On Infinite Dimensional Linear Quadratic Problem with Fixed Endpoints Continuity Question

Munich Personal RePEc Archive

On Infinite Dimensional

Linear-Quadratic Problem with Fixed

Endpoints. Continuity Question

Przyłuski, K. Maciej

2014

Online at

https://mpra.ub.uni-muenchen.de/57430/

On Infinite Dimensional Linear-Quadratic Problem

with Fixed Endpoints. Continuity Question

K. Maciej Przyłuski

e-mail:

[email protected]

To appear in

Internationa Journal of Applied Mathematics and Computer Science

Abstract

In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solutionuto the equationSu= Rzto be continuously dependent onzare given. These conditions are used to study continuity of the minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

Keywords: Minimum norm problem, linear-quadratic control and linear-quadratic econo-mies, controllability and approximate controllability, null-controllability, continuity of opti-mal control.

JEL classification: C61, C65, E60.

1

Introduction

The existing theory of linear-quadratic problem has been successfully applied to the design of many industrial and military control systems (see e. g. Athans 1971). A stochastic ver-sion of this problem plays today an important role in macroeconomics, where the so-called linear-quadratic economies are considered (see e. g. Ljungqvist and Sargent 2004, Sent 1998). These (dynamic stochastic) optimizing models had to havelinearconstraints withquadratic objective functions to get a linear decision rule (see e. g. Chow 1976, Kendrick 1981). How-ever, such stochastic problems are frequently infinite dimensional; see e. g. (Federico 2011), and the references cited therein.

We will consider infinite dimensional linear control systems which can be represented by two linear continuous operators describing the influence of control, and the constraints imposed on all system’s trajectories by given initial and final conditions. The minimum en-ergy and linear-quadratic problem for such systems will be developed. These problems can be studied in an appropriate Hilbert space setting. Then (as it is well known) the existence and uniqueness of optimal solution to the above problems can be easily established, under rather mild assumptions.

dependence of optimal solutions on constraints on values of admissible controls has been considered in (Przyłuski 1981). A much more general approach to such problems is pre-sented in (Kandilakis and Papageorgiou 1992, Papageorgiou 1991).

The paper is organized as follows. In Sections 2 and 3 we consider quite general mini-mum norm problems. The obtained results are next applied (in Section 4) to study a linear-quadratic problem. In the last sections (Sections 5 and 6) the minimum energy problem with fixed endpoints for some classes of linear infinite dimensional (discrete-time and continuous-time) control systems is considered.

The notation used in the paper is standard (see e. g. Aubin 2000, Laurent 1972, Luenberger 1969, Corless and Frazho 2003). In particular, for any Hilbert spaceHandx, y∈H, by(x|y)

we usually denote the inner product ofxandy. Let us recall that the normkxkof anyx ∈H

is defined to be equal to the square root of(x|x). When Mis a subset of a Hilbert space,M

denotes the closure ofM. For any linear subspaceSofH, we denote byS⊥ _{the orthogonal}

complement ofS_{. For arbitrary Hilbert spaces}H_1andH_{2, we write}H₁⊕H_{2for the Hilbert}

sum of these spaces. For h := (h1, h2) ∈ H1⊕ H2, the norm khk := kh1k2 +kh2k2

1/2

. We shall write L(H1,H2)for the (naturally normed) Banach space of all continuous linear

operatorsH1 → H2. When H1 = H2, the symbolL(H1) is used instead ofL(H1,H2). For

any operatorA∈L(H1,H2),kAkdenotes its (operator) norm,KerAdenotes its kernel, and

Im_{Ais its image. The (Hilbert space) adjoint ofAis denoted byA}⋆_{. For any Hilbert spaceH}

we writeℓ2

τ(H)for the Hilbert space of allH-valued sequencesh= (hk)τk−=10, the space being

normed by the norm|·|2defined (as usual) by the formula|h|2 := Pτ

−1 k=0khkk

21/2_.

2

Minimum norm problem

Let H_u, H_vand H_zbe real Hilbert spaces. Let S ∈ L(Hu,Hv)and R ∈ L(Hz,Hv)be fixed

operators. We consider the followingminimum norm problem. For a givenz∈Hz, findub ∈Husuch that

Su_b =Rz (1a)

and

kubk=inf

u {kuk|Su=Rz}. (1b)

We summarize below somewell known resultsconcerning the above described optimiza-tion problem. We first define thespaceZof admissible values ofzin the following way:

Z:= {z ∈Hz|∃u∈Hu:Su=Rz}. (2)

Of course,Z = R−1_(ImS)

(the inverse image ofIm_{SunderR). LetP denote the orthogonal}

projection ofHuonto (KerS)⊥. Assume z ∈ Zisfixed, and letu′ andu′′ be such thatSu′ =

Su′′_{=Rz. ThenSPu}′ _=SPu′′ _{=Rz. In particular,Pu}′_−Pu′′_{∈KerSand thereforePu}′ _=Pu′′_.

It follows that Puis the same for allu ∈ H_{u satisfying the constraint} Su = Rz, with fixed

z∈Z. For anyz ∈Z, we denote suchPubybu(z). Observe that, for anyusatisfyingSu=Rz, we haveu=u(z) + (Ib −P)u, whereIdenotes the identity operator onHu. It follows that

kuk2=ku(z)b k2+k(I−P)uk2≥ ku(z)b k2.

