Optimal control in linear Sobolev type mathematical models

The research of the control problems for mathematical models based on Sobolev type equations is relevant due to the need to study important applied problems, in particular, in the theory of filtration, elasticity, biology and others. When studying mathematical models, it is important not only to understand the properties of the processes being studied, but also to be able to find the optimal regulation (external influence), with the help of which the state of the system takes the required value. The article describes methods and approaches developed in the framework of the scientific direction headed by G.A. Sviridyuk to study the optimal control problems for linear Sobolev type mathematical models with classical and non-classical initial (initial-final) conditions. A wide class of such mathematical models has been studied based on the theory of Sobolev type equations. Consider some of them.

The Barenblatt-Zheltov-Kochina model. Let Q C R ⁿ be a bounded domain with a boundary d Q of the class C “ . In the cylinder Q x R + consider the Dirichlet boundary condition

x(s,t) = 0 , ( s,t ) E d Q x R                              (1)

for the Barenblatt–Zheltov–Kochina equation [1]

( A — A) x = a A x + u                          (2)

that simulates the process of fluid filtration in a fractured porous medium. Here a, A E R characterize the properties of the medium; parameter a E R ₊ , and parameter A can take negative values that do not contradict the physical meaning of the problem, the function u = u(s,t) plays the role of an external exposure (i.e. characterizes the sources (drains) of the fluid). A mathematical model based on equation (2), supplemented by classical or non-classical initial (initial-final) conditions, was studied in various aspects in [2–4].

Mathematical model of an I-beam bugling. Let G = G ( V ; E ) be a finite connected oriented graph, where V = { V i } is a set of vertices, and E = { E i } is a set of edges. Each edge E i has the length l i E R ₊ and the cross section area d i E R ₊ . In each vertice Vi, i = 1,M, of the graph G consider the continuity conditions

xj(0,t)    xk(0,t)    xm(lm,t')    xn (ln,t), where Ej,Ek E Ea(Vi), Em,En E Eш(Vi) (Eа(ш)(Vi) denotes the set of edges with the beginning (end) at the vertex Vi), and flow balance condition

^ d j X js (0,t) -    ^   d k x ks (l k ,t ) = 0                   (4)

E j E E ^a ( V i )                  E k E E “ (V i )

for Hoff equations [5]

Ax jt + X j _tss = a j X j + U j .                               (5)

Conditions (3), (4) and equation (5) form the Hoff mathematical model on a graph that describes the dynamics of deformation of an I-beam construction under constant load A E R ₊ . Parameters a j E R characterize beam material properties; the free term U j = U j (s,t) corresponds to the external (lateral) load on the j -th edge of the graph. Equations (5) on graphs were first studied in [6]. The Hoff mathematical model in the domain or on the graph, supplemented by classical or non-classical initial (initial-final) conditions in various aspects was studied in [7–9].

The Boussinesq-Love model. Let Q C R ⁿ be a bounded domain with a boundary d Q of the class C “ . In the cylinder Q x R consider the Boussinesq-Love equation [10]

( A — A) x tt = в (A — A')x + u                          (6)

with boundary condition (1). Model (1), (6) describes the propagation of waves in shallow water. Parameters β , λ , λ ^′ relate to depth, gravitational constant and Bond number. The function x(t, s ) determines the wave height at time t at the point s , u(t, s ) is a control that defines external forces. The Boussinesq–Lo¨ve mathematical model, supplemented by classical or non-classical initial (initial-final) conditions, was studied in various aspects in [10, 11].

The Dzektzer model. Let Q C R n be a bounded domain with a boundary d Q of the class C “ . In the cylinder Q x R ₊ consider the problem

x(s, t) = A x ( s, t) = 0 , (s, t) E d Q x R ₊                          (7)

for a non-classical partial differential equation [12]

( A — A) X = aAx — в A 2 x + u.                       (8)

The mathematical model (7), (8) describes the evolution of the free surface of the filtered fluid. Here а, в E R ₊ , A E R are the parameters characterizing the fluid, the free term u = u(s,t) characterizes the sources (drains) of the fluid. Mathematical model based on equation (8), supplemented by classical or non-classical initial (initial-final) conditions in various aspects was studied in [13, 14].

