',satisfying the inequality (u(t, is said to be admissible in problem (1.1) if the trajectory that it generates cL)I(I, =T, within ON+'), we shall satisfies the terminal constraints. If these conditions hold to say that the control is s-admissible. An admissible (s-admissible)control is said to be koptimal if the corresponding value of the performance index differs from its optimum by O($+'). Below we shall describe an algorithm which, for a given natural number n, constructs an n-admissible n-optimal control for problem (1.1). The basic idea is to determine the asymptotic behaviour of the switching points of the optimal control. In addition, we shall show how to use the asymptotic approximationscomputed by the algorithm in order to obtain an exact solution of the problem at a given value of the small parameter. We shall use the following notation:

so that problem (1.1) may be written as follows: c'(P)r(t.)-max, +A(p)r+A(p)u, (u(t)1<1,t=T,

2.

The first

basic

(1.2a)

s(0)==zO,

(1.2b)

Hz(L)-g.

problem.

The first step of the algorithm is to solve the following unperturbed optimal control problem: c,'~(t.)-+max,d-&+b,u, In(t where

A,-A,-A,A,-lA,,

bO=-b,-A,A,-‘bj.

t@,

(2.la)

sr(0)--YO,

(2.lb)

~#(~*)-sz,

We shall refer to this as the first

basic

problem.

Assumption 2. Problem (2.1) is solvable and is "simple" in the sense of /6/. Using the support method of /7/ to solve the problem, we obtain: 1) an optimal control and trajectory, u"(t),y"(t),teT; 2) a support {T,,...,z,J, i.e., a set of m, distinct points in the interval ]O,t.[such - called the support matrix - is nonthat the (m,Xm*) matrix C+==(~O(~~),j-l, 2,...,m,) singular, where coo--H,F,(t)b,, teT, (2.2) and

FO(t),t=T,

is an (nrXn,)matrix-valued function satisfying the differential equation PO=-FOfL

3) a vector of potentials

t=T;

F,(t.)--E;

li"'-~o'@,-l, where

(2.3)

yO-(rO(~j), j-l, 2, ....m2) and yo(t)==cI’Fgtt)bo,

4) a cocontrol A,(t)==-$o’(t)bo, t=T, where qO(t), t=T, is a solution of the adjoint system +-A,‘$, qo(t.)-c&12’h0. Note that t=T. Ao(t)-h”‘cp,(t)-r,(t), (2.4) The cocontrol is related to the optimal control by d(t)=-sgnAo(t), teT, and the support times 0 j-1, 2,...,mr, are simple zeros of the cocontrol. Denote.all the zeros of the cocontrol by t,*, . . . ) trot arranged in increasing order. Of course, l>mr.

Assumption 3.

3. t,‘>O, t%t.; Ao(t,“)+O,j-1,2, , . . (1. The second basic problem.

At the second step of the algorithm, the following control problem, in which the duration of the process is variable, is solved:

21 0

c'g(O)-lAo(

~~u(s)+Ilds+max,

(3.la)

*a

da/da-A,z+b,

sgn bo(V)u,

z(s,)=A,-‘b,

where c-c,+(Aa,-‘)‘(c,--Hz%‘).

sgn h,(V),

s
Iu(s)la

(3.lb)

H,a(O)--HrA,-‘A,y’(t.)+g,,

We shall refer to this as the second basic

(3.lc)

probtem.