Hence,for anyz ∈Z,u(z)b is the (unique) solution to our minimum norm problem.

The considerations presented above show that one can define a mappingZ_→Hu, which

Proposition 1. The mappingK:Z_→Huis linear, i. e.K(α1z1+α2z2) =α1Kz1+α2Kz2.

Proof. Let z1, z2 ∈ Z, α1, α2 ∈ R, andz = α1z1+α2z2. Since Z is a linear subspace of Hz,

z ∈ Z. To justify thatKis linear we should prove thatub(α1z1+α2z2) = α1u(zb 1) +α2u(zb 2).

For this, let us observe that

S(α1u(zb 1) +α2bu(z2)) =α1Su(zb 1) +α2Su(zb 2)

=α1Rz1+α2Rz2=R(α1z1+α2z2) =Rz.

Since

α1u(zb 1) +α2u(zb 2)∈(KerS)⊥,

we conclude that

b

u(z) =α1u(zb 1) +α2u(zb 2).

The main result of this section is the following theorem.

Theorem 1. Kis continuous if and only if the spaceZof admissible values ofzis closed inH_z. Proof. Necessity. Letz ∈Z, the closure ofZ. Then there exists a sequence(zn)∞n=1such that

zn∈Zand limzn=z. Letun=Kzn. Of course,Sun=Rzn. Then

kun−umk ≤ kKkkzn−zmk,

and (since(zn)∞n=1is convergent),(un)∞n=1is a Cauchy sequence, and therefore the sequence

(un)∞n=1is also convergent. Letu=limun. If we take the limits of both sides of the equality

Sun=Rznasn→ ∞, we find thatSu=Rz. It means thatz∈Z.

(Sufficiency.) LetZbe closed. ThenZis a Hilbert space with respect to the inner product induced from Hz. Let eR denote the restriction of the operator R to the Hilbert space Z.

Observe thatIm_S⊃Ime_{R. Using the Douglas Factorization Theorem (see e. g. Douglas 1966,}

Rolewicz 1987) we conclude that there exists an operatorKe ∈ L(Z,H_u) such thatSKe = Re. Let P denote (as usual) the orthogonal projection of Hu onto (KerS)⊥. Then, for z ∈ Z,

S(PK)ze = SKze = Rze = Rz. Since PKze ∈ (Ker_S)⊥_{, K := PKe is the mapping which assigns}

to any z ∈ Zthe minimum norm solution u(z)b to the equation Su = Rz. It is obvious that

K∈L(Z,H_u). In particular,Kis continuous.

Remark 1. The existing proofs of the Douglas Factorization Theorem are usually based on the Closed Graph Theorem (see e. g. Douglas 1966, Rolewicz 1987). So it is not surprising that to prove the sufficiency part of Theorem 1 we could have used (instead of the Douglas Factorization Theorem) the Closed Graph Theorem.

Using Theorem 1 one can prove1_{the following remarkable characterization of closedness}

of the spaceZ_{of admissible values of}z.

Corollary 1. The following statements are equivalent. (i) The spaceZof admissible values ofzis closed inHz.

(ii) There existsα ≥ 0such that, for everyz ∈ Z, one can findu ∈ H_usatisfyingSu =Rz and the inequalitykuk ≤αkzk.

(iii) For everyε > 0,z ∈Z, andu∈HusatisfyingSu=Rz, there existsδ > 0such that for every

z′ _{satisfying the inequalitykz−z}′_{k ≤ δand belonging toZ, one can findu}′ _{∈ H}

usuch that

Su′ _=Rz′_andku−u′_{k ≤ε.}

We see that it is important to know when the spaceZ_{is closed. We collect below a few}

simple results in this direction.

Proposition 2. Im_S⊃Im_{Rif and only ifZ=Hz. .}

In particular, ifIm_S⊃Im_{R, the spaceZof admissible values ofzis closed inHz.}

Before formulating our next result we recall that a linear continuous operator acting be-tween Hilbert spaces possesses a linear continuous right inverse if and only if this operator is surjective (employ the Douglas Factorization Theorem, or see e. g. Aubin 2000). Let us re-call also that for any mappingLand any subsetMof its domain,L−1_{(M)denotes the inverse}

image ofMunder the mappingL.

Proposition 3. LetRbe right invertible. Assume thatZis closed. ThenIm_{Sis also closed.}

Proof. LetJbe a right inverse ofR, so thatRJ =I, the identity operator onH_v. ThenIm_S= (RJ)−1_{(ImS) =J}−1_R−1_(ImS)=J−1_{(Z). SinceJis continuous,J}−1_{(Z)(being equal toImS) is}

closed.

Remark 2. The above proposition says thatwhenRis right invertible andIm_S6=Im_{S, the space} Z of admissible values of z cannot be closed, and therefore the corresponding linear mapping K is discontinuous.

Proposition 4. Assume thatIm_{Sis closed. ThenZis closed.}

Let us note that the spaceZ_{of admissible values of}zis always closed, whenIm_{Sis finite}

dimensional (or finite codimensional).

