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FREE END-POINT LINEAR-QUADRATICCONTROL SUBJECT TO IMiPLICITCONTINUOUS-TIME SYSTEMS: NECESSARY ANDSUFFICIENT CONDITIONS FOR SOLVABILITYTon Geerts ~, .

FEw 560 ~ ` 'n y ~(í~ i' ny,:. ~~~s ,T ~

Communicated by Prof.dr. J.M. Schumacher

- 1 -

FREE END-POINT LINEAR-QUADRATIC CONTROLSUBJECT TO IIiPLICIT CONTINUOUS-TINE SYSTEMS:

NECESSARY AND SUFFICIENT CONDITIONS FOR SOLVABILITY

Ton Geerts`

Tilburq University, Dept. of Econometrics, P.O. Box 90153,5000 LE Tilburq, the Netherlands

Abstract.

For an implicit continuous-time system rrith arbitrary constantcoefficients we derive necessary and sufficient conditions forsolvabilíty of the associated free end-point linear-quadraticoptimal control problem. In particular, this problem turns outto be solvable if and only if the underlyinq system is outputstabilizable, as is the case for a stamiard system.

Keywords.

Implicit systems, linear-quadratic problems, regularity,impulsive-smooth distributions, output stabilizability.

1. Introduction and preliminaries.

Given the implicit continuous-time system Z:Ex(t) - Ax(t) f Bu(t), (l.la)y(t) - Cx(t) f Du(t), (l.lb)

Nlth ll(t) E Rm, X(t) E 12n, Y(t) e Rr for all t E R' :- [O. ~).Let k denote the number of equations in (l.la) and let e- rank(E). All matrices involved are real-valued and constant. ile may,and hence will, assume that [E A B] is of full rorr rank. If E isinvertible, then the solutions of (l.la) are

x(t) - exp(E'`At)xo t o1texp(E''A(t-s))E'iBu(r)dr (1.2)

(x, e Rn arbítrary) and hence every x, is consistent, i.e., for

- 2 -

every xo, (l.la) has a solution x with x(0') - x,. If E is notinvertible, however, this need not be the case and inconsistentinitial conditions may give rise to impulsive solutions of(l.la), see e.g. [12], (2]. The most natural way to deal withsuch phenomena is the use of distributions [11], as was doneearlier in e.g. [2]. Instead of (1.1), we will consider itsdistributional interpretation:

Ea(1) ~ x- Ax } Bu t Exa6, (1.3a)y - Cx } Du, (1.3b)

where 6, a(1) denote the Dirac distribution and itsdistributional derivative, respectively, ~ stands forconvolution of distributions, x, e Rn, arbitrary. Horeover, u Ecmmp, the m-vector version of cimp, the commutative algebra(over R) of impulsive-smooth distributions [10, Def. 3.1], [9].A distribution is impulsive-smooth if it can be decomposed(uniquely) in an impulse (any linear combination of ó and its

derivatives ó(1), i~ 1) and a smooth distribution. Adistribution is called smooth if it corresponds to a functionthat is smooth on R' and zero elsewhere. Let C denote thesmsubalgebra of smooth distributions. The distributional

derivative of u e csm, u(1) - ó(1) ~ u, equals u t u(0')ó, whereu e Csm denotes the ordinary derivative of u on R'. Example: Letu e csm correspond to 2exp(t) on R'. Then u(1) - u} 2á. Formore details on Cimp, see [9) -[10], also [6] -[8]; because ofits nice properties we can keep our treatment fully algebralc.

It can be readily shown that, for every real-valued square

matrix H, (I6(1) - Hó) is invertible (w.r.t. convolution); itsinverse corresponds to exp(Ht) on R'. Hence the solutions off1.3a) reduce to the ordinary ones ((1.2)) if E is invertible

and u e Csm; for every pair (x „ u), (1.3a) has exactly onesolution. Also, note that (1.3a) reduces to (l.la) if u and xare smooth. In general, however, the solution set

S(xo, u) -(x e Cimp~[EÓ(1) - Aó] ~ x- Bu f Ex,ól,(1.4)

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may be empty or contain infinítely many elements, see [6]. Weare ready for the definition of the free end-pointlinear-quadratic control problem subject to (1.3).