The Chen-Gurtin model with complex coefficients. Let Q C R ⁿ be a bounded domain with a boundary d Q of class C “ . In the cylinder Q x R + consider the boundary value problem (7) for a non-classical partial differential equation [15]

( A — A) x _t = v A x — idA²x + u.                        (9)

Here the coefficients ν, λ, d ∈ R characterize parameters of the system. The required complex-valued function x ( s,t ) describes the dynamics of the process, and the complexvalued function u ( s,t ) describes an external effect on the system. In the special case of d = 0 equation (9) describes the process of heat conduction with “two temperatures” [15], as well as the dynamics of fluid pressure in a cracks-porous medium [1]. A mathematical model based on equation (9), supplemented by classical non-classical initial (initial-final) conditions in various aspects was studied in [16, 17].

The mathematical models under consideration belong to a wide class of Sobolev type models (i.e., models based on Sobolev type equations). Sobolev type equations currently constitute a significant part of the non-classical equations of mathematical physics. Initially, such equations arose in the works of A. Poincare, C. Rossby, J. Boussinesq and other mechanics in the late XIX – early XX centuries. However, a systematic study of such equations began in the middle of the last century with the work of S.L. Sobolev. A detailed history of this issue can be found in the monograph [18]. Note that various terms are used in literature to denote such equations [18–20]. The term “Sobolev type equations” [3, 21] first appeared in the works of R.E. Showalter [22]. We adhere to this term, considering the rest as synonyms.

The mentioned mathematical models with one or another initial (initial-final) conditions in suitable Banach spaces can be reduced to the corresponding problems for a linear Sobolev type equation

Ax ⁽ⁿ⁾ = Bx + y + Cu,                          (10)

where operators A E L (X; Y ) , B E C l(X; Y ) , C E L (U; Y ) , functions u : I ч U, y : I ч Y ( I C R) , and X , Y , U are Hilbert spaces. To select the unique process under study, the mathematical models under consideration and their abstract interpretation (10) are supplemented by one of the following conditions:

– the Cauchy condition [3]

x(m)(0) = xm, m = 0,.. .n — 1,

– the Showalter–Sidorov condition [23, 24]

P (x(m) (0) — xm) =0, m = 0,..., n — 1,

– the initial-final condition [4, 25]

Pin (x(m)(0) — xm) =0, Pfin (x(m)(t) — xm) =0, m = 0,...,n — 1, where P, Pin , Pfin are some spectral projectors in the space X, which will be defined later. Condition (13) differs from the initial conditions in that one projection of the solution is specified at the initial moment of time, and the other is set at the final moment of the considered time interval. The initial-final condition is a generalization of the Showalter– Sidorov condition, which in turn is a generalization of the classical Cauchy condition. As it is well known (see, for example, [23]), the Cauchy problem for the Sobolev type equation (10) (in case ker A = {0}) is not solvable for arbitrary initial values xm, m = 0,n — 1. To overcome this difficulty, G.A. Sviridyuk proposed the phase space method. The foundations of this concept were laid down in [20], then the concept was developed in [3] and many other works. Another approach to overcome the difficulties associated with non-existence of the solution to (10), (11) is to consider the initial Showalter–Sidorov condition (12) and a more general initial-final condition (13) instead of the initial Cauchy condition (11). We are interested in solving the optimal control problem, which consists in finding a pair (x,u), for which the relation

J (x,u) = min J (x,u) (14) (x,u)eXxUad holds. Here the pairs (x, u) satisfy the Cauchy problem (10), (11) or the Showalter-Sidorov problem (10), (12), or the initial-final problem (10), (13) and J(x,u) is some specially constructed quality functional, Uad is some closed and convex set in the control space U.

