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MSC 47D06, 49J15, 93C25
DOI: 10.14529/mmp140314
OPTIMAL CONTROL OF SOLUTIONS TO THE MULTIPOINT INITIAL-FINAL PROBLEM FOR NONSTATIONARY RELATIVELY BOUNDED EQUATIONS OF SOBOLEV TYPE
M.A. Sagadeeva, South Ural State University, Chelyabinsk, Russian Federation, sam79@74.ru,
A.D. Badoyan, South Ural State University, Chelyabinsk, Russian Federation, badoyanani@mail.ru
We study the problem of optimal control of solutions to an operator-differential equation, which is not solved with respect to the time derivative, together with a multipoint initial-final condition. In this case, one of the operators in the equation is multiplied by a scalar function of time. By the properties of the operators involved, the stationary equation has analytical resolving group. We construct a solution to the multipoint initialfinal problem for the nonstationary equation. We show that a unique optimal control of solutions to this problem exists.
Apart from the introduction and bibliography, the article consists of three sections. The first section provides the essentials of the theory of relatively p-bounded operators. In the second section we construct a strong solution to the multipoint initial-final problem for nonstationary Sobolev-type equations. The third section contains our proof that there exists a unique optimal control of solutions to the multipoint initial-final problem.
relatively bounded operator.
Introduction
Suppose that X, Y, and U are Hilbert spaces, and then take bounded linear operators L £ L(X; Y) and B £ L(U; Y), assuming that the kernel of L is non-trivial. Take also a closed linear operator M £ Cl(X; Y) whose domain is dense in X.
Consider the Sobolev-type equation [1-4]
Lx(t) = a(t)Mx(t) + f (t) + Bu(t) (1)
with a control vector function u : [0,T] ^ U, a vector function f : [0,T] ^ Y of exterior force, and a scalar function a : [0,T] ^ R+, to be specified later, characterizing the change in time of the parameters of (1). The operators L and M generate the analytic resolving group for the homogeneous stationary equation (1), which means that a(t) = 1.
We consider an optimal control problem for (1). Namely, we aim to find a pair (x,U) £ X X Uad with
J (x,U) = inf J (x,u). (2)
(x,«)6XxU0d
Here Uad is a closed convex set of admissible controls in the Hilbert space U of controls, all pairs (x,u) satisfy the multipoint initial-final problem [5] for (1), and J(x,u) is a certain penalty functional in special form.
Previously the authors studied the optimal control problem for solutions to nonstationary Sobolev-type equations (1) with the Showalter-Sidorov condition [6, 7]. In this
paper we study the optimal control of solutions to the multipoint initial-final problem [5], which is a generalized Showalter-Sidorov problem [8] for (1).
Recall the standard notation of the theory of relatively p-bounded operators [3].
Starting with two Hilbert spaces X and Y, take a bounded linear operator L G L(X; Y) with non-trivial kernel and a closed linear operator M G Cl(X; Y) whose domain is dense in X. Consider the stationary equation
LX(t) = Mx(t) + f (t),
called a Sobolev-type equation [3].
Definition 1. The sets pL(M) = G C : (^L — M) 1 G L(Y; X)} and aL(M) =
Definition 2. The operator-valued functions (^L — M) 1, R^(M) = (^L — M) 1L, and LL(M) = L(^L — M)-1 are respectively called the resolvent, right resolvent, and left
L-resolvent of M).
Lemma 1. Given L G L(X; Y) and M G Cl(X; Y), the L-resolvent, right and left L-resolvents of M are analytic on pL(M).
Definition 3. An operator M is called spectrally bounded with respect to an operator L
exist by Lemma 1 for every (L, a)-bounded operator M. The operators P G L(X) and Q G L(Y) are projections [3]. Put X0 = ker P, Y0 = kerQ; X1 = imP, and Y1 = imQ. Denote the restriction of L (M) to Xk by Lk (Mk) for k = 0,1.
Theorem 1. The following claims hold for every (L, a)-bounded operator M:
(i) the operators Lk, Mk : Xk ^ Yk for k = 0,1;
(ii) the operators M0 G L(X0; Y°) and M1 G C¡(X1 ; Y1 );
(iii) there exists operators L-1 G ^(Y1; X1) and M0-1 G L(Y0; X0);
(iv) there exist analytic resolving operator groups {X* G L(X) : t G R} for the homogeneous equation (3) and {Y* G L(Y) : t G R} for the equation RI^(M)ij(t) = M(PL — M)~1y(t), where ¡3GpL(M), which are of the form
C \\ pL(M) are called the L-resolvent set and the L-spectrum of M respectively.
resolvent of M with respect to L (or briefly the L-resolvent, right L-resolvent, and left
(or briefly (L, a)-bounded) whenever 3r0 > 0 G C (|^| > r0) ^ (^ G pL(M)).