Asswqdion

4. Problem (3.1) is solvable. is the equilibrium position of the dynamic system under the The point A,-lbrsgnAo(t,‘) fixedcontrol U(s)--1. Hence the second basic problem is equivalent to the following duration problem: 0 &z(O)-lAp(

(3.2a)

ju(s)ds+max, I* sgnh,(t?)u,

d&is-A,z+b,

s
(3.2b)

lu(s)lG&

z(s*)=Ar-‘b, sgnb,(t?),H,z(O)=H~A~-'A,y'(t.)+g,,

(3.2~)

where S* is a sufficiently small negative number. Indeed, if sIO is an optimal initial time in problem (3.1), then the optimal control U.(S) in problem (3.2) over the interval [810, 01 will be the same as the optimal control in the second basic problem, while for sts,’ we have U’(S)=-i. In addition to Assumption 4 we have

Assumption

5. Problem (3.2) is "simple". Reducing problem (3.2) to a terminal control problem and solving it by the direct support method, we obtain: 1) an optimal control and optimal trajectory U*(S), Z*(S),SE[s*,O]; 2) a support kl,...,%J, i.e., a set of ml distinct points in the interval Is., 01 and a corresponding non-singular support matrix ne-(ncp(c,),i-1,2,..., ml), where (3.3)

ncp(d)-R,G(s)blsBn~a(ko)r and

G(s),s
(n,Xn,) matrix-valued function satisfying the differential equation dGlds--GA,,

G (0) -E;

(3.4)

3) a vector of potentials 9, which is a solution of the non-degenerate system of linear algebraic equations vO'IIrp(o~)-Il~(ol), i==l, 2, . . ..m., where rIr(s)-c'G(s)b,sgnAo(tlO)-lA,(t.)I, 60; is a sol4) a cocontrol IIA(s)--n*'(s)bi sgn A,(V) +(A&.) 1, s= [s’, 01, where IIlp(s),s
-&w - -Ai’IIq(s),

l-@(O)- C-.Hl’VO.

Note that l-IA(s)-~"II(P(s)-nr(s). The cocontrol is related to the optimal control by has the following property:

l-IA@<)-0,

-$IIA(cJ+&

(3.5) u'(s)--sgnnA(s),s~[s*,O],and it

t--1,2,. . . , ml.

Let srO,...,srO denote all zeros of the cocontrol, numbered in increasing order. Since they include the support times, clearly p>m,.

Assumption

6. 0,020,

$IA(s:)+O.

t-i,2

,...(

p.

s* is chosen to be sufficiently small, then s,' is an optimal initiaJ time, 9,'. . .. . the optimal control switching points in problem (3.1),z'(s,a)-A~-'b~sgnA~(~rO),~'(~)---l sO if &a,@.

If

SPQ for over

are

4. M&n theorem. Our further calculations are based on the following assertions:

22

If Assumptions 1-6 are valid and )Jis sufficiently small, then problem (1.2) Theorem 1. has an optimal control of the following form:

where t+,(p) (l-1,2,...,i) and sc=sl(W) (i=1,2,..., p) can be expanded in asymptotic series:

(4.2) 2. Let (n'(&&),h'(p)), (nE@*,&@?*) be the Lagrange vector correspondingby the Maximum Principle /l/ to an optimal control. Then the vectors v(W)=n(m)/p,h(W) can be expanded in asymptotic series

14.3) and, together with the optimal control switching points, they form a solution of the following system of algebraic equations: Hz(t*, t,, . . . , t,, s,, . . . , sp, p)-g=o, Ip'(Qv, h,~)~(~)=o,

(4.4a) (4.4bf

j-l,2,...,t,

2,...,p, $ft.+ust, v,& ir)b(p)==O, i--l,

(4.4c)

where z(t,t,, ....tl, ~9,.+. , sp, p), t=T, is the trajectory generated by a control u(& 1,,...,&, s,,..., I,, IL), t=T, of the form (4.1), and rp(t, v,L, m),t=T, is a solution of the adjoint system ~@)-c(P)-~'%lQ*

+---d'(P)%

J.'(P)-=fP',h').

