Existence and Uniqueness of the Optimal Control in Hilbert Spaces for a Class of Linear Systems

doi:10.4236/iim.2010.22016 Published Online February 2010 (http://www.scirp.org/journal/iim)

Existence and Uniqueness of the Optimal Control

in Hilbert Spaces for a Class of Linear Systems

Mihai Popescu

Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy Bucharest, Romania

Email: ima [email protected]

Abstract

We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear sys-tems. We are interested in the minimization of time, energy and final manifold in transfer problems. The state variables space X and, respectively, the control variables space U, are considered to be Hilbert spaces. The linear operator T(t) which defines the solution of the linear control system is a strong semigroup. Our analy-sis is based on some results from the theory of linear operators and functional analyanaly-sis. The results obtained in this paper are based on the properties of linear operators and on some theorems from functional analysis.

Keywords: Existence and Uniqueness, Optimal Control, Controllable Linear Systems, Linear Operator

1. Introduction

A particular importance should be assigned to the analysis of the control of linear systems, since they represent the mathematical model for various dynamic phenomena. One of the fundamental problems is the functional optimization that defines the performance index of the dynamic product. Thus, under differential and algebraic restrictions, one determines the control corresponding to functional extremisation under con-sideration [1,2]. Variational calculation offers methods that are difficult to use in order to investigate the exis-tence and uniqueness of optimal control. The method of determining the field of extremals (sweep method), that analyzes the existence of conjugated points across the optimal transfer trajectory (a sufficient optimum condi-tion), proves to be a very efficient one in this context [3–5]. Through their resulting applications, time and energy minimization problems represent an important goal in system dynamics [6–11]. Recent results for con-trollable systems express the minimal energy through the controllability operator [11–13]. Also, stability con-ditions for systems whose energy tends to zero in infi-nite time are obtained in the literature [13,14]. By using linear operators in Hilbert spaces, in this study we shall analyze the existence and uniqueness of optimal control in transfer problems. The goal of this paper is to pro-pose new methods for studying the optimal control for exactly controllable linear system. The minimization of time and energy in transfer problem is considered. The

minimization of the energy can be seen as being a par-ticular case of the linear regulator problem in automat-ics. Using the adjoint system, a necessary and sufficient condition for exact controllability is established, with application to the optimization of a broad class of Mayer-type functionals.

2. Minimum Time Control

2.1. Existence

2.1.1. Problem Formulation

We consider the linear system

 

_A,B :

, )

( ), ( )

(t Bu t xt0 x0 H

x A t d

x

d _{   }

(1)

where H is a Hilbert space with inner product

  

 , and norm  .

H H A D

A: ( )  is an unbounded operator on H

which generates a strong semi group of operators on

H,

 

() _0 

 

tA _t0

t e

t

S .

H U

B:  is a bounded linear operator on another Hilbert space U , for example BL



U,H



.

H

u:[0,] is a square integrable function repre-senting the system control (1).

0 , ) ( ) ( 0 ) (

0 

_{e x}



_e  _{Bu s ds t}

t x t A s t A

t ₍₂₎

The control problem is the following one: Given t₀I ,x(t0),zX

U

and the constant ,

let us determine such that

0



M

u u M and



t xt x t ii z t x i    ) ( inf ) ( ) ( ) ( 1 1



(3) Here,X andUare Hilbert spaces.

Theorem 1. The optimal time control exists for the above formulated problem.

Proof

Let



be a decreasing monotonous sequence such

that . Also, let us consider the sequence



n

t t tn



)

1



u_n( , with u_n u(t_n)U. We have , ) ( ) ( ) ( ) ( ) ( 0

0 





 n t t n n n

n St xt S t Bu d

t

x    (4)

which gives





     n t t n n t t n n n n d u B t S d u B t S t x t S t x 1 1 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ₀       (5) ) (t

S being a continuous linear operator, it is, there-fore, bounded. It follows that we have:

) ( ) ( ) ( )

(t xt₀ S t₁ xt₀

S _n  (6)

0 ) ( ) ( 1  



n t t n n Bu d

t

S    (7) If we consider the set of all the admissible controls



uU u M





 (8)

then,  is a weakly compact closed convex set. As U is weakly compact, any sequence

  ) (u  _ u

possesses a weakly convergent subsequence to

an element . Thus, for and any

(the dual of ), we have

 