(LQCP)-: For all x,, determine

J'(x,) :- inf{,~y~ydtlu E cm , x E S(xo, u) n Cn !,sm sm (1.5)and if, for every x„ J'(x,) ~~, then compute ( if possible)

optimal controls u E csm and associated optimal state

trajectories x E S(xo, u). The problem (LQCP)- is ~olvable ifboth requirements are met.

In the sequel we will need several subspaces of interest. Let

'r(F) :- Ix, E Rn ~3 m 3 n: lim ru(t) ~- 0}U E Csm X E S(Xo, U) fl CSm t~ LX(t)

9'C(i) :- IXo E Rn ~3U E Cm 3X E S{Xo, U) fl Cn ~ y- 0.sm sm

x(0') - xol ,

c(E) '- (x, E Rn~3u E cm 3x E S(x „ u) n cn : lim y(t) - 01sm sm t-oo (1.6)

and let r'B(E), JB(E) denote those subspaces of ~(i) and o(E),for which u and x in the respective definitions are of the Bohltype (a Bohl function is any linear combination of functions

tkexp{~t), k~ 0). For YC(E) we have the following result.

Proposition 1.1 [7, Prop. 3.5, Theorem 3.6].

YC(z) is the largest subspace ~ c Rn for which there exists a

matrix F E RXn such that (A } BF)x c E:~, (C t DF): - 0.

If, moreover,

Y(E) :- Ixo E rtn~3u E Cm 3x E S{xo, u) n en ' y- 01,sm sm 11.7)

then (7, Prop. 3.4] tells us that

Y(Z) - YC(F) f ker(E). (1.8)

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In [10], [7] a point x, e v~(E) is called weakly unobservable; we

establish that all points in YC(E) are also consistent. Let, forany subspace T and q any complex row vector of compatible síze,qT stand for [qt~t e TI. The next result is stated in [3].

Proposition 1.2.

Let E be invertible. Then ~(Z) f v(E) - a(z) -{xo e rtn~J'(x,) c~l, vB(E) - v(z), rB(E) - ~(E) and o(E) - Rn if and only if, forall A E e with Re(,~) ~ 0,

q[nE - A, - B] - 0 and pEv(z) - 0 only if q- 0. (1.9)

If in Proposition 1.2, C- I aad D- 0, then Y(Z) - 0 and wereobtain the well-known statement that s(E) - Rn if and only ifE is (stare) stabilizable. We will say that Z is output

stabilizable if r(I) - Rn.

Now, we consider z with arbitrary E. From [6, Theorem 4.5] weborrow

Proposition 1.3.

vXp E Rn 3u E Csm 3X E S(X„ U) fl Cgm ~

im(E) f im(B) t A(ker(E)) - R1. (1.10)

2. Iisin results. Yithout loss of qeaerality, we may rewrite z inthe form

(I~ 0~16(i) ~rXa, - [A:i A22] [x:J } C~JU } ~ OJ lxos,a~

YL - [c, C~] [XL'~ t Du. (2.1)Assume that (1.10)~is satisfied, i.e., that [A,Z B2] is of full

row rank. Let T z ~~~ E R(ntm-e)x(ntm-k) of full column rank,Ci 2

be such that [Aii Bi] - 0. Set N:z AzzA21' } BZBi' ) 0, L:-T~T ~ 0. Then

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~Q:- BZ?~ T~l is invertible, 4-' -~- L-Ol4'. (2.2)

If F denotes the stJandard systeml" J

611) ' z- Az t Bv t zoa, (2.3a)rr - Cz t pv, (2.3b)

with

A:- Aa~ -[A~z B1 ] (B::'1N'`A~1, B:- [A1s B1]T,l: J,C:- C1 -[C, D] A~' N-'A21, D:- [C, D]T, (2.3c)

Bz'then it turns out that all solutions for (1.3) caa be expressedin solutions for (2.3) and vice versa.

Theorem 2.1.