(4.5)

Proof. Using Cauchy's formula to write down the solution of the singularly perturbed system for the control m(t,t,,..., tI, s,, . . . , s,, a) t taTI we obtain

Hz(t., k, . . . , tr, Sl,. . . , smPI - g=HF (0, p)z” +

(4.6)

1, sgn&(t,O i cp@,p)dt+... +senA,w)

P t.+*r*

s q#,p)dt2-r t.+lU,

. ..-+(-w-’

II

f q&&&it+ %+cl~.,

1.

t-W

i dtt

W]-g,

L+YS,

where F(t,p),t~T, is an equation

[(m*+m*)X(~~+~)]matrix-valued function satisfying the differential P--FR(P),

F(L)-E,

(4.7)

and q(&p)-~~(f,m)b(p),t-T. Write the solution of the singularly perturbed matrix differential Eq.(4.7) form:

in block

where F&, p), WT, i-i; 2,3,4, are mstrices of dimensions n,Xnr,nlXnr,nrXnr,. n,Xn, respectively. Using the boundary function method /0/, we can expand these matrices in asymptotic series: *

~~ft,pl--f:)I*rP,~(t)+~,(r)l, I-P

8 -(t-t.)/&

tfe.

(4.8)

23

We emphasize that these are uniform asymptotic eXpSnSiOnS, a constant C,,>O, independent of t, such that

i.e., for any

natUral

Tl

there iS

It is also essential that the functions &F',(s),sGO,called the boundary terms, satisfy the estimates (4.9) lB-W*(SH~akexP(Blb), f-i,2,3,4, k-0, i,.. . , where a+,Bk are positive constants. We will list a few of the first coefficients of the expansions (4.8): F,,=O,

F,--A,-‘A,F,(f),

&‘,,-=A,-‘A#&f)A&-‘,

IW"I=G(S), W.-O,

P,*=O,

(4.2Oa)

FPPo(f),

F,,--F,(f)AdA,-‘,

(4.10bI

t=T, l-IpFa=O,

W'r=G(SM,-'A,,

(4.1Oc)

SGO,

IV,-=A&-‘G(s),

(4.10d)

are solutions of Eqs.(2.3), (3.4). be vector-valued functions whose co~~nents axe respectively the first m, components and the last ?nS components of cp(f,p),f=T. Then, as is obvious from the definition of this vector-valued function and from formulae (3.31, 14.81 and (4.10), we have the following uniform asymptotic expansions:

where

Fo(t),t=T, GM,

Let

sQ0, ~~(f,~),~*(f,~),f~T,

(4.11b)

where

Note

that by (2.2) and (4.10), (4.i3a) (4.13b). Let

P(&, . . . , k, 81, . . . , 6, p) -HI2 (to,fi, f . *, t I, R(L . * +, b, sr, . . ..a.,IrwQ/(f*,t,,

8 1,. . . ,

sr, k+-81,

. . . . tr,s,,..., s,,p)-gr.

Froa (3.31, (4.6) and (4.8)-(4.13)we obtain the following asymptotic expansions: ”

P(f,, . . . , trr II,.

. . , S?,

p) * ;t: p%(t*, . . . ;tt, k-_)

St, * * . ,

8.1,

(4.W

s )r

(4.14bl

”

R(t,,...,tr,sr,....s,,B)~

zi

pv?&, . . *t frrSt,

.“I

P

1-o

where

(4.15)

(4.16)

except that H, is replaced by &, Fs by There is an analogous formula for R,k>l, &-,cp, by cp:r,&-SQZ, respectively. by FM, and cp~,, F,, FY p)+k'cpr(t, p)-c'(p)F(t, p)b(~),=T. As follows Put A(& Y,li, P)--$'(t,v,h, ~)b(p)==~'q~(t, from (4.81, (4.10), (4.11) and (4.13), this function admits of a-uniform asymptotic expansion

A-

(4.17)

$[A,(t,v,h)+ l-LA(s,v,A)I,

where A,--L'cpo(t)-s'F,(l)b,, noA~v'ncp(s)sgni\,(trO)-(cr'+c,'Ad,-'A’R&A,-‘)G(s)b,, Ar=v'q,.~-IS IIlA=v'n*-rcp,+l'nrcpz-c,'(n,F,b,+n,-,F,b,)hfcp,-c,'(F,rb,+F1,r-,~l)-c,'(F,,r+,b,+ F,,b,), k=i, 2,.. . , s'(II,+,F,b,+ng,b,), k--l, 2,.... Note that by (2.4) and (3.51, (4.18a)