The article provides an overview of the results developed in the framework of the direction headed by G.A. Sviridyuk on the optimal control of the solutions to the initialfinal problem and, in particular, the Showalter–Sidorov and Cauchy problems for linear Sobolev type equations. The first who began to study the controllability problems and the optimal control problem for linear Sobolev type equations with the Cauchy condition were G.A. Sviridyuk and A.A. Efremov [2, 13, 26]. In these papers, the optimal control problem with a quadratic quality functional was studied in case n = 1 with (A, p )-bounded or (A, p )-sectorial operator B and the Cauchy condition, the necessary and sufficient conditions for the existence and uniqueness of a solution were obtained. G.A. Sviridyuk suggested moving from considering the classical solution x E C ^J; X ) of (10), (11) to the strong solution x E H ^p+1 ( X ) of this problem, which allowed to set the optimal control problem (10), (11), (14) and to use the technique of Hilbert spaces for its research. These studies formed the basis of a number of works by G.A. Sviridyuk’s disciples and followers on the study of optimal control problems for linear Sobolev type equations based on the theory of degenerate resolving (semi)groups of operators [3]. Since [2, 3, 13], when considering the classical Cauchy condition, due to the degeneracy of the equation, it was necessary to reconcile the initial data with the control action, then G.A. Sviridyuk suggested an idea to use more general initial Showalter–Sidorov condition (initial-final condition), which made it possible to remove the restriction on the set of optimal controls in the subsequent works of his disciples and followers and opened the way to a whole class of problems on this subject [8, 27]. In [10] the necessary and sufficient conditions for the existence and uniqueness of the solution of optimal control problems for high-order Sobolev type equations with an initial-final condition were obtained. The ideas and methods developed by G.A. Sviridyuk and A.A. Efremov on controllability of linear abstract Sobolev type equation opened the way to the study of more general controllability problems [28].

The article consists of introduction, 6 sections and conclusion. The first Section gives the main points of the theory of relatively bounded operators, the complete proofs of which can be found in [3]. It contains theorems on existence and uniqueness of classical solution of optimal control problems for an abstract Sobolev type equations, as well as on existence and uniqueness of strong solution to the initial (initial-final) problem for such equations. In the second Section, the abstract results are applied to specific Sobolev type models, namely, the Barenblatt–Zheltovaya–Kochina model, the Hoff model on the graph, and the Boussinesq–L¨ove model. The third Section contains the results of the theory regarding relatively sectorial operators for the first-order Sobolev type equation. In the fourth Section, the Dzektzer model and the pressure evolution model on the graph are reduced to an abstract Sobolev type equation with initial (initial-final) conditions, and then the abstract results are applied to the study of optimal control problem for them. The fifth Section contains the main results of the theory with relatively radial operators. The optimal control in the Chen–Gurtin model based on the Sobolev type equation with a relatively radial operator is studied in the sixth Section.

1. Relatively p-Bounded Operators.Strong Solutions. Optimal Control

In this section definitions and results of the theory of relatively bounded operators are given [3,29]. Let X , Y and U be Banach spaces. Operators A,B E L (X; Y ) , operator C E L (U; Y ) . The set

PA(B) = {m E C :(pA - B)-1 E L(Y; X)} is called a resolvent set of operator B with respect to operator A (an A-resolvent set of the operator B). The set C\pA(B) = aA (B') is called a spectrum of operator B with respect to operator A (an A-spectrum of operator B). The operator function

(mA - B) ^- , R A = (M — B) ^-1 A,  L ^A = AEA — B ) ^-1

of a complex variable with domain p ^A (B ) is called a resolvent, a right resolvent, a left resolvent of operator B with respect to operator A ( an A-resolvent, a right A-resolvent, a left A-resolvent of operator B ).

Definition 1. The operator B is called polynomially bounded with respect to operator A (or simply (A, a)-bounded), if

3 a >  0 V p E C : ( | д | > a) ^ (m E p ^A (B)).

Lemma 1. [29] If the operator B is (A, a)-bounded, then the following operators

P =

P [ R A (B^-'AdM, 2ni J    ^

Γ

Q =

2ni J

Γ

pn-AAR A n (B )dM

are projectors, moreover P : X ч X and Q : Y ч Y . Here Г = { A E C : | A ⁿ | = r > a } .

Set X ⁰ = ker P , Y 0 = ker Q , X ¹ = im P , Y A = im Q . By A k denote the restriction of the operator A, and by B k denote the restriction of the operator B to X k , k = 0 , 1. The theorem of operator actions’ splitting is true.

Theorem 1. [3, Ch. 4] Let the operator B be (A,a)-bounded. Then

• (i)    operators A k , B k : X k ч Y k , k = 0 , 1;

• (ii)    there exists an operator B - E L ( Y ⁰ , X ⁰ );

• (iii)    there exists an operator A -¹ E L (Y 1 , X ¹ );

• (iv)    operator B 1 E L (X 1 , Y ¹ ) .