Let (XO2~ e Rn, u e Cmmp and (X21 e S(fXó~l, u). Then

lx, - Z(X,,, V), lu'1 - IBL22J~~N-'(-l A,11) (Z(Xa,. v)) t TVll ll :~

xith v- L-`[Tl~x2 t T,~uJ e Cimp-k. Moreover, y- M(xol, v).

Conversely, let zo e Re, v E CimD-k, and z- z(z,, v). Then u-

- B2'N-`A „ z t T,v e Cimp and, for all x,~, (X'1 e S(fX" ~, u)l ~dJ L o2

Nith x~ - z and x, -- A„'N-`A21z t T,v. In ad ition, y-W(za, v).

Proof. First half. If in (2.3a) Mith zo - xo, Me inaert v as

prescribed, then a(1) ~ z- Az t[A1z B,JQ~-i L-~IIQ' (~21 t

IA~~xI~~ t xoaa ~ Àz t[A,: Bi] lu2J t(All ~`-' À)xl t1 8olla -J Az t

(`8(1) , x, - A,ixl - x,la)t(Alllll

- A)x~ f xc,a - a(1) ~ xt t

Á(z - x,), by (2.1) -(2.2). Hence [Iea(1) - 11a] ~ Iz - x,) - 0

1 Iu: l - M-11u:~ ~~8::'~M''(-Apixl) f?v,and z- x - 0. Since I` J K5L f1`

the rest is clear. The second hal is noK trivial.

Observe that if in (2.1), e- k(i.e., B is of full roM rank),then T is invertible and A - A,,, C- C, in (2.3). Here is ourfirst main result.

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Theorem 2.2

If the system ( 1.3) satisfies (1.10), then f(E) t Y(i) - a(Z) 3

Ixo e Rn~J'{x,) ~~I, ~B(E) - f(E) and aB(z) - a(a). Noreover,(1.3) is output stabilizable if and only if (1.9) - (1.10) aresatisfied.

Proof. Consider (2.1) - (2.3). Then [q, n,] r~I - A „ -A12 -B~jL - A21 -A,~ -B~J

- 0 if and only if q,[~Ie - A, - B] - 0 and q2 equals -

ql[A1: B,] ~A2t~JN'', for every a E C. Since ker(E) is containedBi,in all subspaces involved, both claims fo11oN immediately fromProps. 1.2, 1.3 and Theorem 2.1.

NoN, let us consider ( LQCP)". By Theorem 2.2, it is obvious thatoutput stabilizability is ne~~~essary for solvability. Thus,assume that z is output stabilizable. It folloMS from Theorem2.1 that (LQCP)' is solvable if and only if the correspondingproblem subject to ( 2.3) is solvable. Since Z is outputstabilizable, it is known that the optimal cost for the latterstandard problem can be represented as a quadratic form [3] -(4]. Noreover, for every initial condition z, there exist aunique optimal control v and a unique optimal state trajectoryz, both of the Bohl type, if ker(D) - 0, i.e., if the problem isregular [4]. In other words, output stabilizability of E is alsosufficient for solvability of the free end-point LQCP subject to(2.3) if the problem is reqular. If the problem is singular,

then for every zo optimal coatrols and state trajectories stillexist in the sense that the infimum of the cost criterion (1.5)is actually attained - yet, in qeneral they are distributionsrather than functions [13], [5]. Note that, in terms of (2.1) -

(2.3), ker(D) - 0.~ ker(fC22 D'1) - 0. Nence, by Theorem 2.1,l: Jwe arrive at

- 7 -

Theorem 2.3.

For every xo e~n, J'(x,) ~~ if and only if (1.9) - ( 1.10) aresatisfied. Assume this to be the case. Then there exists aunique real symmetric matrix P' ~ 0 with ker(E) c ker(P'), suchthat, for all xo, J'(x,) - x,~P'x,. If

ker( ~ Dl) n[A B]''im(E) - 0, {2.4)

then for every x, there exists a unique optimal control u and aunique optimal state trajectory x E S(x „ u), both of the Bohltype. If ( 2.4) is not satisfied, then for every x, there exist uE cmmp and x E S(x,, u) such that y E cgm and J-(xo) -,~y~ydt.