We end this section with the following two general remarks.

Remark 3. Let us recall (see e. g. Luenberger 1969) that theMoore-Penrose pseudoinverseS†_of Sexists if and only if the image ofSis closed. The assumption thatIm_S=Im_{Ssignificantly}

simplifies the minimum norm problem since then the mapping K which maps z ∈ Z to the minimum norm solution bu(z)to the equationSu = Rz is equal to the restriction of the continuous linear operatorS†_{Rto the (closed) subspaceZofH}

z.

Remark 4. Consider the special case where H_z = H_v and R = I, the identity operator. Assume thatIm_{Sis a proper dense subspace ofHz(i. e.}Im_{S = Hv 6=} Im_{S). Then, only for} v ∈ Im_{S, there exists a (unique) solution to our minimum norm problem. When v /∈} Im_S,

one can consider a relaxation of this problem. One of possible approaches is to solve the (unconstrained) problem of minimizing kuk2 _+ρkSu−vk2_{, for large positive ρ. Another}

possibility is to study the (constrained) minimization problem of findingu∈Huof minimal

norm and such thatkSu−vk ≤η, for small positiveη. These approaches are closely related. For details, the interested reader should consult (Kobayashi 1978), or (Emirsajłow 1989).

3

Extended minimum norm problem

LetH0be a real Hilbert spaces andR0 ∈L(H0,Hv)be a given operator. We consider below

the followingextended minimum norm problem. For givenz0∈H0andzv∈Hv, findub∈Husuch that

and

kubk=inf

u {kuk|Su=R0z0+zv}. (3b)

One can reduce the above problem to the minimum norm problem defined by relations (1). For this, let I denote the identity operator on Hv and Hz := H0 ⊕Hv (as usual, ⊕

denotes the direct sum of Hilbert spaces). Let z = (z0, zv) and R =

R0 I

, so that Rz = R0z0+zv, andR∈ L(Hz,Hv). We see at once that relations (3) can be rewritten in the form

used to define our standard minimum norm problem, with R as above. Note that for the extended minimum norm problem, by the space of admissible values ofzwe should mean the following subspace ofH0⊕Hv:

Z={(z0, zv)∈H0⊕Hv| ∃u∈Hu:Su=Rz0+zv}. Proposition 5. The spaceZdescribed above is closed if and only ifIm_{Sis closed.}

Proof. We know from Proposition 4 thatZis closed, ifIm_{Sis closed. Assume now that}Im_S

is closed. SinceR=R0 I

is right invertible, one can use Proposition 3 to deduce thatZis closed.

Proposition 6. LetR=R0 I

. AssumeIm_S⊃Im_{R0. Then(z0, zv)∈Zif and only ifzv∈}Im_S.

Proof. Letzv ∈ImS. Thenzv =Suv, for someuv ∈Hu. Letz0 ∈H0. SinceImS⊃ ImR0, one

can findu0 ∈ Husuch that R0z0 = Su0. HenceS(u0+uv) = R0z0+zv. It follows that any

z= (z0, zv)withzv∈ImSbelongs toZ.

Conversely, let(z0, zv)∈ Zso thatSu=R0z0+zv, for someu∈ Hu. SinceImS ⊃ImR0,

one can findu0 ∈Husuch thatR0z0 =Su0. ThenS(u−u0) =zv, i. e.zv∈ImS. Corollary 2. Let R = R0 I

. Then Im_{S ⊃} Im_{R0 if and only if}Z = H₀⊕Im_{S. In particular,} Z=H0⊕Hvif and only ifSis surjective.

We know that, for anyz ∈ Z, there exists a (uniquely defined) solutionu(z)b to the considered extended minimum norm problem. Since z = (z0, zv), we also write u(zb 0, zv) instead of u(z)b .

In virtue of Proposition 1, the mapping (z0, zv) 7→ u(zb 0, zv)is linear. It is a consequence of

Theorem 1 and Proposition 5 thatthis mapping is continuous if and only ifIm_{Sis closed.}

Unfortunately, the assumption thatIm_{Sis closed is rather restrictive. Our next result is}

dealing with the extended minimum norm problem forSwhich image isnotclosed.

Theorem 2. Assume that

Im_S⊃Im_{R0 and} Im_S6=Im_S.

Letu(zb 0, zv)be the solution to the extended norm minimization problem. Then

b

u(z0, zv) =K0z0+Kvzv,

where K0 is linear and continuous (i. e. K0 ∈ L(H0,Hu)), and Kv : ImS → Hu is linear, but it

cannot be continuous.

Proof. In view of Corollary 2,u(zb 0, zv)is well-defined for all pairs(z0, zv)such thatz0 ∈ H0

andzv∈ImS. In particular,(z0, 0)and(0, zv)are inZ. Observe thatu(zb 0, 0)is the minimum

norm solution to the equationSu=R0z0, whereasu(0, zb v)is the minimum norm solution to

the equationSu=zv. Sinceu(zb 0, 0)andbu(0, zv)belong to(KerS)⊥, and

S(u(zb 0, 0) +u(0, zb v)) = R0z0+zv,

we have the equalityu(zb 0, 0) +u(0, zb v) =u(zb 0, zv). It means that

K0z0 =u(zb 0, 0), and Kvzv=u(0, zb v).

