Solution of the Synthesis Problem in Hilbert Spaces

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1 Solution o the Synthesis Problem in Hilbert Spaces Valery I. Korobov, Grigory M. Sklyar, Vasily A. Skoryk Kharkov National University 4, sqr. Svoboda 677 Kharkov, Ukraine Szczecin University 5, str. Wielkopolska 745 Szczecin, Poland {vkorobov, sklyar, skoryk}@univer.kharkov.ua {korobow, sklar}@sus.univ.szczecin.pl Keywords: synthesis problem, positional control, controllability unctional method. Abstract The problem o synthesis o restricted positional controls or a linear equation with bounded operators in Hilbert spaces is considered. It is shown that any non-increasing non-negative unction having suicient number o points o decrease generates a control solving the local synthesis problem. I, in addition, the operator is skew-adjoint and the unction has a monotone non-decreasing derivative then the control solves the global synthesis problem. The local synthesis problem or a nonlinear equation is solved with respect to the irst approimation. Introduction Consider the problem o synthesis o restricted positional controls or the controllable process described by the ollowing equation d = A + Bu, X, u Ω U, int Ω, where X, U are Hilbert spaces, the operator A with the domain DA generates the strongly continuous group o operators { e At}, < t < +, and B is a bounded operator acting rom U to X, i.e. B [U, X]. The problem consists in constructing o a restricted control u in the eed-back orm positional control u = u Ω such that or any belonging to a certain neighborhood Q o the origin the solution t o the Cauchy problem d = A + Bu, = satisies the condition t = or T <. The t T synthesis problem is global i Q = X and local otherwise. The synthesis problem or a inite time is contiguous to the stabilization problem. However, in contrast to the problem o stabilization, the synthesis problem cannot be solved with the help o a linear eed-back control u = P. For eample, or any P [X, U] a solution t = e A+BP t o the equation d = A + BP satisies the inequality t ce λt, λ >, and hence the control u=p does not solve the synthesis problem. The eception is the case ln e At when ω A= =. t t Without loss o generality we assume that equation is eactly -controllable at a ree time. We investigate the synthesis problem by using the controllability unctional method which is similar to the Lyapunov unctional method or the stability problem. For the irst time this approach had been proposed by V.I. Korobov in [] or the inite-dimensional controllable systems and developed or the ininite-dimensional case in [, 3] in cooperation with G.M. Sklyar. The initial point or the present work is the ollowing Theorem []. Consider the controllable process described by the equation d = A + ϕ, u, X, u Ω U, where X and U are Banach spaces, the operator A generates the strongly continuous group o operators {e At }, < t <, the unction ϕ, u is continuous in X U and in any domain < ρ + u ρ satisies the Lipschitz condition ϕ, u ϕ, u L ρ, ρ + u u. Let in any closed domain Q, intq, there eists a continuous unctional satisying the ollowing conditions: > at and = ; i then, i.e. or any ε > there eists δ > such that rom < δ it ollows that < ε;

2 3 there eists C > such that the set Q = { : C} is bounded and Q intq ; 4 is continuously dierentiable in the Frechét sense in the domain Q ecept the point = ; 5 there eists a control u, X \ {}, satisying the restriction u Ω, Q \ {}, and such that in any Kρ, ρ = { : < ρ ρ < } the Lipschitz condition u u L ρ, ρ is satisied; 6 the unction Φ, t = t is continuously dierentiable with respect to t where t is the generalized solution o the Cauchy problem d = A + ϕ, u, = Q \ {}, and the unctional Ψ = Φ t, is continuous at in Q \ {}; 7 there eist α > and β > such that or DA Q \ {} the ollowing inequality holds Ψ =, A + ϕ, u β α. 3 Then i i α < then the generalized solution t o equation with the control u or any Q \ {} satisies the conditions t=, where < T t T α β α, and t Q or all t [, T ; ii i α = then t as t and t Q or all t. The main result o the present paper is the description o the set o restricted controls solving the synthesis problem or equation in the case when A is a bounded operator. Construction o controls Let s be an arbitrary non-increasing non-negative unction on the semiais [, + such that ln s = > s s in particular, the constant may be equal to +, or eample, when is a inite { unction. ω Denote λ = ma, A }. For any λ > λ let us introduce the operator N λ by the ormula N λ = λte At BB e A t, X. 4 From the eact controllability at a ree time o equation it ollows that the operator N λ at λ > λ is positively deined and consequently is invertible. Following the approach proposed in [] we consider the equation with respect to a = N,, a >. 5 Let us show that equation 5 has the unique positive solution = or any rom a certain closed domain Q, intq. For a ied let us consider the unction G = a N,, a >, 6 deined or all, λ < < when λ =. Then G =a + Ñ where Ñ λ = N, 7 te At BB e A t dλt. 8 Choose < λ such that or < < the operator Ñ eists and is positively deined. We have G a > or <. Let us show that G as. Indeed, since N as, we have N, N + as. From here we have that G as. Now choose such a number R > that N, < a or R and put Q = { : R }, or eample, it is suicient to choose R < a / N. Then G > or Q \ {}. Thereore, or all Q \ {} the equation 5 has the unique positive solution. The both parts o equation 5 are continuously dierentiable by and and G = or =. Then by the implicit operator theorem [5] = deined by 5 eists and is continuously dierentiable in Q \ {} and its Frechét derivative has the orm = Thereore a + Ñ N N a N. 9. Complete the deinition o putting =. Note that the unctional satisies conditions and 4 o Theorem. Our nearest goal is to prove the continuity o at = and the act that satisies conditions and 3.