1

u u₁



 



 ui u  u u

ilim , 1, (9)

Since u₁, it follows that

M

u₁  (10) Also, we have

. ) ( , : U x t S U X B

n 



 

 (11)

So,

. )

(t x U

S

B  _n_   ₍₁₂₎ For every xX , let us evaluate the difference

x d u B t S x d u B t S F n t t t t n n , ) ( ) ( , ) ( ) ( 1 1 1 1 1





          (13) By writing , , 1 , ) ( 1 1

0 Bd k n

t t

S _k

t

t

k  



 L (14)

the Expression (13) becomes

x u x u x u x

un n

n  L  Ln  L1

L , 1 1, , 1, (15)

or . , , , , ₁ x u x u x u x u F n n n     _{  }  n 1 1 1 L L L L (16) Therefore,





















d x t S B x t S B u d x t S B u u x u x u u F t t n n t t n n n





                    1 0 1 0 ) ( ) ( ), ( ) ( ), ( ) ( , , 1 1 1

1 L1 Ln L1

(17)

By using the properties of the operator , one ob-tains ) (t S



( )



. ) ( ), ( ) ( ), ( ) ( 1 0 1 0 1 1 1 1        d x x t t S t S B u d x t S B u u F t t n n t t n





            (18)

Because the sequence un converges weakly to , the first term in (18) tends to zero for

n u

1

u n

(see(9)). On the other hand, is a strongly

continuous semigroup and, hence, from its boundedness, it follows that there exists a constant such that



 ) (t₁  S







x x t t S K

x x t t S t

S B

n

n

 

   



 



) (

) ( ) (

1

1 1 

(19)

So, it follows that the second term in (18) tends to

zero as . The sequence is weakly

convergent to if, for any ,

we have

 

n



x(t_n)



X

x X

z t

x(₁)  X X

, ), ( ), (

lim xt_n x xt₁ x

n  (20)

which gives

x d u B t S t x t S x z

t

t

, ) ( ) ( ) ( ) ( ,

1

0

1 1

0

1 











 (21)

From (21), one gets

z d u B t S t x t S

t

t

 





1







0

) ( ) ( ) ( )

(_{1 0 1 1} (22)

and, hence, u₁ is the optimal control.

An important result which is going to be used for proving the uniqueness belongs to A. Friedman:

Theorem 2 (bang-bang). Assuming that the set 

is convex in a neighborhood of the origin and is the control of optimal time in the problem formulated in (2.1.1), ([2, 10]) it follows that , for almost all

.

) (t u

 

) (t u ]

, [t₀ t₁ t

2.2. Uniqueness of Time-Optimal Control

2.2.1. Rotund Space

Let  be the unity sphere in the Banach space U and let  be its boundary [11].

The space is said to be rotund if the following equivalent conditions are satisfied:

U

a) if x₁x₂  x₁  x₂ , it follows that there ex-ists a scalar



0, such that x₂



x₁;

b) each convex subset has at least one ele-ment that satisfies

U

K

; ,

, u K z K

z

u   

c) for any bounded linear functional on there exists at least an element

f U





x such that

; ) (

,f f x f

x  

d) each x



is a point of extreme of . Examples of rotund spaces:

1) Hilbert spaces.

2) Spaces lp, _Lp, _{1 p1.}

3) If the Banach spaces are rotund,

then the product space U is rotund,

too.

n U U U₁, ₂,,

n U U    ₂  1

4) Uniform convex spaces are rotund, but the con-verse implication is not true.

2.2.2. Uniqueness

We assume that u₁ and u₂ are optimal, 2

, 1 ,  U i

ui . Therefore,

z d u B t S t x t S

t

t

 





  

1

0

) ( ) ( ) ( )

(_{1 0 1 1} (23)

z d u B t S t x t S

t

t

 





  

1

0

) ( ) ( ) ( )

(_{1 0 1 2} (24)

By adding Equations (23) and (24), one obtains



u u



d z B

t S

t x t S

t

t

 

 



   

1

0

) ( ) ( 2 1 ) (

) ( ) (

2 1 1

0 1

(25)

It follows that u₁(



) ,u₂(



), 12



u₁(



)u₂(



)



are optimal. By using Theorem 2, we have





. 1 2

1 , 1

) ( ) ( 2 1 ), ( ), (

2 1

2 1 2

1

  

   

u u u

u u u

u

k

   



Since the condition (i) is satisfied, is a rotund space and

U

2

1 u

u 



. Thus,



1



2 2

2

1u   u 

u  (26)

and, therefore

2 1

1u u



 (27) which ends the proof of the uniqueness.