Ao(t,vO,hO)=A&), t=T, II,A(s, vO,ho)=-IIA(s)sgn&(t,O)-A,(L)=-n$(s)b,, S40. Define g(s,v,h,~)-A(t.+~s,v,h,~),s
(4.18bl t=t.

+ps, that (4.19) where 8r=IInA(s,v, a)+ lx .f=A&t.,v,a), ,_(I iI dt’

and

Let h-((t,, . . ..k. s,,...,s,,v,,...,v~,, 5 (j-i,2,...,mr) those of

k-0,1,.

.. .

(4.20)

where v, (i=l,2,...,m,)arethe components of VT

L,...,h,)‘,

L. Then system (4.4) may be written as

WY

(4.21)

CL)'0,

where P (tu *

N&N=

I

4

.,

t,,

R (&, . . ., t,, A((tj,V,h,P), 8(si,V,a,p),

S1’ * *

-3

S&i

P)

81,. . -9S,1P)

As follows from (4.141, (4.17) and (4.19) and the estimates (4.9), we have the asymptotic

25 expansion (4.22)

where (4.23)

Put Iti4F(tq0,

N(h,O)-No(h). Then the vector-valued function N(h, p) is continuous in the domain 2,...,I, IsI-spI
j=1,

ficiently small positive numbers), together with its partial derivatives with respect to the components of k. u”(t), t=T, is admissible in the It follows from (4.16) and the fact that the control first basic problem, that

Since

it follows from (4.13) and (4.15) that

Po(hO,.. . , tl”, s:, . . . ( spy= -N,A,-‘A,yO(t.) - g, -Id.0

H,G(s,‘)A,-‘b,sgni\,(t,‘)+.

+(-l)p’+*

FIIcp(s)ds-... 00

JrItp(s)ds--H,z’(O)

- H,A,-‘A,yO(t.)-

g,==o.

‘P”

Finally, by (4.18) and (4.20), we have A,($“, vo7h”)=&(t,“)=O, h')+A.(t.)=~A(~,~)sgnAo(t~~)-0, i= i,2,...,p.

i=l, 2 ,... ,l,

bo(~p, Q, h”)E~IoA(~,o, vO,

Referring to formula (4.23), we see that N(h,,O)=No(h,)=O, where h,- (t,',...,t~,~~',.~.,s~', v,O,. . . 1vm,“,hlO,.*.,L10)‘. One sees by direct differentiationthat the Jacobian I,-0N(h,,O)/ah has the form (4.24)

where

B,=(2~la(t,‘)sgn’Ap(t,‘),

. . . , I), &.-(cpa(t~), are

j-l,

i==l, 2,. . . ,1), B,=(2(-l)‘$(srO),

2,. . . , I)‘,B,-(ncp(scO)sgndo(tlO),

i-1,

I-I, 2,.

2, . . ., P), B,=(2~po(eo)sgnho(t10), i-l, 2, . . ,p)‘, &==ULcptbt9+cp&~), i-l, 2 ,... ,p)’