The condition (2.4) can be interpreted as a system property forE. In [8, Theorem 3.2] it is proven that ( 2.4) holds if and onlyif

y E Csm Cy u E Csm, X E S(Ro, U) Í~ ~,Sm. (2.5)In other words, ( 2.4) stands for the property that outputs for Eare functions only if associated controls and state trajectoriesare futtrtions as Mell. Therefore (LQCP)' is called reguLer in[8] if (2.5) is satisfied. Observe that ( 2.4) reduces to theclassical ker(D) - 0 if E is invertible. The linear-quadraticproblems considered in [1] -[2] are reQular in the sense of(2.4), since it is assumed there that ker( ~ Dl) - 0. An exampleof a regular linear-quadratic problem for phich ker( ~ Dl) ~ 0is given in [8]. Also, note that, unlike in [1] -[2JJ1,` xe1 allowthe state trajectories to diverqe.

- 8 -

If F is output stabilizable and (2.4) is not satisfied,0

then we may still assume A B to be of full column rank. LetC D

this be the case. Now the distríbutional optimal controls andstate trajectories for (LQCP)' (see Theorem 2.3) are in generalnot unique. This follows from Theorem 2.1, since it is proven in[5] that optimal controls and state trajectories for (LQCP)'subject to a standard system E are unique if and only if E isleft invertible [10, Theorem 3.26], i.e., if in (1.3) with Einvertible, y- 0 and x, - 0 imply that u- 0(and hence also x- 0). Noreover, the smooth parts of these unique optimalcontrols and state trajectories are of the Bohl type.

Two different concepts for left-invertibility for tAtplicit

systems are qiven in [7]. There, a system (1.3) is defined leftinvertible in the stronq sense if xo - 0 and y- 0 imply that u- 0 and Ex - 0(aad left invertible in the weak sense if inerelyu- 0), see [7, Defs. 4.1, 4.10]. Under the above-mentioned rankcondition, it is proven in [7, Corollary 4.15] that E is leftinvertible in the stronq sense if and only if xo - 0, y- 0imply that u- 0, x- 0. Hence, aqain by Theorem 2.1, Z is leftinvertible in the strong sense if and only if (2.3) is leftinvertible in the sense of [10] and thus

Corollary 2.4.

0Let E be output stabilizable and ker( A Bl) - 0. Then for every

C DJxo there exists exactly one (possibly distributional) u and

exactly one ( possibly distributional) x such that y e Cr andsm,~~ydt - J"(xo) if and only if Z is left invertible in thestronq sense. Noreover, if u,, x2 denote the smooth parts of uaad x, then u2 and x, are of the Bohl type.

' Supported by the Dutch orqanisatáon for scientific research(N.W.O).

- 9 -

References.

[1J Bender, D.J. and A.J. Laub (1981). The linear-quadraticoptimal regulator for descriptor systems. IEEE Trans. Aut.Ctr., AC-32, 672-688.

[2] Cobb, D. (1983). Descriptor variable systems and optimalstate regulation. IEEE Trans. Aut. Ctr., AC-28, 601-611.

[3] Geerts, A.H.A. and M.L.J. Hautus (1990). Theoutput-stabilizable subspace and linear optimal control. InRobust Control of Linear Systems and Nonlinear Control,Proqress in Systems and Control Theory, 4, Birkhaeuser,Boston, 113-120.

[4] Geerts, T. (1988). A necessary and sufficient condition forsolvability of the liaear-quadratic control problem withoutstability. Syst. Contr. Lett., 11, 47-51.

[5] Geerts, T. (1989). All optimal controls for the singularlinear-quadratic problem without stability; a newinterpretation of the optimal cost. Lin. Alg. Appl., 116,135-181.

[6] Geerts, T. Solvability conditions, consistency and weakconsistency for linear differential-algebraic equations andtime-invariant sinqular systems: The general case. Lin.Alg. Appl., to appear.

[7] Geerts, T. Invariant subspaces and invertibility propertiesfor sinqular systems: The general case. Lin. Alg. Appl., toappear.