3. Minimum Energy Problem

3.1. Problem Formulation

Let (see(1)) be a controllable system with finite dimensional state space

B A,



X [6,10,12, 14]. LetI [0,t₁],x(0)aX

) , ( 2 _{I U}

L

, be given. Let be

the Hilbert space . The minimum norm control

X

problem can be formulated as follows: determine such that for some ,

U t u()

t x(₁

I t₁

1) )b,

2) u is minimized on [0,t₁], where  represents the norm on _L2(_I,_U).

Let us fix t10. Consider the linear operator

1 2₍₀

:L H

n 

L ,t1;U)



1 s)B

t

1

aL_t

H

(28) defined by





1

0

) ( (

:

t

nu S u s ds

L (29)

We have

. )

( )

(t₁ S t₁ u b

x   (30)

Theorem 3. Let be a linear mapping between a Hilbert space and a finite dimensional space

U

t:  L

U

H[11]. Then, there exists a finite dimensional

space such that the restriction of

to

U



M M

t

L Lt

M is an injective mapping. Proof

Let , be a basis for the range of

in

 

ei i1,,n Lt

H . Given any uU,L_tu can be written as

,

i ie



1





 n

i tu

L (31)

where each



_i can be expressed by

.

,u f_iU U

f_i

i 

 (32)

Then,

.

1





 n

i tu

L f_i,u e_i (33)

i

f are linearly independent and generate a

dimen-sional subspace .

n

U M 

From the properties of Hilbert spaces (the projection theorem), it follows that UMM.

Let uM. Then, f_i,u 



_i 0. Thus,



u U _tu 0



ker _t.

M  L   L ₍₃₄₎

Let ukerL_t. We get

. , 0

1







 n

i

i i

tu f u e

L (35)

Since e_i are independent, f_i,u 0,i1,,n

and, so, M kerL_t.L_t maps M bijectively to the range of and hence is an injective mapping into

t

L M

t

L

H. Let

. 0 , ,

, _{1 2 2}

2

1   

_{u u u M u M} _u

u (36)

Because Ltu2 0,

H x u u M_t

t L 1 

L (37)

Since u₁,u₂ 0, we have

2 2 2 1 2 1 2 1 2

,u u u u

u u

u      (38)

Then, a unique minimum norm exists and

is the minimum norm element satisfying

 

LM_t 1x x

u M

t

L



1

x u

L .

Remark 1. The unique solution of the equa-tion

) (t u x

u

L , with minimum norm control, is the

pro-jection of u(t) onto the closed subspace



f1,,fn



M  .

Hence, from (30), there exists a control transferring to in time if and only if

) ; , 0 ( )

( _L2 _{t1 U}

u  a b t₁



bS(t1)a



ImLt1. The control which

achieves this and minimizes the functional





1

1

0 2

) (

t

t u s ds

E , called the energy functional, is



b S t a u _t1 (₁)

1 

L



(39) Define the linear operator

0 , ) ( ) (

0

 



_{S r BB}_{S r dr t}

Q

t

t (40)

We have the following results (see [10–12]):

Proposition 1.

1) The function , is the unique solution of the Equation [1,13]

0 ,t Q_t

I Q A D x

x B x x Q x x Q t d

d

t t

 

 

  

0 2

, ) (

, 2 ,

(41)

where





. ) ( ,

, , )

(

A D x x C

y x A R C H y A D

H

H

 

 

 

 



2) If A generates a stable semigroup, then

Q Q_t

tlim  (43)

exists and is the unique solution of the equation

, ) ( ,

0 ,

2 QAx x  Bx 2  xD A (44)

The proof of Proposition 1 is given in [10,11]. The following theorem gives general results for the functionals , the minimal energy for

transfer-ring to in time , and , where

,t (see [12]). ) , (ab E

b

0



a

H



1

t E(0,b)

b a, ₁

Theorem 4.