matrices of orders m,Xl, m,Xp, m&l, Mm,, pXm,, pXm I, respectively;B,=(diag&(t!),j--l, 2,..., p) are diagonal matrices of orders 7,and p, respectBI=diag(dnA(sP)aBnho(trO)/ds, f=l, 2,..., ively. It is readily seen that, since the support matrices of the basic problems are nonsingular, the same is true of the Jacobian I,. Thus, system (4.211, or, what is the same, system (4.4), satisfies all the conditions of the Implicit Function Theorem, according to which there is a right neighbourhood of zero, in which there are well-defined continuous functions t,(p)(j-1,2,.. . ,0, c(p) (i-1, OQ<~,r 2,...,P), satisfying (4.4), such that moreover tj(0)==tjD(j-l,2,...,1), s,(O)=s:(i-1, V(P), A(cc)* 2,...,p), v(O)-@, h(O)--ho. But this means that for sufficiently small v problem (1.2) has an admissible control U'(t, p),@M, of type (4.1) and there is a Lagrange vector (p~'(p)~ A'(W)) such that the switching points of u'(t, p),td, are zeros of the function A(t, p)--#(t, k)b(p). where $(t, p), t=T, is a solution of the adjoint system (4.5) with v=v(~),b==-h(p). t=T, Since the left-hand sides of system (4.4) can be expanded asymptotically in integer powers of p, it follows /9/ that expansions (4.2) and (4.31 are valid. Since Aft,Ir)-Ak v(V),h(p),p),t=T, it follows from (4.17) and (4.18) that there is a constant G-0 such that IAO(t)i-nA(s)sgni\,(t,O)A,@.)-A(t,p)IGCp, S-(t--t.)&, t@T. Relying on this relationship and Assumptions 3 and 6, and using the Implicit Function Theorem, it is not hard to prove that for sufficiently small w the cocontrol A(t,II), tcT, has no zeros other than the switching points of the control W(t, CL), ET, and moreover d(t, CL)=--sgnA(t, p), ~2’. But this means that 0,

26 for sufficiently small n the admissible control u*(t, IQ,@@, satisfies Pontryagin's Maximum Principle Ilf with Lagrange vector (ova, A'(p)),so that it is optimal. This completes the proof. 5. Asyriptotic expandon. n),t=T, of type (4.1) with O-t? (]=I,2,...,1), s,=s: The control ~l(~)(t, Q-4 %Lt_T’P) is O-admissible and the O-optimal control for problem (1.1). To construct an n-admissible j-l, ;;,qp;;ma18;;;tr;; @>I), it will suffice to determine the coefficients 4: (k--l, 2,...,n; of the expansions (4.2). n, tP+i, 2,. . . ,p) -9 Wi shall set'ip'systemsof linear algebraic equations to calculate these coefficients. Let

Expand the vector?valued function

Y

in powers of p up to order n inclusive and equate the coefficients of the expansion to zero (beginningwith the coefficient of n). This yields non-degeneratesystems of linear equations for the successive determination of the coefficients hl,, k-l, 2,. . . , n:

We note that in view of the structure 14.24) of the Jacobian I,, each of systems (5.1) splits: one first finds the components of the vectors v',h", and then uses them to calculate the coefficients t,* (j-i, 2,..., I), 6: (t-1, 2,...,p) independently. If the cocontrols of the basic problems have no zeros other than the support points (I=-m,,p=-ml), the original system (4.4) splits: the switching points of the control (4.1) can be found independentlyof the Lagrange multipliers. This naturally implies a correspondingdecomposition of systems (5.1). Successively solving systems (5.11, we determine the coefficients &' (l-1,2,...,&k-4, 2,..., n), 8: (i-1,2,..., p, ‘k-l, 2,... , n), sk (k- i, 2,..., n), P (k-i, 2,..., n). of expansions (4.21, (4.3) and construct polynomials