[8] Geerts, T. Reqularity and singularity in linear-quadraticcontrol subject to implicit continuous-time systems.Circts., Syst. and Sign. Proc., to appear.

[9] Hautus, M.L.J. (1976). The formal Laplace transform forsmooth linear systems. Lect. Notes in Econ. and Math.Syst., 131, 29-46.

[10] Hautus, M.L.J. and L.M. Silverman (1983). System structureand singular control. Lin. Alq. Appl., 50, 369-402.

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[11) Schrrartz, L. (1978). Theorie des Distributions. Heraann,Paris.

[12] Verghese, G.C., B.C. Levy and T. Railath (1981). Aqeneralized state-space for sinqular systems. IEEE Trans.Aut. Ctr., AC-26, 811-831.

[13] Nillems, J.C., A, Kitapci andL.M. Silverman (1986).Sinqular optimal control: A qeometric approach. SIAM J.Ctr. Opt., 24, 323-337.

IN 1991 REEDS vERSCHENEN466 Prof.Dr. Th.C.M.J. van de Klundert - Prof.Dr. A.B.T.M. van SchaikEconomische groei in Nederland in een internationaal perspectief467 Dr. Sylvester C.W. Eijffinger

The convergence of monetary policy - Germany and France as an example468 E. Nijssen

Strategisch gedrag, planning en prestatie. Een inductieve studiebinnen de computerbranche

469 Anne van den Nouweland, Peter Borm, Guillermo Owen and Stef TijsCost allocation and communication470 Drs. J. Grazell en Drs. C.H. Veld

Motieven voor de uitgifte van converteerbare obligatieleningen enwarrant-obligatieleningen: een agency-theoretische benadering471 P.C. van Batenburg, J. Kriens, W.M. Lammerts van Bueren andR.H. Veenstra

Audit Assurance Model and Bayesian Discovery Sampling472 Marcel Kerkhofs

Identification and Estimation of Household Production Models473 Robert P. Gilles, Guillermo Owen, René van den Brink

Games with Permission Structures: The Conjunctive Approach474 Jack P.C. Kleijnen

Sensitivity Analysis of Simulation Experiments: Tutorial on Regres-sion Analysis and Statistical Design475 C.P.M. van Hoesel

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476 Stephan G. VannesteA Markov Model for Opportunity Maintenance

477 F.A. van der Duyn Schouten, M.J.G. van Eijs, R.M.J. HeutsCoordinated replenishment systems with discount opportunities478 A. van den Nouweland, J. Potters, S. Tijs and J. ZarzueloCores and related solution concepts for multi-choice games479 Drs. C.H. Veld

Warrant pricing: a review of theoretical and empirical research480 E. Nijssen

De Miles and Snow-typologie: Een exploratieve studie in de meubel-branche481 Harry G. Barkema

Are managers indeed motivated by their bonuses?

ii

482 Jacob C. Engwerda, André C.M. Ran, Arie L. RijkeboerNecessary and sufficient conditions for the existgnce of a positivedefinite solution of the matrix equation X. ATX- A- I

483 Peter M. KortA dynamic model of the firm with uncertain earnings and adjustmentcosts

484 Raymond H.J.M. Gradus, Peter M. KortOptimal taxation on profit and pollution within a macroeconomicframework

485 René van den Brink, Robert P. GillesAxiomatizations of the Conjunctive Permission Value for Games withPermission Structures

486 A.E. Brouwer 8~ W.H. HaemersThe Gewirtz graph - an exercise in the theory of graph spectra

48~ Pim Adang, Bertrand MelenbergIntratemporal uncertainty in the multi-good life cycle consumptionmodel: motivation and application

488 J.H.J. RoemenThe long term elasticity of the milk supply with respect to the milkprice in the Netherlands in the period 1969-1984

489 Herbert HamersThe Shapley-Entrance Game

490 Rezaul Kabir and Theo VermaelenInsider trading restrictions and the stock market

491 Piet A. VerheyenThe economic explanation of the jump of the co-state variable

492 Drs. F.L.J.W. Manders en Dr. J.A.C. de HaanDe organisatorische aspecten bij systeemontwikkelingeen beschouwing op besturing en verandering