The inclusionIm_S⊃Im_{R0implies (see Proposition 2) thatR}−1

0 (ImS) =H0, and therefore

K0 is continuous. On the other hand, since ImS 6= ImS, Kv is discontinuous, in view of

4

Linear-quadratic problem

Let H_w, H_{y be real Hilbert space, and}W ∈ L(Hu,Hw), L1 ∈ L(Hu,Hy), L2 ∈ L(Hz,Hy)

be given operators. We always assume thatW is an injection with closed image. For Hilbert spaces, such operators are characterized (see e. g. Aubin 2000) by the existence of a positive constantγsuch thatkWuk ≥ γkuk, for allu. This inequality is equivalent to positive defi-niteness (also called coerciveness) of the self-adjoint operatorW⋆_{W. It follows thatW is an}

injection with closed image if and only if W⋆_{W is positive definite. Since W}⋆_{W is always}

nonnegative definite,W⋆_{Wis positive definite if and only if the operator is invertible.}

In this section we consider the followinglinear quadratic problem. For a givenz∈Hz, findub ∈Husuch that

Su_b =Rz (4a)

and

kWu_bk2+kL1bu+L2zk2

=inf

u

kWuk2_+kL

1u+L2zk2|Su=Rz . (4b)

Let us observe that for anyu∈H_uandz∈H_z,

kWuk2_+kL

1u+L2zk2

= u|(W⋆_W+L⋆

1L1)u

+2 u|L⋆

1L2z

+kL2zk2. (5)

Let

Q:= W⋆_W+L⋆

1L1.

Of course,Q∈L(Hu,Hu). SinceWis an injection with closed image, the operatorQabove

defined is always (i. e. independently of L1) positive definite, hence invertible. Moreover,

there exists a unique positive definite square rootQ1/2_{ofQ. Observe that the first term on}

the right-hand side of (5) can be written askQ1/2_uk2_{. SinceQ}1/2_{is positive definite, it is also}

invertible. The inverse ofQ1/2will be denoted byQ−1/2

.

Our purpose is to reduce the considered linear quadratic problem into a norm mini-mization problem. For this, let us compute the norm of Q1/2_(u+Q−1_L⋆

1L2z). After easy

calculations we obtain the following equality:

kQ1/2(u+Q−1_L⋆

1L2z)k2

=kQ1/2uk2+2 u|L⋆

1L2z

+kQ−1/2

L⋆

1L2zk2. (6)

It follows (compare equations (5) and (6)) that

kWuk2+kL1u+L2zk2

−kQ1/2(u+Q−1_L⋆

1L2z)k2

=kL2zk2−kQ

−1/2_L⋆

1L2zk2

We see that the difference betweenkWuk2_+kL

1u+L2zk2 andkQ1/2(u+Q−1L⋆1L2z)k2 is

inde-pendent ofu. It means that instead of the linear-quadratic problem defined by (4), one can consider the problem in which (for fixedz) we are minimizing with respect tou(foru∈H_u

satisfyingSu=Rz) the norm

kQ1/2(u+Q−1_L⋆

1L2z)k. (7)

Let

q:=u+Q−1_L⋆

Then (7) takes the form kQ1/2_{qk, and the constraint Su = Rz should be replaced by the}

equalitySq = (R−Q−1_L⋆

1L2)z. Now, let us define onHua new inner product(·|·)Qby the

formula(x|y)_Q := (x|Qy), where x, y ∈ Hu, and(·|·) is the original inner product ofHu.

Since Qis a positive definite operator, (x|y)_Qis a well-defined inner product on Hu. For

the induced by this inner product norm k·kQ, we have kqkQ = kQ1/2qk, for all q ∈ Hu.

SinceQis positive definite, thenormsk·kQandk·k(i. e. the original norm ofHu) are equivalent.

Let us recall that continuity of functions defined onH_{uand closedness of subsets of}H_uare

independent of the assumed norms onHu, if these norms are equivalent.

On account of the considerations presented above, one can formulate a minimum norm problem reflecting all properties of the studied in this section linear quadratic problem as follows.

For a givenz∈Hz, findqb∈Husuch that

S_bq= (R−SQ−1_L⋆

1L2)z (9a)

and

kqbkQ=inf q

kqkQ|Sq= (R−SQ

−1_L⋆

1L2)z , (9b)

whereQ=W⋆_W+L⋆

1L1, andWis an injection with closed image.

It is immediate that,for a givenz, the above minimum norm problem has a solution if and only if our original linear-quadratic problem defined by relations (4) is solvable. Then, the solutionsbq

andubto these problems are related by (8).

Let, for the minimum norm problem defined by (9), Zq denote the counterpart of the

spaceZof admissible values ofz, defined in Section 1 by (2), i. e.

Zq:=

z∈Hz|∃q∈Hu:Sq= (R−SQ−1L⋆1L2)z .

It follows from our considerations of Section 1 that, for every z ∈ Z_{q, there exists a uniquely}

defined solutionbqto the minimum norm problem (9), andbqis a linear function ofz. This function, to be denoted byKq, is a continuous functionZq→Huif and only ifZqis closed inHz(see Theorem

1).