3 Note that or = the unction N, monotonically decreases with respect to. Then, ε, or ε. N N For an arbitrary number ε > let us choose a number δ > satisying δ < a ε / N and consider any such that < δ. Assuming ε we get a ε a N ε < a ε what gives the contradiction. Thereore, < ε and the continuity o the unctional at = is proved. Now prove that satisies the condition o Theorem. As ar as a = we have N ε, a N N, or any Q \ {}. Since N as then rom the last inequality it ollows that as. It is easy to see that or any C < R /a N condition 3 o Theorem holds. Deine the control u or Q \ {} by the ormula u = B N. The control u is a continuous and dierentiable unction in Q\{}. Let us prove that u satisies the Lipschitz condition see condition 5 o Theorem. To this end, we continue u to the set Q \ {}, i.e. consider the unction v = B N, Q \ {}. We now show that the unction v satisies the Lipschitz condition in each Kρ, ρ = { : ρ ρ }, < ρ < ρ < R. Considering the Frechét derivative v we have v = B N N Ñ N + B,, where the operator Ñ λ is deined by 8 and the Frechét derivative o the unctional is given by 9. On the other hand rom the monotone decrease o the unction and rom the inequalities N a N M ρ, a N M, we obtain that in Kρ, ρ the unctional satisies the inequalities < M = ρ a M < < M = M ρ a. 3 Taking into account and 3 we obtain rom that the Frechét derivative v in Kρ, ρ is bounded and consequently the unction v satisies the Lipschitz condition in Kρ, ρ. From here we make a conclusion that the control u rom satisies the Lipschitz condition with a constant Lρ, ρ in any domain Q Kρ, ρ Lρ, ρ + as ρ. Consider the unction Φ, t = t, where t is the solution o equation with the control u corresponding to the initial state Q \ {}. Introduce the unctional Ψ = Φ t, and Ψ =, A + Bu, DA Q \ {}. To prove that it satisies condition 6 o Theorem we use the ollowing Lemma. The operator N λ rom 4 maps the set DA to the set DA and or any DA the ollowing equality holds AN λ + N λa = N λ + BB, 4 where the operator N λ is given by the ormula N λ = Proo. Consider the it Since then e At BB e A t dλt. 5 e A E N λ, X. e A N λ = N λe A + λt+e At BB e A t e A + [λt+λt]e At BB e A t e A, e A E N λ = N λ ea E + [λt+ λt]e At BB e A t e A + λt + e At BB e A t e A. 6

4 Since λt + e At BB e A t e A BB as, [λt+λt]e At BB e A t e A N λ as and the operator N λ is invertible, then the it e A E N λ eists i and only i there eists the ollowing it e A E. According to [4] that means that N λ DA i and only i DA = DA. At the same time ea EN λ = AN λ, Hence, 6 yields ea E = A. AN λ = N λa N λ + BB, DA. The proo o the lemma is completed. From Lemma and 4 it ollows that the operator N maps DA to DA and or any DA satisies the equality N A + A N = N N = N +N BB N +. 7 Besides, rom 6 on the basis o 7 we have a, or Q \ {}. N Consequently, by Theorem [3] the unctional Ψ is deined and continuous or Q \ {}. The unctional Ψ in DA Q \ {} has the orm [[ Ψ = N A + A N ], BB N /[ N N ] /, + Ñ ], =, 8 where the operator Ñ λ is deined by ormula 8. From 8 and taking into account equality 7 we obtain that [ Ψ = N N ] / + Ñ /[ N N, + where the operator N λ is deined by 5. 3 Local synthesis ], =, 9 Further we suppose that the operator A is bounded. Our goal is to show that the control u is bounded in a certain neighborhood o the origin and solves the synthesis problem or equation. Since equation is eactly controllable then due to the V.I. Korobov and R. Rabah criterion [6] there eists a certain integer number m such that Span BU, ABU,..., A m BU = X. Let us introduce the Hilbert space U m+ = U... U, m =,,..., }{{} m+ with elements o the orm u, u,..., u m, u k U, k =,,..., m, with the norm u, u,..., u m = m k= u k and the scalar product given by the ollowing