Let us construct the operators H = B-¹ A ₀ E L (X ⁰ ) and S = A-¹B ₁ E L ( X 1 ). Then

(цА - B ) ^-1

= (- E / h k}

\ k=0      /

∞

B - 1 (I - Q ) + E^ - k ^s k -' A -¹ Q.

k=1

Definition 2. The point ∞ is called:

• (i)    a removable singular point of the A-resolvent of the operator B, if H = O;

• (ii)    a pole of order p of the A-resolvent of the operator B, if H ^p = O ,H ^{p +1} = O , p E N;

• (iii)    an essentially singular point of the A-resolvent of the operator B, if H q = O , ∀ q ∈ N .

Definition 3. (A, a)-bounded operator B is called (A,p)-bounded for some p E { 0 } U N , if the point ж is a pole of order p E { 0 } U N of the A-resolvent of operator B.

Definition 4. The vector-function x E C ⁿ (R; X), n E N , satisfying (10) is called a classical solution of this equation.

Consider linear homogeneous (y = u = 0) Sobolev type equation (10).

Definition 5. The operator-function V • E C “ (R; L (X)) is called a propagator of homogeneous equation (10), if for arbitrary v E X the vector-function x(t) = V t v is the solution of this equation.

Theorem 2. [29] If the operator B is (A, a)-bounded, then formulas

t

m

=-[

2ni J

n - m -1 ^µ

(p n A — B) ¹ Ae " dp, m = 0,n —

1 ,

Г

define propagators of a homogeneous equation (10) for t ∈ R .

Consider the inhomogeneous equation

Ax ⁽ⁿ⁾ = Bx + y                              (16)

and sets

• ^p         nqq+m

  M m = { x e X : (I - P)x =

  
  

  ^-E ^H q ^{B -} d q  ^{( l} - ^У ^{(0) }} , m = 0 7^-1 .

q =0

The results on existence and uniqueness of a classical solution to (11), (16) were obtained in [10, 29].

Theorem 3. [29] Let the operator B be ( A, p ) -bounded for some p E { 0 } U N , the function y : I ч Y ( I C R) be such that y ⁰ = (I - Q)y E C ^{np + n} (I; Y ⁰ ) and y ¹ = Qy E C ( I ; Y ¹ ) .

Let the initial values x m E M m , m = 0 , n — 1 . Then there exists a unique classical solution to the Cauchy problem (11), (16) given by

x(t) =

p                              n -1

— £ Hq B   — Q)y

t

I Xn-siA-¹Qy(s)ds, t E I.

There were obtained results [10] on the existence and uniqueness of a classical solution to problem (12), (16). Note that when using the Showalter–Sidorov condition, the assumptions of Theorem 3 can be weakened without requiring initial data being agreed with the right side of equation (16).

Theorem 4. [29] Let the operator B be (A, p)-bounded for some p E {0}U N, the function y : I ч Y be such that y⁰ = (I — Q)y E C^np+n(I; Y⁰) and y¹= Qy e C(I; Y¹). Then for arbitrary x_mE X, m = 0,n — 1, there exists a unique classical solution to Showalter-Sidorov problem (12), (16) given by (18).

Proceed to consider a more general than the Showalter–Sidorov initial-final condition. Introduce an additional condition aA(B) = aA(B) U aA(B), ^ B) = 0,k = 0,1;

and contour y₀ is the boundary of domain D E C such that          (B)

D n &A(B) = &A(B), D n aA(B) = 0.

Then there exists an operator

Pfin = Л / R^AAn(B)pn^-1Adp E L(X). 2ni

γ0

Lemma 2. [29] Let the operator B be (A,p)-bounded for some p E {0} U N, let condition (B) be satisfied. Then Pfin is a projector, and PfinP = PPfin = Pfin.

Construct an operator Pin = P — Pfin E L(X). By Lemma 2 operator Pin is a projector.

Consider operators

Xf '     2_n ix B)(p“^-

γ0

^m-1A — B)e^ dp.

Note that Xfin is a propagator of homogeneous (y = 0) equation (16). Introduce a family of operators

ХШ = X —

Xfn (t),   k = 0,n — 1.