It happens that Zqis closed inHzif and only ifZ =R−1(ImS)is closed. More precisely, we

can prove the following elementary result, saying in particular thatZq=Z. Proposition 7. For any linear mappingF:H_z→H_u,

R−1_{(ImS) = R+SF}−1

(Im_S.

Proof. Of course,z∈R−1_{(ImS)if and only if there existusuch thatSu=Rz. ThenSu+SFz=}

Rz+SFz, andS(u+Fz) = (R+SF)z. Now it is obvious thatz ∈ R+SF−1(Im_S.

Conversely, assume thatz ∈ R+SF−1(Im_{S. Then there existsusuch thatSu = (R+} SF)z, for somez. ThenS(u−Fz) =Rz, and thereforez ∈R−1_(ImS).

It should be clear now that the linear-quadratic problem studied in this section possesses a solution if and only if z ∈ Z = R−1_{(ImS). The solution is uniquely determined by z, and}

will be denoted (as usual) bybu(z). LetK :Z :_→H_ube the mappingz 7→ u(z)b . From (8) we

conclude that

K=Kq−Q

−1_L⋆

1L2,

Theorem 3. Consider the linear quadratic problem defined by relations (4). Assume thatW is an injection with closed image. Then the linear mapping K : Z _→ H_u above defined is (well-defined and) continuous if and only ifZ=R−1_{(ImS)is closed inH}

z.

One can also generalize Theorem 2.

Theorem 4. Consider the linear quadratic problem defined by relations (4), with R = R0 I

(see Section 3). LetWbe an injection with closed image. Assume also that

Im_S⊃Im_{R0 and} Im_S6=Im_S.

Letu(zb 0, zv)be the solution to the considered linear quadratic problem. Then (as in Theorem 2)

b

u(z0, zv) =K0z0+Kvzv,

where K0 is linear and continuous (i. e. K0 ∈ L(H0,Hu)), and Kv : ImS → Hu is linear, but it

cannot be continuous.

We end this section with the following remark.

Remark 5. The fact that any linear-quadratic problem can be reduced to an appropriate min-imum norm problem is well known forcontrol systems described by differential equations. This reduction requires solving a Riccati-type differential or integral equation (for finite dimen-sional systems, see e. g. Brockett (1970); for infinite dimendimen-sional systems consult e. g. Curtain (1984)). A bit more general treatment of this topic is presented in (Chap. 4 of Porter 1966). Our approach to this reduction seems to be new.

5

Minimum energy control problem for infinite dimensional

discrete-time control systems

Consider alinear discrete-time control systemdefined by the difference equation

xk+1=Axk+Buk, (10)

where k runs through the set of non-negative integers. We assume that A ∈ L(X), B ∈

L(U, X), where thestate spaceXas well as thecontrol spaceUare real Hilbert spaces. Let

x0 ∈Xbe aninitial stateandu:= (uk)τ

−1

k=0be acontrolling sequence, whereτdenotes afixed

positive integer (“final time”). Then

xτ=Aτx0+ τ−1 X

k=0

Aτ−k−1_Bu k.

For discrete-time systems, we formulate the following fixed endpoints minimum energy con-trol problem.2

For given x0 ∈ X, xfinal ∈ X, and τ being a fixed positive integer, find a controlling sequence

b

u:= (u_bk)τ

−1

k=0such that

x_final=Aτx0+ τ−1 X

k=0

Aτ−k−1_B

b

uk (11a)

2_{In view of our results of Section 4 there is no need to consider explicitly a more general linear quadratic}

and

τ X

k=0

kubkk2

1/2

≤

τ X

k=0

kukk2

1/2

, (11b)

for any controlling sequenceu= (uk)τk−=10satisfying

x_final =Aτx0+ τ−1 X

k=0

Aτ−k−1_Bu

k. (11c)

In order to reformulate the fixed endpoints minimum energy control problem defined by (11) as an extended minimum norm problem discussed in Section 3, we putHu := ℓ2τ(U)so

that the norm of u ∈ Huwill be|u|2. We also assume thatH0 := X, Hv := X,Hz := X⊕X.

Let

R0:= −Aτ, (12)

S:=Aτ−1_{B, A}τ−2_{B, . . . , AB, B. (13)}

Let us note thatR0 ∈L(H0,Hv),S∈L(Hu,Hv), and

Su=

τ−1 X

k=0

Aτ−k−1_Bu k,

for anyu = (uk)τ

−1

k=0 ∈ Hu = ℓτ2(U). Of course, the operatorsR0 andS are depending onτ.

The image ofSis known as theτ-controllable subspace.

It is clear that the considered fixed endpoints minimum energy control problem for sys-tem (10) takes the following form.

For givenx0 ∈H0 =Xandxfinal ∈Hv =X, find (if it is possible)ub = (ubk)τ_k−=10 ∈Hu =ℓ 2 τ(U)

such that

Su_b =R0x0+xfinal,

and|u|_b2is not greater than the norm|u|2, for anyu= (uk)kτ−=10∈HusatisfyingSu=R0x0+xfinal,

whereR0andSare defined by (12) and (13), respectively.