ormula u, u,..., u m, v, v,..., v m = u k, v k. k= Let S m be a bounded operator acting rom the space U m+ to the space X by the rule S m u, u,..., u m = A k Bu k. k= Then relation yields that the image o the operator S m coincides with the space X. Since the space U m+ is Hilbert then the operator S m determines the isomorphism o the subspace KerS m and the space X. Hence, the operator S m has a right inverse operator, i.e. there eists the operator P [X, U m+ ] such that S m P = E X E X is the identical operator in X. From this we have that there eist operators P k [X, U], k =,,..., m, such that A k BP k = E X. Hence, k= Pk B A k = E X. k=

5 From we get that or any integer l the ollowing equality is true. Then B A m+l = B e A t = l= k= B A m+l Pk B A k k= t k k! B A k = t k k= k! E U + k t m+l m + l! B A m+l Pk B A k, where E U is the identical operator in U. Further we need the ollowing lemma which is a generalization o Lemma rom [7]. Lemma. Let X and U be Hilbert spaces, A [X, X] and B [U, X] be arbitrary operators and let µs, s [, be a monotone unction having at least m points o increase and such that the condition s m+ e s A dµs < at < µ is true the constant m is deined by the condition. Then there eist a constant c µ > and certain positive constants l µ and L µ < lµ < Lµ such that or < c µ the ollowing inequalities are true l µ m k= k B A k L µ m k= B e A t dµ t k B A k, 3 where X. Proo o Lemma. From equality we obtain that t B e A t k = k! E U + r k t B A k, k= where r k t [U, U], k =,,..., m, are the operators o the orm and r k t = k l= t m+l m + l! B A m+l P k 4 r k t R k t m+ e t A, R k >, t [, +. 5 Actually, rom equality 4 we have r k t t m+ l= t l l! B A m+l P k = = B A m+ P k t m+ l= t l l! A l R k t m+ e t A, k =,,..., m. Thereore we have = = e At BB e A t dµ t, = B e A τ, B e A τ dµτ = m τ k k= k! E U + r kτ k z k, τ k k! E U + r kτ k z k dµτ = k= where D m = = D m F µ,m D m z, z + G m z, z, 6 m δkj k! E U k,j= δ kj is the Kronecker sym- m bol, F µ,m = τ k+j dµτe U, z U m+ is k,j= the vector with the components z k = k B A k U, k =,..., m, G m = g k,j m k,j=, g k,j = τ k rj τ k! j+ + τ j r k τ j! k+ + + r kτrj τ k+j+ dµτ, k, j =,..., m. Then on the basis o 5 we have g kj τ k+m+ R j mj + τ j+m+ R k mk + k! j! + R k R j τ m+ mkj+ e τ A e τ A dµτ, and hence the operators g kj, k, j =,..., m, are bounded or < µ. Since the matri dµτ... τ m dµτ [F µ,m ] = τ m dµτ... τ m dµτ is a moment matri and is positively deined due to our assumptions see, [8, 9] then the operator F µ,m is positively deined. This act and the boundedness o the operator

6 G m yield rom 6 that there eists a constant c µ such that or any < < c µ t e At BB e A t dµ, λ min G m z l µ z, where λ min is the minimal eigenvalue o D m F µ,m D m and l µ >. Similarly, we have e At BB e A t dµ t, λ ma + G m z L µ z, where λ ma is the maimal eigenvalue o D m F µ,m D m. Lemma is proved. Denote by F m A the class o all non-increasing nonnegative unctions on the semiais [, + having at least m points o decrease and such that the ollowing conditions hold: lns = >, s + s s m+3 e s A ds < or <, where the number m is deined by condition. Note that the second condition implies the irst condition in the case when the operator A is not trivial. The ollowing theorem gives the solution o the local synthesis problem. Theorem. Consider equation, where X, U are Hilbert spaces, A [X, X], B [U, X]. Let the condition be satisied and let F m A. Then there eists the constant c > such that Q = { : c } intq and or any Q \ {} equation with the control and the initial condition = has the unique solution t deined on the semiinterval [, T. The solution t satisies the condition t = and t Q or t [, T, where t T T β, β >. Besides, or any number