493 Paul C. van Batenburg and J. KriensApplications of statistical methods and techniques to auditing andaccounting

494 Ruud T. FrambachThe diffusion of innovations: the influence of supply-side factors

495 J.H.J. RoemenA decision rule for the (des)investments in the dairy cow stock

496 Hans Kremers and Dolf TalmanAn SLSPP-algorithm to compute an equilibrium in an economy withlinear production technologies

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501 Harry Barkema and Sytse DoumaThe direction, mode and location of corporate expansions

502 Gert NieuwenhuisBridging the gap between a stationary point process and its Palmdistribution

503 Chris VeldMotives for the use of equity-warrants by Dutch companies

504 Pieter K. JagersmaEen etiologie van horizontale internationale ondernemingsexpansie

505 B. KaperOn M-functions and their application to input-output models

506 A.B.T.M. van SchaikProduktiviteit en Arbeidsparticipatie

507 Peter Borm, Anne van den Nouweland and Stef TijsCooperation and communication restrictions: a survey

508 Willy Spanjers, Robert P. Gilles, Pieter H.M. RuysHierarchical trade and downstream information

509 Martijn P. TummersThe Effect of Systematic Misperception of Income on the 5ubjectivePoverty Line

510 A.G. de KokBasics of Inventory Management: Part 1Renewal theoretic background

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512 R. PeetersOn Coloring j-Unit Sphere Graphs

iv

513 Drs. J. Dagevos, Drs. L. Oerlemans, Dr. F. BoekemaRegional economic policy, economic technological innovation andnetworks

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515 Drs. E. SchalingEuropean central bank independence and inflation persistence

516 Peter M. KortOptimal abatement policies within a stochastic dynamic model of thefirm

51~ Pim AdangExpenditure versus consumption in the multi-good life cycle consump-tion model

518 Pim AdangLarge, infrequent consumption in the multi-good life cycle consump-tion model

519 Raymond Gradus, Sjak SmuldersPollution and Endogenous Growth

520 Raymond Gradus en Hugo KeuzenkampArbeidsongeschiktheid, subjectief ziektegevoel en collectief belang

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522 A.G. de KokBasics of inventory management: Part 3The (b,Q)-model

523 A.G. de KokBasics of inventory management: Part 4The (s,S)-model

524 A.G. de KokBasics of inventory management: Part 5The (R,b,Q)-model

525 A.G, de KokBasics of inventory management: Part 6The (R,s,S)-model

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52~ A.G.M. van Eijs, M.J.G. van Eijs, R.M.J. HeutsGecoSrdineerde bestelsystemeneen management-georiënteerde benadering

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529 Stephan G. VannesteMaintenance optimization of a production system with buffercapacity

530 Michel R.R. van Bremen, Jeroen C.G. ZijlstraHet stochastische variantie optiewaarderingsmodel

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Vi

IN 1992 RSEDS VERSCHENEN532 F.G, van den Heuvel en M.R.M. Turlings

Privatisering van arbeidsongeschiktheidsregelingenRefereed by Prof.Dr. H. Verbon

533 J.C. Engwerda, L.G. van WilligenburgLQ-control of sampled continuous-time systemsRefereed by Prof.dr. J.M. Schumacher

534 J.C. Engwerda, A.C.M. Ran 8~ A.L. RijkeboerNecessary and sufficient conditions for the existence of a positivedefinite solution of the matrix equation X f AwX-lA - Q.Refereed by Prof.dr. J.M. Schumacher

535 Jacob C. EngwerdaThe indefinite LQ-problem: the finite planning horizon caseRefereed by Prof.dr. J.M. Schumacher

536 Gert-Jan Otten, Peter Borm, Ton Storcken, Stef TijsEffectivity functions and associated claim game correspondencesRefereed by Prof.dr. P.H.M. Ruys

~i";~ Jttck P.C. Kletjnen, Gustnv A. A]inkValidation of simulation models: mine-hunting case-studyRefereed by Prof.dr.ir. C.A.T. Takkenberg