There is no doubt that one can employ the results of Section 3 when studying the fixed endpoints minimum energy control problem for system (10). For this, let us note that, for the considered discrete-time system, the space Z =R−1_{(ImS)(as defined in Section 3) is as}

follows:

Z=(x0, xfinal)∈X⊕X| ∃u= (uk)τk−=10∈ℓ 2 τ(U):

x_final=Aτx0+ τ−1 X

k=0

Aτ−k−1_Bu

k . (14)

This space is depending onτ.

Let us observe that the minimum energy control problem specified by relations (11) is well-defined if and only if (x0, xfinal) ∈ Z, with Z given by (14). Let K (see Proposition 1)

denote the linear mapping which maps (x0, xfinal) ∈ Z to u(xb 0, xfinal) ∈ Hu = ℓ2τ(U), the

Theorem 5. Consider the fixed endpoints minimum energy control problem specified by relations (11), and the linear mappingK : (x0, xfinal) 7→ u(xb 0, xfinal). ThenKis continuous if and only if the

τ-controllable subspaceIm_{Sis closed.}

Let us recall (see e. g. Fuhrmann 1972) that a linear discrete-time system is said to be

exactly controllable in τ steps, if for any xfinal ∈ X, one can find a controlling sequence

u= (uk)τk−=10such that

x_final=

τ−1 X

k=0

Aτ−k−1_Bu k,

so that when x0 = 0, xfinal = xτ, for some u. In other words, the considered discrete-time

system is exactly controllable inτsteps if and only ifIm_S=X.

Corollary 3. The domain ofKis equal toX⊕Xif and only if system (10) isτ-exactly controllable. ThenKis continuous.

Proof. In view of Corollary 2 and Theorem 5, it is sufficient to observe that the spaceZ(see (14)) coincides withX⊕Xif and only theτ-controllable subspace is equal toX.

The assumption that a system is exactly controllable (or this that its τ-controllable sub-space is closed) may be too demanding for some infinite dimensional control systems. One can relax this assumption using Theorem 2 of Section 3. To formulate some results in this direction we introduce below two additional concepts of controllability; they are weaker than that of exact controllability. These concepts are well known; see e. g. Fuhrmann (1972) or Curtain and Zwart (1995).

We say that system (10) isapproximately controllable inτstepsif for eachx_final∈Xand anyε > 0, there exists a controlling sequenceu= (uk)τ_k−=10such that

kx_final−

τ−1 X

k=0

Aτ−k−1_Bu

kk ≤ε,

so that whenx0 =0, the norm kxfinal−xτkdoes not exceedε, for someu. It means that the

consideredsystem is approximately controllable inτsteps if and only if itsτ-controllable subspace is dense inX.

We also need the concept of null-controllability. It is said that system (10) isnull-controllable inτstepsif for everyx0∈Xthere exists a controlling sequenceu= (uk)τ

−1

k=0such that

Aτx0+ τ−1 X

k=0

Aτ−k−1_Bu k=0,

so that, for eachx0 one can findusteeringx0to the origin. In other words, the considered

system is null-controllable inτsteps if and only if Im_R0⊂Im_{S, i. e.}

Im_Aτ_⊂Aτ−1_{B, A}τ−2_{B, . . . , AB, B.}

Let (as usual) K denote the linear mapping which maps (x0, xfinal) ∈ Zto bu(x0, xfinal) ∈

H_u=ℓ2_τ(U). SinceKis linear, we have

b

u(x0, xfinal) =K(x0, xfinal) =K0x0+Kfinalxfinal,

for appropriate linear mappingsK,K0andKfinal.

Theorem 6. Consider the fixed endpoints minimum energy control problem specified by relations (11). Assume that the considered system is null-controllable in τ-steps, and that its τ-controllable subspace (i. e. Im_{S) is not closed. LetK0 and Kfinal be as above. Then K0is continuous, i. e. K0 ∈} L X, ℓ2

τ(U)

, andK_final:Im_S→ℓ2 τ(U)

is linear, but discontinuous. We have also the following.

Corollary 4. Assume that system (10) is null-controllable in τ-steps. Let the system be approxi-mately controllable inτsteps, but not exactly controllable. Then the conclusion of Theorem 6 is valid, i. e.K0is continuous, andKfinalis discontinuous.

6

Minimum energy control problem for infinite dimensional

continuous-time control systems

We will consider continuous-time systems. In what follows, we denote byT afixedpositive real number. Let a linear continuous-time control system be described by the differential equation

˙

x(t) =Ax(t) +Bu(t), (15)

where t runs through the set of non-negative real numbers. We assume that A is is the infinitesimal generator of a strongly continuous semigroup of continuous linear operators

Φ(t)_t≥0,B∈L(U, X), where thestate spaceXas well as thecontrol spaceUare real Hilbert spaces. We write L2_{((0, T);U) for the Hilbert space of all (equivalent classes of)}

square-integrable functions[0, T]_→U, normed in the usual way. Letx0 ∈Xbe aninitial stateand

u(·)∈L2((0, T);U)be acontrolling function. Then we say that

x(t) =Φ(t)x0+ Zt

0

Φ(t−s)Bu(s)ds (16)

is amild solution of equation (15) on[0, T]. The above formula makes sense for allx0 ∈ Xand

u(·) ∈ L2((0, T);U), and it can be shown thatx(·) ∈ L2((0, T);X). At this point we refer the reader to Balakrishnan (1981) or Curtain and Pritchard (1978), for details and the very clear exposition of various properties of mild (and weak) solutions of differential equations.