tJ”(&d=-

lw,

j-1,2,.*.,1,

The control u'"'(t, p), teT, of type (4.11, where tp-tJn(p), f-1, 2,...,&~,-s,~(g), i-1, 2,...,p, is n-admissible and n-optimal for problem (1.1). Note that if insteadof s?(p), i-i, 2, . . . . p, one takes ~*~(W),I-i,2,... ,p, the result is a control for which the performance index deviates by O(p*") from optimal. As to the terminal constraints for the trajectory, the condition for the "slow" variable # holds to within O(pR+') and for the "fast" variable E to within O(V). The asymptotic approximationsobtained above can be used for the exact solution of problem (1.1) when the small parameter is fixed. To that end one should use the "finishing" procedure of /7/, using Newton's method to find the root8 of system (4.4) or, equivalently, of system (4.21), taking h,,(p) as the initial approximation. When this is done, the integration of "stiff" systems can be avoided by replacing the matrix aN(h, p)/ah with an asymptotic expansion, whose coefficients can be calculated from those of expansion (4.22). 6. Rzumple. Consider a material point of small mass u , moving along a horizontal straight line under the action of a control force u whose magnitude cannot exceed a certain constant C>O. Without loss of generality, we may assume that C--l. The point is moving in a medium exerting

2-l Supa resistive force proportional to the velocity, with coefficient of proportionalityk. pose that at the initial time the point was in position zr and has a velocity vO. It is required to find a law regulating the magnitude of the force in time, so that at a given time t. the point will be at a maximum distance from the initial position and its velocity at that time will be zero. This reduces to the following terminal control problem: ui=-kz+u,

&=I,

lu(t)l
4,

(&la)

z(O)==v0, u(O)-s,

ab)=O,

(6.lb)

P(t*)-+max.

The first basic problem here is 1 #'--I& k

la(t) 16 1, ST,

Il(O)=zo,

I/ (t.) + max

in which the right endpoint of the trajectory is free. There is an optimal control u’(t)-& for this problem, and the corresponding cocontrol is A,(t)=-l/k,tET. t=T, The solution of the second basic problem da/&=-kz-u,

I44 16 1,

s
I [u(8)+

‘do,)-Ilk,

2 (0) =o,

i]du + mln,

Ilk. is the control u*(s)=~,sE[~,~, 01, where s,"--&2)/k,, and the control is IIA(s)=[i-2exp(ks) In this case Assumptions l-6 are satisfied (problems in which the right endpoint of the trajectory is free are always "simple"). Thus, for sufficiently small u, the optimal control in problem (6.1) will be

and we have an asymptotic expansion _ s,(u)-8,O + z $8,". L-l It is not hard to see that in this case P~(s,)=O,k--l, 2,.... But then Hence it follows that for any natural number n the control 1, u'O'(t,N){ -1, is

d-0,

k-i,

2,. . . ,

t 40, t.+w,“[, te[t.+p,‘, t.1,

n-admissible and n-optimal.

The presence of a small parameter as the coefficient of the "fast" variables in the performance index of problem (1.1) has a significant effect on the asymptotic behaviour of the solution. Examples show that in the absence of such a parameter the optimal control switching points cannot usually be expanded in asymptotic series in integer powers of u. Assumption 4, of course, presupposes that b,+O. The case Cl=0 can be investigated using the scheme proposed in /5/. Extension of the algorithm to systems with multidimensionalcontrol and to time-dependent systems with sufficiently smooth coefficients does not involve any serious difficulties. REFERENCES 1.

2. 3. 4. 5. 6.

PONTRYAGIN L.S., BOLTYANSKII V.G., GAMKRELIDZE R.V. and MISHCHENKO E.F., Mathematical Theory of Optimal Processes, Nauka, Moscow, 1983. RAKITSKII YU.V., USTINOV S.M. and CHERNORUTSKII I.G., Numerical Methods for Solving Stiff Systems, Nauka, Moscow, 1979. VASIL'YEVA A.B. and DMITRIYEV M.G., Singular perturbations in optimal control problems. Itogi Nauki i Tekhn., Matem., Analiz, 20, 3-77, VINITI, Moscow, 1982. SAKSENA V.R., O'REILLY J. and KOKOTOVIC P.V., Singular perturbations and time-scale methods in control theory: Survey 1976-1983. Automatica, 20, 3, 273-293, 1984. KALININ A.I. and ROMANYUK G.A., Optimization of linear singularly perturbed systems. In: Constructive Theory of Extremal Problems, Izd. Belorus. Univ., Minsk, 1984. GABASOV R., KIRILLOVA F.M. and KOSTYUKOVA O.I., Optimization of control systems using quadratic supports. In: Constructive Theory of Extremal Problems, Izd. Belorus. Univ., Minsk, 1984.