538 V. Feltkamp and A. van den NouwelandControlled Communication NetworksRefereed by Prof.dr. S.H. Tijs

539 A. van SchaikProductivity, Labour Force Participation and the Solow Growth ModelRefereed by Prof.dr. Th.C.M.J. van de Klundert

540 J.J.G. Lemmen and S.C.W. EijffingerThe Degree of Financial Integration in the European CommunityRefereed by Prof.dr. A.B.T.M. van Schaik

541 J. Bell, P.K. JagersmaInternationale Joint VenturesRefereed by Prof.dr. H.G. Barkema

542 Jack P.C. KleijnenVerification and validation of simulation modelsRefereed by Prof.dr.ir. C.A.T. Takkenberg

543 Gert NieuwenhuisUniform Approximations of the Stationary and Palm Distributionsof Marked Point ProcessesRefereed by Prof.dr. B.B, van der Genugten

vii

544 R. Heuts, P. Nederstigt, W. Roebroek, W. SelenMulti-Product Cycling with Packaging in the Process IndustryRefereed by Prof.dr. F.A. van der Duyn Schouten

545 J.C. EngwerdaCalculation of an approximate solution of the infinite time-varyingLQ-problemRefereed by Prof.dr. J.M. Schumacher

546 Raymond H.J.M. Gradus and Peter M. KortOn time-inconsistency and pollution control: a macroeconomic approachRefereed by Prof.dr. A.J. de Zeeuw

54~ Drs. Dolph Cantrijn en Dr. Rezaul KabirDe Invloed van de Invoering van Preferente Beschermingsaandelen opAandelenkoersen van Nederlandse Beursgenoteerde OndernemingenRefereed by Prof.dr. P.W. Moerland

548 Sylvester Eijffinger and Eric SchalingCentral bank independence: criteria and indicesRefereed by Prof.dr. J.J. Sijben

549 Drs. A. SchmeitsGeintegreerde investerings- en financieringsbeslissingen; Implicatiesvoor Capital BudgetingRefereed by Prof.dr. P.W. Moerland

550 Peter M. KortStandards versus standards: the effects of different pollutionrestrictions on the firm's dynamic investment policyRefereed by Prof.dr. F.A. van der Duyn Schouten

551 Niels G. Noorderhaven, Bart Nooteboom and Johannes BergerTemporal, cognitive and behavioral dimensions of transaction costs;to an understanding of hybrid vertical inter-firm relationsRefereed by Prof.dr. S.W. Douma

552 Ton Storcken and Harrie de SwartTowards an axiomatization of orderingsRefereed by Prof.dr. P.H.M. Ruys

553 J.H.J. RoemenThe derivation of a long term milk supply model from an optimizationmodelRefereed by Prof.dr. F.A. van der Duyn Schouten

554 Geert J. Almekinders and Sylvester C.W. EijffingerDaily Bundesbank and Federal Reserve Intervention and the ConditionalVariance Tale in DM~S-ReturnsRefereed by Prof.dr. A.B.T.M. van Schaik

555 Dr. M. Hetebrij, Drs. B.F.L. Jonker, Prof.dr. W.H.J. de Freytas"Tussen achterstand en voorsprong" de scholings- en personeelsvoor-zieningsproblematiek van bedrijven in de procesindustrieRefereed by Prof.dr. Th.M.M. Verhallen

viii

556 Ton GeertsRegularity and singularity in linear-quadratic control subject toimplicit continuous-time systemsCommunicated by Prof.dr. J. Schumacher

557 Ton GeertsInvariant subspaces and invertibility properties for singular sys-tems: the general caseCommunicated by Prof.dr. J. Schumacher

558 Ton GeertsSolvability conditions, consistency and weak consistency for lineardifferential-algebraic equations and time-invariant singular systems:the general caseCommunicated by Prof.dr. J. Schumacher

559 C. Fricker and M.R. JaibiMonotonicity and stability of periodic polling modelsCommunicated by Prof.dr.ir. O.J. Boxma

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