For continuous-time systems, we will consider the following fixed endpoints minimum energy control problem.3

For givenx0 ∈X,xfinal∈X, andT being a fixed positive real number, find a controlling function

b

u(·)∈L2((0, T);U)such that

x_final=Φ(t)x0+ ZT

0

Φ(T−s)Bu(s)dsb (17a)

and

ZT

0

ku(s)b kds

1/2

≤

ZT

0

ku(s)kds

1/2

, (17b)

for any controlling functionu(·)satisfying

x_final =Φ(t)x0+ ZT

0

Φ(T −s)Bu(s)ds. (17c)

Like in the case of problem (11), the above fixed endpoints minimum energy control problem can be rewritten as an extended minimum norm problem of Section 3. For this, it sufficient to putHu:= L2((0, T);U),H0:= X,Hv:=X,Hz:=X⊕X. Let

R0:= −Φ(T)x0, (18)

Su(·) :=

ZT

0

Φ(T −s)Bu(s)ds, (19)

for any u(·) ∈ L2_{((0, T);U). Then, for the considered continuous-time system, the space}

Z=R−1_{(ImS)(as defined in Section 3) is as follows:}

Z=(x0, xfinal)∈X⊕X| ∃u(·)∈L2((0, T);U):

x_final=Φ(t)x0+ ZT

0

Φ(T −s)Bu(s)ds . (20)

Let us note that R0 ∈ L(H0,Hv), andS ∈ L(Hu,Hv). In this section we assume thatR0, S

andZare given by formulas (18), (19) and (20), respectively. It is clear that the operatorsR0,

S, and the spaceZare depending onT.

The image of the above defined operator S is named the T-controllable subspace. For a broad class ofinfinite dimensional continuous-time systems, theT-controllable subspace (i. e.

Im_{S) cannot be closed, and therefore}Im_{Sis a proper subspace ofX. It takes place whenBis}

compact, orΦ(·)is a compact semigroup. Then the operatorSis compact and has (usually) infinite dimensional image. This important fact is well known (see Balakrishnan (1981), Curtain and Pritchard (1978), Kobayashi (1978), or Triggiani (1975a)).

In a similar manner like for discrete-time systems, one can define (see e. g. Curtain and Pritchard 1978, Curtain and Zwart 1995) the concepts of exact controllability, approximate controllability, and null-controllability for a continuous-time system.

Let us recall that a linear continuous-time system is exactly controllable on [0, T], if for everyx_final ∈X, one can find a controlling functionu(·)∈L2_{((0, T);U), such that}

x_final=

ZT

0

Φ(T −s)Bu(s)ds,

so that whenx0=0,xfinal=x(T), for someu(·). In other words, the considered

continuous-timesystem is exactly controllable on[0, T]if and only ifIm_S=X.

System (15) is said to be approximately controllable on[0, T], if for eachx_final ∈ X and anyε > 0there exists a controlling functionu(·)∈L2_{((0, T);U)such that}

kx_final−

ZT

0

Φ(T−s)Bu(s)dsk ≤ε,

so that when x0 = 0, the norm kxfinal −x(T)k does not exceed ε, for some u(·). It means

that the considered system is approximately controllable on[0, T] if and only if itsT-controllable subspace is dense inX.

The important concept of null-controllability for continuous-time systems is defined as follows. We say that system (15) isnull-controllable on[0, T]if for everyx0∈X, there exists

a controlling functionu(·)∈L2_{((0, T);U)such that}

Φ(t)x0+ ZT

0

so that, for eachx0one can findu(·)steeringx0to the origin. In other words, the considered

system is null-controllable on[0, T]if and only ifIm_R0⊂Im_S.

Various important results concerning the above introduced concepts of controllability have been obtained by Triggiani (1975a, 1975b, 1976).

We know that the minimum energy control problem described by relations (17) is well-defined if and only if (x0, xfinal) ∈ Z, with Zgiven by (20). Then (see Proposition 1) there

exists a linear mappingKwhich maps each(x0, xfinal)∈Ztobu(x0, xfinal)∈Hu=L2((0, T);U),

the (unique) solution to the considered fixed endpoints minimum energy problem, so that

b

u(x0, xfinal) = K0x0+Kfinalxfinal, for suitable linear mappings. It is obvious that the results

analogous to those obtained for our discrete-time problem (11) remain true,mutatis mutandis, for the continuous-time fixed endpoints minimum energy problem defined by relations (17). We record only the following result.