d > the coeicient a can be chosen such that the control u satisies the restriction u d or Q \ {}. Proo. Deine the constants c µi using Lemma or µs = µ i s, i =,, 3, o the orm µ s = s, µ 3 s = s τdτ. s τdτ, µ s = Put c = min{c, c µ, c µ, c µ3 }, where the constant C has been deined in Section. Consider the domain Q = { : c }. Since Q Q then Q intq. Let us make use o Theorem. To do this, it remains to complete the proo o condition 5 and to prove condition 7. At irst let us prove estimate 3 or the unctional Ψ o the orm 9 in the domain Q \ {}. Denote ψ = N. Then rom 9 or Q \ {} we have N ψ, ψ Ψ = N, 7 ψ, ψ + Ñ ψ, ψ where = and the operators N, N and Ñ have the ollowing representation N λ ψ = N λ ψ = Ñ λ ψ = e At BB e A t ψdµ λt, 8 e At BB e A t ψdµ λt, 9 e At BB e A t ψdµ 3 λt. 3 Then rom Lemma and 7 we obtain that there eist positive constants l µ, L µ, L µ3 such that the inequality l µ Ψ β, Q \ {}, 3 L µ + L µ3 is valid, i.e. estimate 3 is proved with α = and β = l µ /L µ +L µ3. Let us establish the boundedness o the control u or Q \ {}. From and 5 we have = u = B ψ = 4 = B ψ a N ψ, ψ a B ψ, N ψ, ψ = where the operator Nλ is deined by 8. Then rom inequality 3 or µ = µ we have u l µ a. Choosing the coeicient a satisying the condition we obtain that < a a = l µ d, 3 u d or Q \ {} 33 what proves condition 5. Then due to Theorem we have that the time o movement T rom the point to the origin satisies the inequality T L µ +L µ3 l µ, t Q or t [, T and t T t =. That means that the control u solves the local synthesis control problem in the domain Q.

7 4 Global synthesis In this section we assume the operator A rom equation to be skew-adjoint, i.e. A = A. It is well known that any skew-adjoint operator A generates a contractive semigroup o operators { e At}, t. By the Stone theorem [], this contractive semigroup can be etended to a unitary group { e At}, < t < + []. In this case the operators N λ, N λ, Ñ λ are deined on the positive semiais λ >. Equation 5 deines the unctional or all X \ {} because the unction G rom 6 monotonically increases on, + and G =, + + G = +. Then the control and the unctional Ψ are also deined or all X \ {}. In terms o Section that means that Q = Q = X. In this section we give the conditions on the unction and the constant a under which the control solves the global problem o synthesis o a restricted positional control. From the eact controllability o equation it ollows that there eists T > such that or any T > T the operator N T deined by the ormula N T = T e At BB e A t is a positive deined operator [], i.e. N T, δ, δ >. 34 Denote by F ma the subclass o all unctions rom F m A which satisy the ollowing conditions: i the unction has a monotone non-decreasing derivative on the semiais [, + ; ii there eists T > T such that T /c. That means that in the case when is a inite unction its support includes the point T /c the constant c has been chosen in the proo o Theorem. Statement. For any unction F ma and c the inequality k= kt γ, γ >, 35 is true. Since τ is a monotone non-increasing nonnegative unction then or c we have: T < T T T c = T T c τdτ = T τdτ T t. From here on the basis o the inequality T we obtain that t T k= kt T k= T kt c γ. In the proo o Theorem we have shown that the unctional Ψ satisies the inequality 3 in the domain { : < c } = Q \ {}. Let = c. Now we establish inequality 3 or certain α > and β > in a global sense, i.e. or all X \ {}. We have Ñ ψ, ψ = t B e A t ψ d t B ψ t d t = I B ψ, 36 N ψ, ψ = B ψ t B e A t ψ τdτ = I B ψ, 37 where I = τdτ < is a positive constant. Further we obtain lower estimates or N ψ, ψ and N ψ, ψ. Taking into account conditions on the unction and its derivative on the basis o inequalities 34, 35 we have that N ψ, ψ = = k= δ T k= k= k+t kt B e A t ψ d t = B e A τ e A kt ψ τ+kt dτ k+t N T e A kt ψ, e A kt ψ k= N ψ, ψ = k+t e A kt ψ δγ ψ, 38 k= k+t kt t B e A t ψ =