Put y = {^ G C : |^| = r > r0}. The Riesz-type integrals
Theorem 1 implies the existence of the operators H = M0 1L0 G L(X0) and S L-1Mi G ¿(X1).
Definition 4. An (L, a)-bounded operator M is called
(i) (L, 0)-bounded whenever the point to is a removable singularity of the L-resolvent of M, that is, H = O;
(ii) (L,p)-bounded whenever the point to is an order p G N pole of the L-resolvent of M, that is, Hp = O and Hp+1 = O;
(iii) (L, TO)-bounded whenever the point to is an essential singularity of the L-resolvent of M, that is, Hq = O for all q G N.
Take two Hilbert spaces X and Y- For two operators L G L(X; Y) and M G Cl(X; Y), where M is (L,p)-bounded for p G {0} U N, introduce the condition
rf(M), n G N, and af& j=o ___
and Yj = dDj, where Dj D af(M), such that Dj fl af(M) = 0
°L(M) = U j (M), n G N, and ) = 0, there is a closed loop Yj С C
and Dk fl Di = 0 for all j,k,l = l,n,k = l. Define the operators Pj G L(X) and Qj G L(Y) for j = j, n as 1 f „r............................. ~ 1
Pj = 2nij Ri(m№, Qj = 2ni J l\\:(m)dp, j = i,n thanks to the relative spectral theorem [9], and moreover, the results of [9], and the
operators Po = P - Pj, Qo = Q - Qj ■
Consider the multipoint initial-final problem
Pj (X(Tj) - Xj) = ï0, (Tj < Tj+i) j = 0,n (5)
for (3)- Applying to (3) the projections I — Q and Qj for j = 0,n yields the equivalent system
Hx0 = x0 + M-1f0, (6)
= Sijx1 + L-1fh (7)
where H = M-1L0 G L(X0) is a degree p G {0} U N nilpotent operator, the operator
S1j = L-lMj G Cl(Xl) has the range a(Sj) = ajf(M), while f0 = (I — Q)f, j = Qjf,
x0 = (I — P)x, and xl = Pjx for j = 0, n.
Put N0 = {0} U N and construct the space
Hp+1(Y) = {£ G L2(0,r; Y) : £(p+1) G L2(0,T; Y), P G N0}
P+1 T
which is a Hilbert space with the inner product [£, n] = £ a (i),n(<,)) y dt.
Definition 5. A vector-valued function x G H1 (X) is called a strong solution to the multipoint initial-final problem (3), (5) whenever it satisfies (3) and the terms of Pj(x(Tj) — xj) = 0 for j = 0,n almost everywhere.
Lemma 2. If an operator M is (L,p)-bounded, with p G No, then for every vector function f0 G HP+1(Y°) there exists a unique solution x0 G H^X0) to (6):
x° (t) = Hq Mo-—f 0(t).
Lemma 3. Under the assumptions of Lemma 2, if condition (4) is fulfilled then for every vector Xj G X and every vector function f) G H(Y)) there exists a unique solution x) G
H 1(X)) to the problem Pj(x(rj) — Xj) = 0 for (7): x)(t) = Xj TxTj — f Xj~sL-j f)(s)ds.
Theorem 2. Given vectors xj G X for j = 0,n and a vector function f : [0, t] - Y satisfying the assumptions of Lemmas 2 and 3, there exists a unique solution x G H^X):
p n i Tj
x(t) = — £ HM-1 d^f 0(t) + £ xr, — ! X‘-L-1f}(s)ds | ■
q=0 j=0 \\ Jt
For a Hilbert space X consider the equation
Lx(t) = a(t)Mx(t) + f (t) + Bu(t) (8)
with operators L G L(X; Y), M G Cl(X; Y), and B G L(U; Y), a scalar function a : [0,t) — R+, as well as vector functions u : [0,t) — U and f : [0,t) — Y to be specified later.