1) For arbitrary t₁0 and a,bH

 



( )



. )

,

(a b Q12 1 S t₁ a b 2

E_  _   (45)

2) If S(t) is stable and the system



_A,B



is null controllable in time t₀0, then

 

Q b b H b

E_(0, ) 12 1 2,  (46)

3) Moreover, there exists C_t0 0 such that

 

 

1 0

2 1 2 1

2 1 2 1

, ,

) , 0 (

0

1

t t H b b Q C

b E b Q

t

t

  

 

 

(47)

4. Numerical Methods for Minimal Norm

Control

4.1. Presentation of the Numerical Methods

Let _A,B be a dynamic system with _URm,

n

R X ,

1

R

I  [9,10,13]. Given x00, , , , determine

such that

0

t t1 b

U



)

t u(

1) x(t1)b,

2) up is minimized for p[1,)

Now,





 

 

 

1

0 1

0

) ( ) (

) ( ) (

1 1

t

t t

t

d u t

d u B t S b e

  

  

(48)

In order to make Equation (48) true, must be chosen on the interval . Consider the i-th compo-nent of the vector :

) (t u

] , [t₀ t₁

e

i

e

) ( , )

( ) (

1

0

1 i i

t

t i

i t u d u f

e 



        (49)

where _i is the row of the matrix and is the unique functional corresponding to the inner product (Rieszrepresentation theorem).

 f

Let _i be an arbitrary scalar. Then,

, ) ( )

( _{i i i}

i i

ie



f



f







  ₍₅₀₎

Since Rn is a Hilbert space, the inner product

e f

e

n

i

i i n

i i

i ( ) ,

1 1

    







 

(51)

 being a vector with arbitrary components 1,,n.

If Equation (51) is true for at least different linearly independent,

n

n i

i, 1,,

 then

Equation (48) is also true. We have

. ,

1 1

u f

n

i i i n

i i

i





 

     

  

  

 (52)

By Hölder’s inequality,

. 1 1 1

, ,

1 1

   

 

 







 

q p

u u

q n

i i i p n

i i

i 



(53)

From Equations (51) and (53),

. ,

1

1 1

1 1

q n

i i i

q n

i i i n

i i i

q n

i i i n

i i i

p

e

e f

u













 



  _{ }

   

  



  

 

 

  

(54)

Every control driving the system to the point must satisfy Equation (54), while for the optimal (minimum norm) control, Equation (54) must be satisfied with equality (Theorem 2). Numerically, one must search for

vectors

b

Equa-tion (54) takes its maximum.

4.2. A Simple Example

We consider a single output linear dynamic system , where

B A,



1 1

1_{, X R , I R}

R

U   

Because t₁I fixed, then S(t1)B()



  

1

0

. ) ( ) (

t

t

d u b

e    (55)

Therefore,





  

 

1

0 1

0

) ( ) ( )

( ) (

t

t t

t

d u d

u b

e       (56)

and from Hölder’s inequality,

, )

( ) (

1

0

u d

u

t

t

  



   (57)

where we have assumed that . The

mini-mum norm control, if exists, will satisfy

) ( ) (t L2 I



.



 e

u (58)

From (56) and from (57), we have

] , [ , )) ( ( sign )) ( (

sign u t   t t t₀ t₁ (59) and, respectively,

, ] , [

, ) ( ) (

1 0t

t t

t u h

t q p

 



 ₍₆₀₎

h arbitrary constant.

The control u satisfies the relation

p q p

q

p _{t k t}

h t

u() _ 1 _() _{ }() ₍₆₁₎

or

. ) ( (u(t)) sign )

(t k t q p

u   (62)

From condition (59),

. ) ( ) ) ( ( sign )

(t k t t q p

u    (63)

By substituting Equation (63) into Equation (55), the constant k can be determined.

4.2.1. The Particular Case pq2

In this case, the Equation (55) represents the inner product in L2(t₀,t₁;U),

u

b , (64) The relation (63) becomes

. ) ( ) ) ( ( sign )

(t k t t

u    (65)

Then,

. )

( 2

2

1

0

  





k d k b

t

t



 (66)

Thus,

2



 b

k (67)

and the optimal control is given by

. ) ( )

( ₂

  b t

t

u (68)

5. Exact Controllability

5.1. Adjoint System

Let S(t),t[0,t₁], be the fundamental solution of an homogeneous system associated to the linear control system _A,B(see(1)) [4,5,13] .