8 T = k= k= δ k= Since obtain T B e A τ e A kt ψ τ+kt dτ k+t N T e A kt ψ, e A kt ψ k+t ψ = δ ψ t T k= δ N ψ, ψ ψ T δ T ψ T k= kt. 39 kt then rom 39 we T τ dτ I c t δ T ψ, 4 where I c = τdτ. Then on the basis o 36, 37, 38 T c or Ψ we have the estimate Ψ δγ I B β, { : c }, 4 β >. Taking into account 3 and 4 we inally obtain that Ψ satisies the estimate Ψ min{β, β }, X \ {}. Further we prove that the control u is bounded or { : c }. Using 5 and 4 we have: u = 4 B ψ = = B ψ a N ψ, ψ T B ψ I c δ ψ a = T B δi c a. 4 Choosing the coeicient a according to the condition < a a = rom 4 we obtain that δi c d T B, u d or { : c }. 43 Taking into account 33 and 43 we have that or the coeicient a such that < a ã = min{a, a }, the control u satisies the restriction u d or X \ {}. Thereore, we have proved the ollowing theorem which gives the solution o the global synthesis problem. Theorem 3. Let in addition to the conditions o Theorem the operator A be skew-adjoint operator and F ma. Then the control u deined by solves the global synthesis problem and or any number d > the coeicient a can be chosen such that u satisies the restriction u d or X \ {}. Remark. The statements o Theorems and 3 in the case when the unctional at is deined by the equation a ν = N also true. Remark. In the case when s =,, a >, ν, are { s q, s,, s >, we have Ψ = and consequently the time T o motion rom to equals in the statements o Theorems and 3. Remark 3. To deine constructively the solution t corresponding to the control u it is suicient to solve the ollowing Cauchy problem d = A BB N d = N N /[ N, + + Ñ N, ], t t= =, t t= =, / 44 where is the solution o equation 5 at =. Thereore, to ind t we need to solve equation 5 only once. 5 Synthesis with respect to the irst approimation Consider the controllable process described by the nonlinear equation d = ξ, u, X, u Ω U, intω, where X, U are Hilbert spaces, ξ, u is a twice continuously Frechét-dierentiable unction on X U and ξ, =. This equation can be represented as d = A + Bu + g, u, 45 where A = ξ,, B = ξ u,, and g, u is a continuous unction on X U. In this section we show that the positional control which solves the local synthesis problem or the linear equation

9 d = A + Bu also solves the local synthesis problem or the nonlinear equation 45. Theorem 4. Consider the controllable process 45, where X, U are Hilbert spaces, A [X, X], B [U, X]. Suppouse that equation is eactly controllable and the unction g, u in each domain {, u : < ρ ρ, u d} satisies the Lipschitz condition g, u g, u L g ρ, ρ + u u and the inequality g, u c s + c s u s 3 + c 3 u s4, 46 c, c, c 3, s >, s 4 >, s + s 3 >. Let F m A. Let the unctional at = be deined by the equation a ν = N,, a >, 47 { } where ν +m ma s + s, s +s 3+ s +s 3, s 4 + s 4 ; =. Then there eists the constant ĉ such that the control u o the orm solves the local synthesis problem or the equation 45 in the domain Q = { : ĉ}. Besides, or any number d > the coeicient a can be chosen such that this control satisies the restriction u d or Q \{}. Proo. Put ĉ = min{c, }, where c was chosen in Theorem. Then Q = { : ĉ} intq. As it has been noted in Remark the control u o the orm solves the local synthesis problem or the linear equation also in the case when the unctional is deined by equation 47. To apply Theorem or to nonlinear equation it remains to obtain the inequality o the orm 3 or Ψ o the orm Ψ =, A + Bu + g, u. In the domain Q \ {} we have /[ ν N [ Ψ = N ψ, ψ ψ, g N ψ, B ψ ] / ψ, ψ + Ñ ψ, ψ ], =, 48 and N λ, N λ, Ñ λ are where ψ = N rom 8, 9, 3. Applying Lemma we have a ν = N µ ψ, ψ l µ m k= k+ B A