28

R. and KIRILLOVA F-M., Constructive Optimization Methods, II: Control Problems, Izd. Belorus. Univ., Minsk, 1984. 8. VASIL'YEVA A.B. and BUTUZOV V.F., Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973. 9. VAINBERG M.M. and TRENOGIN V.A., Theory of Branching of Solutions of Non-linear Equations, Nauka, Moscow, 1969. 7.

GABASOV

Translated by D.L.

U.S.S.R. Comp.MUthS.Math.phyS.,Vo1.30,No.2,pp.28-33,1990 Printed in Great Britain

$10.00+0.00 Pergamon Press plc

0041-5553/90

01991

THE COMPLEXITY OF THE COMPUTATION OF THE GLOBAL EXTREMUM IN A CLASS OF MULTI-EXTREMUM PROBLEMS* A.G.

PEREVOZCHIKOV

Unconditional global minimization of a Lipschitz function in En is considered, and a difference method for solving the problem, accurate to within e&-O, is proposed. In the general case this involves computing O(eo-") values of the function. A special class of functions is introduced, with the property that the measure of the set of eoptimal points increases at a power rate O(s'*), r=(O, 11. It is shown that for a successive search algorithm of the branch-and-bound type, the total number of computed values of a function in the class may be reduced to O(e-i(l-") if r-z1 and to O(lneO-')if r=l. It has been shown /l/ that the estimate O(e,-*) for the number of computations of the function necessary to achieve so accuracy in unconditional global minimization problems cannot be improved in the class of Lipschits functions with constant L>O . In order to reduce the order of the bound, therefore, one has to investigate narrower classes of functions /2-51. One of these classes, which we shall consider in this paper, is that of the functions f(z) such that the measure of the set X,(e) of s-optimal points has bounded growth. It consists of those functions f@.,ip(X,L), where X is a fixed parallelepiped in En, such that mes&(e)r c'q(e) for any e=(U,e'],where c',e'>G are constants. We denote this class by S=+(X, L,C’,e’, q(s)). The function cp:]O,e']+E', q(s)>0 is assumed to be increasing and continuous on the right. It will be shown that if q(6)-e'" and r=(O, 11, then the total number of computations of a function f(*)NF in a global extremum search can be reduced to O(S~~"*) if r< 1. and to O(lneo-')if r-l. These estimates are achieved in a certain successive search algorithm of the branch-andbound type /6/. The search takes place over a tree whose nodes are the points of an e-mesh of X of varying mesh size; as the need arises, the search branches as new information about the computed values of f is obtained and in accordance with the prescribed accuracy so. The subclass of sin which these estimates are sharp- we shallrefer to as the class of functions with power growth of the measure of the set of e-optimal points, denoting it by - is quite large. It contains, inter alia, all uniformly convex funcF,-QT,(X, L, C’, e’, r) tions (with power convexity function)/7/, all functions with a sharp minimum and minima of a finite number of such functions, as well as functions bounded below by functions of these types under certain conditions. The subclass thus contains many types of multimodal functions that arise in computational practice. 1. Statement of the problem. Consider the unconditional global minimization problem f(x) - min,

x=X&",

(1.1)

where X is a parallelepiped in E" which is known to contain an unconditional minimum point, We may assume without loss of generality that it is the cube with side d>O, centre at OeE” and axes parallelto the coordinate axes. We shall assume that f=r(X, L, CL, e',cp(.)) for fixed L,C’, e’>O,cp :[O,e']+E'. To solve problem (1.1) to within accuracy e+-0, we introduce a &-mesh X%=X in X, %h.uychi&32.Mat.t7kzt.Fi2.,30,3,379-387,1990