Thus, we have

] , 0 [ , ) 0 ( ), ( ) (

1

t t I S t S A t d

t S

d _{  }

(69)

From



() 1( )



 0.

t d

I d t S t S t d

d

(70)

one obtains

. ) ( )

( 1

1 _{t S t A}

S t d

d  _  ₍₇₁₎

which implies that



S1(t)



A



S1(t)



. (72) Since



_S1(_t)

 

_{ S}(_t)



1. ₍₇₃₎

the system becomes



_S (_t) 1



_A

 

_S (_t) 1,_t [0,_t1]

t d

d _{    }  _

(74)

It follows that

 

S(t)1,t[0,t₁]_{, is the fundamental}

solution for the adjoint System (69).

We consider the class of linear control systems in vecto-rial form [1,3,4,8,13,15]

0

) 0 (

, x x

C u B x A t d

x

d _{    (75)}

Let be any internal point of the closed bounded convex control domain .

0

u

U

This domain contains the space of variables . We take

m

E

) , ,

(u₁ u_m

u 



u u

u   (76) By this transformation, system (75) becomes



Bu C.

u B x A t d

x

d _{  } _



₍₇₇₎

Thus, one transfers the origin of the coordinates of space in . At the same time, the origin of the coordinates of space is an internal point inside the domain . Let us denote by the solution of sys-tem (75) which corresponds to the control .

m

E

U



u

m

E

) (t x

0



u

This control is admissible, as the origin of the coordi-nates belongs to the domain and satisfies the initial

condition . As a result,

U

0

) 0

( x

x 

. ) ( )

( _{Ax t C}

t d

t x

d  _  _{ (78)}

Let , be any control and be the

trajectory corresponding to system (75).

1 1(t),0 t t

u   x₁(t)

We have

. ) 0 ( , ) ( ) ( ) (

0 1 1

1

1 _{Ax t Bu t C x x}

t d

t x

d _{   }

(79)

We take

) ( ) ( )

(t x₁ t x t

x    (80) Then, from (78) and (79), one obtains

. ) ( ) ( ) (

1 t

u B t x A t d

t x

d _{ }

Hence, for uu₁(t), the system (75) becomes

, 0

) 0 ( :

,B n

A _x x R

u B x A x

 

 

  

  (81)

Thus, the linear control system belongs to the class defined by (81).

B A,



5.3. Optimal Control Problems

For a given , let us determine the optimal control t₁ u~

that extremises the functional of final values





_





 n

i i ix t

c t

x F J

1 1

1) ( )

( (82)

and satisfies the differential constraints represented by the system (81).

The adjoint variable yR satisfies equation

x H y

   

 (83)

where H is the Hamiltonian associated to the optimal problem

x y

H , (84) Since the final conditions are free and the final time has been indicated from the condition of transver-sality, one obtains

) (t₁ x

1

) (t₁ yt

y  .

It follows that the adjunct system becomes

   

  

 

1

) ( :

1 ,

t B

A _{yt y}

y A y

(85)

with solution

. ,

) ( )

(t S t₁ t y₁ y₁ H

y    _{t t}  ₍₈₆₎

Proposition 2.

For the class of optimum problems under considera-tion, the following identity holds true:







1

1 1

0

, ,

t

U H

t

t y u B y dt

x (87)

Proof

Assuming that uC1



 

0,t₁ ,U



and it follows that

) ( ) (t₁ D A y

 



t U



C y_ 1 0,₁ ,

x, .

Integrating by parts, since x(0)0 , y(t1) yt₁ ,

, , we have

H U

B:  B:HU

















 



 



 

 

 

 

1

1 1

1

0 1

1 1

1

0

0 0

0

0 0

0

,

, ,

, ,

, ,

, 0

t

U t

H t

H t

U t

H t

H t

H

t d y B u

t d y A x t d y A x

y x t d y B u

t d y x A t d y x

t d y u B x A x

t 



One obtains

0 ,

,

1

1 1

0

 





t

U H

t

t y u B y dt

x (89)

The identity has been demonstrated.

This result can be extended for arbitrary

and .

) ; , 0

( ₁

2 _{t U}

L

u y(t₁)H

An important result referring to the exact controllabil-ity of the linear system (81) is stated in

Theorem 5. The system is exactly controllable if only if the following condition is satisfied

B A,



H y

y c t d y t S

B _U

t

U

 





 

0

2 0 0

2

0 ,

) (

1

(90)

Proof

“” We assume that is exactly controllable.