Take a Hilbert space Z and an operator C G L(X; Z)- Consider the penalty functional
J{u) = ^ I \\\\z(q — zd^llZdt + I (Nqu(q,u(q^Udt, z = Cx, (9)
q=0 0 q=0 0
where 0 < k < p +1- The operators Nq G L(U) for q = 0,1,... ,p +1 are self-adjoint and positive definite, while zd = zd(t, s) is an observation from some space of observations Z-Note that if x G H 1(X) then z G H 1(Z)- By analogy with Hp+1(Y), define the space Hp+1(U), which is a Hilbert space because so is U- We distinguish a convex and closed subset Hapd+1(U) of the space Hp+1(U), called the set of admissible controlsDefinition 6. A vector function v G Hap++1(U) is called an optimal control of solutions to problem (5), (8) whenever
J (v) = min J (u), (10)
(x(u),u)€£xHp+1(tt)
where the pairs (x(u),u) G X x H1pd+1(U) satisfy (5), (8)By Theorem 2, a unique solution x G H 1(X) to problem (5), (8) exists for all vectors xj G X for j = 0,n, vector functions f G HP+1(Y), u G Hp+1(U) and a function a G Cp+1([0,T); R+) separated from zero:
x(t)=—t Hq M-1(I—Q)( aL I) ‘ +
xj«>-.4( .)L-1Qj (f (s)+Bu(s))dsj (11)
by analogy with [6]- Here A(t) = J a(q)dq- We now fix xj G X for j = 0,n and f G
Hp+1(Y) and consider (11) as a mapping D : u — x(u)Lemma 4. Given Hilbert spaces X, Y, and U, take an (L,p)-bounded operator M, with p G N0, a function a G Cp+1 (R+; R+) separated from zero, and fix vectors xj G X for j = 0,n and f G Hp+1(Y). Then the mapping D : Hp+1(U) —— H 1(X) defined by (11) is continuous.
Proof. Since B G C(Hp+1(U); Hp+1(Y)) and (11) is the solution to (8), this lemma holds by the properties of the operator group XJ and the continuity of a(t) for t G R+, by analogy with the proof of Theorem 2□
Theorem 3. Take an (L,p)-bounded operator M with (p G N0) and a function a G Cp+1 ([0,t); R+) separated from zero. Then for all vectors xj G X for j = 0,n, f G Hp+1(Y), and zd G Z, there exists a unique solution v G H^pl+1(U) to the optimal control problem (5), (8)-(10).
Proof. Using the mapping D of Lemma 4, we see that the functional (9) becomes
J(u) = \\\\Cx(t; u) — zd\\\\2Hi(3) + [q,u],
where n(k&)(t) = Nku(k) for k = 0,... ,rp +1- Therefore,
J(u) = n(u,u) — 29(u) + \\\\zd — Cx(t; 0)\\2Hi(z),
where n(u,u) = \\\\C (x(t; u) — x(t;0))\\\\2H ^ + [n,u] is a coercive continuous bilinear
form on Hp+1(U), and
is a continuous linear form on HP+1(U)- Thus, the theorem is valid by analogy with [6]□
References
Semigroups of Operators- Utrecht, Boston, Koln, VSP, 2003- 216 pDOI: 10-1515/9783110915501
xA{t)-A{T3_
Received May 15, 2014ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ РЕШЕНИЯМИ МНОГОТОЧЕЧНОЙ НАЧАЛЬНО-КОНЕЧНОЙ ЗАДАЧИ ДЛЯ НЕСТАЦИОНАРНЫХ ОТНОСИТЕЛЬНО ОГРАНИЧЕННЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА
М.А. Сагадеева, А.Д. Бадоян
В статье рассматривается оптимальное управление решениями начально-конечной задачи для операторно-дифференциального уравнения, неразрешенного относительно производной. При этом в уравнении один из операторов умножен на скалярную функцию переменной Ь, и свойства операторов таковы, что стационарное уравнение обладает аналитической разрешающей группой. В статье строится сильное решение начально-конечной задачи для нестационарного уравнения соболевского типа в случае относительной ограниченности. Используя построенное решение, доказывается существование единственного оптимального управления решениями указанной задачи. Статья кроме введения и списка литературы содержит три части. В первой из них приводятся необходимые сведения теории относительно ^-ограниченных операторов, во второй — строится сильное решение многоточечной начально-конечной задачи для нестационароного уравнения соболевского типа. Наконец, в третьей части доказывается существование и единственность оптимального управления решениями начальноконечной задачи для нестационарного уравнения соболевского типа.
- N.Y.; Basel; Hong Kong: Marcel Dekker, Inc, 1999. - 236 pp.
- 216 p.
Минзиля Алмасовна Сагадеева, кандидат физико-математических наук, доцент, кафедра «Уравнения математической физики», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), sam79@74.ru.
Ани Давидовна Бадоян, аспирант, кафедра «Уравнения математической физики:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), badoyanani@mail.ru.
Поступила в редакцию 15 мая 20 Ц г.