k ψ, 49 or Q \ {}. The eact controllability o linear equation yields that the operator A k BB A k is positively k= deined and consequently the inequality m A k BB A k ψ, ψ γ ψ, γ>, 5 k= is valid. Since then 49 implies the ollowing inequality a ν l µ m+ γ ψ what gives a ψ νm, ĉ. 5 γl µ Using Lemma and 46 rom 48 we obtain that Ψ νl µ + L µ3 m k B A k ψ l µ m k= k= k B A k ψ c c s 4 ψ s + s 3 c c s 4 + ψ s s3 B ψ s 3 s 4 c 3 s4 B ψ s 4 ψ, 5 where c 4 = N /ĉ. Denote l =s +, l =s +s 3 +, l 3 =s 4 +. Applying the inequalities 5, 5 we have γ ψ l i ψ li k B A k ψ m γ k= a γl µ γ li a γl µ l iνmm li, i =,, 3, since l i ν m m. Then rom 5 we obtain the estimate Ψ l µ s νl µ + L µ3 γ c c s a 4 γl µ γ s3 c c s 4 s 3 B s 3 a γl µ γ s4 c 3 s4 B s4 a γl µ s+s 3 s4 Let a number ã > be such that the inequality γl µ > c c s 4 ã γl µ s +. 53

10 s +s 3 + s3 c c s 4 s3 B ã s3 + γl µ + s 4 c 3 s4 B s 4 ã γl µ s 4 is true. We choose the coeicient a rom the condition < a ã. From 53 it ollows that Ψ β, Q \ {}, β >, i.e. inequality 3 is proved or β = l µ νl µ + L µ3 γ c a c s 4 γl µ γ s 3 c c s 4 s 3 B s 3 γ s4 c 3 s 4 B s 4 a γl µ a γl µ s s +s 3 s4. Choose inally the coeicient a rom the condition < a â = min{a, ã }, where a has been deined in Section 3. Then the control solves the local synthesis problem or 45 in the domain Q \{} and satisies the restriction u d. Remark 4. Note that the statement o Remark 3 remains valid or the nonlinear equation up to the corresponding changes in the orm o the Cauchy problem. 6 Conclusion Let us give the principal moments o the algorithm o solving the local synthesis problem or equation. Deine the integer number m rom condition. Choose the unction rom the class F m A. Deine the solvability domain Q o the problem and the constants l µ, L µ, l µ, L µ3 by Lemma. For a given number d > choose the coeicient a satisying condition 3. Then the control u o the orm with deined by 5 solves the local synthesis problem. Besides, given Q \ {}, the solution t, =, corresponding to this control can be ound as the solution o the Cauchy problem 44. [] Korobov V.I. and Sklyar G.M. The solution o the synthesis problem using a controllability unctional or systems in ininite-dimentional spaces. Dokl. Acad. Nauk Ukraine SSR, Ser. A, No. 5, pp. 4, 983. [3] Korobov V.I. and Sklyar G.M. The synthesis o the control in equations containing an unbounded operator. Teor. Functsii Functional. Anal. i Prilozhen., No. 45, pp , 986. English transl. J. Soviet Math., Vol. 48, No. 4, pp , 99. [4] Balakrishnan A. Introduction to the theory o optimization in Hilbert space. Mir, Moskow, 974. [5] Trenogin V.A. Functional analysis. Nauka, Moskow, 98. [6] Korobov V.I. and Rabah R. Eact controllability in the Banach space. Di. Uravneniya, Vol. 5, No., pp. 4 5, 979. English translation in Dierential Equations, Vol. 5, pp , 98. [7] Korobov V.I. and Sklyar G.M. On a set o positionaly restricted controls solving the problem o synthesis. Dokl. Acad. Nauk SSSR, Vol. 3, No. 6, pp , 99. English transl. in Soviet Math. Dokl., Vol. 4, No. 3, pp , 99. [8] Krein M.G. and Nudel man A.A. The Markov moment problem and etremal problems. Nauka, Moscow, 973. English transl. Amer. Math. Soc., Providence, 977. [9] Akhiezer N.I. The classical problem o moments. Fizmatgiz, Moskow, 96. [] Laks P., Fillips R. Theory o dispersion. Mir, Moskow, 97. [] Yosida K. Functional analysis. Mir, Moskow, 967. [] Megan M. and Hiris V. On the space o linear controllable system in Hilbert spaces. Glasnik Matematicki, No., pp. 6 67, 975. Reerences [] Korobov V.I. A general approach to the solution o the boundary control synthesis problem in a controllability problem. Math. Sb., Vol. 95, No. 48, pp , 979. English translation in Math. USSR-Sb., Vol. 37, 98.