Let and

B A,



t y

) ; , 0

( ₁

2 _{t U}

L

u (₁)H. We consider that the application

) ( :

1 U xT

Lt  (91)

is well defined.

Let be the inverse of . From

Theorem 3 it follows that there exists a finite dimen-sional subspace

) ; , 0 (

: _HL2 _t1U

 L_t1

H

M  , such that the

restriction

1

kerL_t

M

   

 _{ }

1 ker 1 1

t L

t M

t L

L (92)

is injective.

Since it

follows that transfers 0 in for the system .

) (₁

1u xt

L_t   1



(₁)

 

(₁)

1 xt xt

L

u _t 

) (t₁

x _A,B



We choose y(t₁)Handx(t₁)y(t₁),u



x(t₁)



. It follows that



_ 

 

1

1 1

1 1

0 2

), ( ,

t

U t

t t H

t y y x B y dt

y (93)

For , B, using Hölders’s

ine-quality, one obtains

2

) (xt₁ uL

 _B_yL2





 

 

2 1

0 2 2

1

0

2

0 0

1 1

1

1

1 1

1

) (

) ( )

(









  

 

 

 

t t

t

t

t t

t

t d y B t

d x

t d y B x t

d y B x

(94)

From (93) and (94), for x_t1 y_t1, we have

2 1

0 2

2 1

0 2 2

1

1

1

1

1 ( )

    

   

     

    





 

t

U H

t

t

U t

H t

t d y B y

t d y B y

y

(95)

or

2 2 0

2

1 1

1

H t t

Udt y

y B

 



 ₍₉₆₎

By changing tt1t

( S

becomes and

is transformed into .

0

y )

1t

1

t

y

) (t

S t

Therefore, we get one equivalent relation of controlla-ble system in which y_t1 substitutes y₀.

2 2

2

0

2 0 1 0

2 0

1 1

1 1

1

)

(

)

(

H t H t

t

U t

U

y

c

y

t

d

y

t

t

S

B

t

d

y

t

S

B















   

(97)

The direct implication has been demonstrated. “” We assume that condition (90) is fulfilled.

Then

2

0 2

1 1

H t t

Udt c y

y

B 



 ₍₉₈₎

For any yt1H we take the set



u(t) B y(t)







(y(t) is solution of _A,B ).

We consider the solution for which

cor-responds to the above mentioned control . Let us define the bounded operator

) (t

x A,B

u(t)



B y



H L

t x H

y_t    _t  

_{: ( )}  _{( )}

1

1 1 (99)

Since u(t)By, the identity (89) becomes

2

0

2

1 1

1

1, () t _H

t

U H

t

t y B yt dt c y

y  





 (100)

Therefore, there exists a constant for which (100) is satisfied. It follows that

0



c

H

y_t1,y_t1 is posi-tively defined.

This resumes the conclusion that the system is controllable.

B A,

Thus, since is inversable, for any , state

is such that there exists a control which transfers 0 in . The theorem has been demonstrated.

 x_t1H

) ( ₁

1

1 t

t x

y 

y B t

u()  y_t1

6. Conclusions

The main contribution in this paper consists in proving existence and uniqueness results for the optimal control in time and energy minimization problem for controlla-ble linear systems.

A necessary and sufficient condition of exact control-lability in optimal transfer problem is obtained. Also, a numerical method for evaluating the minimal energy is presented.

In Subsection 5.2 the nonhomogeneous linear control systems are transformed into linear homogeneous ones, with a null initial condition. For the control of such a class of systems, one needs to consider the adjoint sys-tem corresponding to the associated optimal transfer problem (Subsections 5.1 and 5.3).

The above theory can be used for solving various problems in spatial dynamics (rendez-vousspatial, satel-lite dynamics, space pursuing), [3,4,8,9]. Also, this the-ory can be successful applied to automatics, robotics and artificial intelligence problems [16–18] modelled by lin-ear control systems.

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[12] G. Da Prato, A. J. Pritchard, and J. Zabczyk, “On mini-mum energy problems,” SIAM J. Control and Optimiza-tion, Vol. 29, pp. 209–221, 1991.

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