Proposition 8. Consider the fixed endpoints minimum energy control problem given by relations (17). Assume that system (15) is null-controllable on[0, T]. Let the system be approximately control-lable on[0, T], but not exactly controllable on[0, T]. LetK0andKfinalbe defined as usual, so that the

optimal solutionubto (17) can be written asu(xb 0, xfinal) =K0x0+Kfinalxfinal. ThenK0is continuous,

i. e.K0 ∈L X, L2((0, T);U)

, andK_final:Im_S→L2_{((0, T);U)is linear, but discontinuous.}

We end this section with the following example of a distributed parameter system.4 Example 1 (nosign). We consider, fort ∈ [0, T] and ξ ∈ [0, 1], the (one-dimensional) heat equation

∂θ

∂t(ξ, t) = ∂2_θ

∂ξ2(ξ, t) +h(ξ, t), (21a)

subject to theboundary condition

∂θ

∂ξ(0, t) = ∂θ

∂ξ(1, t) =0; (21b)

here θ(ξ, t) denotes the temperature at time t at position ξ. Then relations (21) describe a (thin homogeneous) metal rod of length one, with (perfectly) insulated endpoints, with some additional heat source that can increases (or decreases) the temperature at each point

ξalong the rod, at a given rateh(ξ, t), known also as theheat source density.

Our aim it to find a heat source density h such that the initial temperature distribution

θ(ξ, 0) will be changed to a given (desired) temperature distribution θ(ξ, T), at time T, and the energy used for this, i. e.

ZT

0 Z1

0

h(ξ, t)2dξdt, (22)

will be as low as possible.

It is well known (see e. g. Balakrishnan 1981, Curtain and Zwart 1995) that equations (21) can be rewritten as a differential equation of form (15), with suitableAand B. For this, let

X =U =L2_{((0, 1);}

R). Letx(t) := θ(·, t), andu(t) := h(·, t), so that (for eacht ∈ [0, T]),x(t)

andu(t)are real-valued functions of the (spatial) variableξ∈[0, 1]. Observe that

x(0) =θ(·, 0)and x(T) =θ(·, T)

are representing the initial temperature distribution and its desired (final) distribution at

t =T, respectively. For that reason, x(0)will play the role ofx0, andx(T)will be ourxfinal; see

relations (17).

4_{Distributed parameter systems are usually described by partial differential equations. For basic theory of}

The left-hand side of the considered heat equation (i. e. equation (21a)) can be identified with ˙x(t), the derivative of x with respect to t. The second term of the right-hand side of equation (21a) can be represented by u(t). It follows that when expressing relations (21) as a differential equation ˙x(t) = Ax(t) +Bu(t), we should assume that B = I, the identity operatorU_→X(=U).

To describe the operator A, let us consider any x ∈ X. Such x is a function of the spa-tial variable ξ ∈ [0, 1]. The right-hand side of (21a) contains the term (∂2_θ/∂ξ2_{)(ξ, t), i. e.}

the second derivative ofx with respect to ξ. It follows that Ais an ordinary second order differential operator, i. e. the operator defined by the formula

Ax= d

2_x

dξ2.

The domain dom_{A ofA should reflect differentiability conditions, and also the boundary}

condition imposed by (21b). It is known (and not very difficult to check) that the appropriate domain ofAcoincides with the linear subspace ofX=L2_{((0, 1);}

R)containing all absolutely continuous functionsxof the (spatial) variableξ, which first derivative (with respect toξ) is absolutely continuous, the second derivative belongs toL2_{((0, 1);}

R), and such that bound-ary condition (21b) is satisfied, i. e. (dx/dξ)(0) = (dx/dξ)(1) = 0. One can check that the above described linear operatorA : dom_{A → Xis the infinitesimal generator of a strongly}

continuous semigroup. Moreover, A belongs to the class of Riesz-spectral operators, and the semigroup Φ(t)_t≥0generated byAcan be written in an explicit form. For details, the interested reader should consult Theorem 2.3.5 and Examples 2.1.1, 2.3.7 in (Curtain and Zwart 1995).

We see that the considered heat equation (21) can be represented as a linear continuous-time control system described by a differential equation ˙x(t) =Ax(t) +Bu(t), withX, Uand

A, Bdescribed above. Therefore one can reformulate the problem of minimizing energy (22) as a fixed endpoints minimum energy control problem (17). ThenHu=L2 (0, T);L2((0, 1);R)

. Sinceu(t) :=h(·, t)) , for anyu∈H_u, we have

kuk2 ₌ ZT

0 Z1

0

h(ξ, t)2dξdt,

the norm kuk of ubeing evaluated inHu. Hence, the problem of minimizing energy (22)

falls into the framework we know from the beginning of this section.

It remains to check whether or not the linear continuous-time control system ˙x(t) = Ax(t) +Bu(t)representing heat equation (21) is exactly controllable, approximately control-lable, or null-controllable. It happens that (for arbitrary positiveT) the considered continuous-timesystem is approximately controllable on[0, T], null-controllable on[0, T], but it is never exactly controllable. These facts are well known, and can be justified with the aid of various argu-ments. The simplest way to prove them is to use controllability criteria presented in (Chap. 4 of Curtain and Zwart 1995). It has been done in the existing literature. In particular, Example 4.1.10 of (Curtain and Zwart 1995) proves that this system is never exactly controllable on

[0, T], but it is null-controllable. To prove that this system is approximately controllable on

[0, T], one can use the duality between observation and control. Example 4.1.15 of (Curtain and Zwart 1995) contains all necessary details.

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