Theorem 5. [29] Let the operator B be (A, p)-bounded for some p E {0}UN. Let the vector function y : I ч Y be such that y0 = (I — Q)y E Cnp+n(I; Y0) and y1 = Qy E C(I; Y1). Then for arbitrary xm ,xm E X, m = 0,n — 1, there exists a unique classical solution to problem (13), (16), for t ∈ I given by p                                   n-1                    n-1

x(t) = — e hq (B0)^-1(i—Q)y™(.t) + e Xm (t)Pn xm + e Xm, (t)Pf.n xm+ q=0                          m=0               m=0

tτ

+ f X^(t — s)A-PintWds — J f'(t — s)A-Pf_mV(s)ds. 0

Thus, a general theory that allows one to find classical solutions for (16) with initial (initial-final) conditions is constructed. Now turn to the study of control problems. It should be noted that in such problems the technique of Hilbert spaces is traditionally used, which requires consideration of other types of solutions. Further we consider X, Y and U being Hilbert spaces. Consider space Hk(Y) = {v E L2 (0,t; Y) : v(k) E L2(0,t; Y)}. The space Hk(Y) is Hilbert with inner product kτ

[v,w] = ^ (v^(q),w^(q))Y dt q=°^J0

Definition 6. The vector-function x E Hⁿ(X) = {x E L₂(I; X) : x^n E L₂(I; X)} is called a strong solution to (16), if it turns the equation to an identity almost everywhere on interval I. A strong solution x = x(t) of (16) is called a strong solution of the Cauchy problem, if condition (11) holds; a strong solution of the Showalter–Sidorov problem, if condition (12) holds; a strong solution of the initial-final problem, if condition (13) holds.

Note that classical solution (18) is also a strong solution to equation (16) by virtue of the continuity of the embedding Hⁿ(X) ^ C^n-1(I; X).

Theorem 6. [30] Let the operator B be (A,p)-bounded for some p E {0} U N. Then for arbitrary y E H^np+n(Y), x_mE Mm,m = 0,n — 1, there exists a unique strong solution to the Cauchy problem (11), (16).

Theorem 7. [31] Let the operator B be (A, p)-bounded for some p E {0} U N. Then for arbitrary xm,xm E X,m = 0,n — 1, and y E H^np+n(Y) there exists a unique strong solution to the initial-final problem (13), (16).

Note that in the case of (aA = a^A(B),c^A_in = 0) the Showalter-Sidorov problem can be considered as a special case of the initial-final problem. The following result follows from Theorem 7.

Corollary 1. Let the operator B be (A,p)-bounded for some p E {0} U N. Then for arbitrary x_mE X,m = 0,n — 1, and y E H^np+n(Y) there exists a unique strong solution to the Showalter–Sidorov problem (12), (16).

Definition 7. The pair (x,u) is called a solution to the optimal control problem (10), (11), (14) if relation (14) is satisfied and all pairs (x,u) E H^np+p(X) x Uad are solutions of problem (10), (11). A vector function U E Uad is called an optimal control of solutions to (10), (11).

Consider the penalty functional nτ

J^(x,u)= p ^ I^||x(q) q=°l

-

x^\\²dt + v^ [ {N_q u^(q\u^(q\ dt. q=0 0

Here p,v > 0, p + v = 1, N_qE L(U), q = 0, 1, ..., np + n, are self-adjoint positively defined operators, and x(t) is the target state of the system.

Remark 1. By Theorem 7 on the existence of a unique solution for all y E H^np+n(Y) and u E Uad the solution x = x(u). Therefore, the quality functional depends only on u : J(x,u) = J(u). Thus, the set of feasible solutions to problem (10), (11), (14) is not empty.

When using the Cauchy condition, we pass to consideration of the subspace of controls _onp+n

H (U) = {u E L₂(0,t;U) : u^(np+n)E L₂(0,t;U),u^(q)(0) =0,q = 0,np + n - 1}.

_onp+n_onp+n

In the space H (U) we single out a closed convex subset Uad C H (U), which will be called the set of admissible controls. Results on existence and uniqueness of optimal control of solutions to the Cauchy problem (10), (11) were obtained in [30].

Theorem 8. [30] If the operator B is (A,p)-bounded for some p E {0} U N, then for arbitrary y E H^np+n(Y) and x_me Mm, m = 0,n — 1, there exists a unique optimal control of solutions to problem (10), (11).

Introduce the control space H ^np+n(U). Single out a closed convex subset U_adC H ^np+n(U). Results on existence and uniqueness of optimal control of solutions to initialfinal problem (10), (13) were obtained in [31].

Theorem 9. [31] If the operator B is (A,p)-bounded for some p E {0} U N, then for arbitrary y E H^np+n(Y) and xm,xm E X, m = 0,n — 1, there exists a unique optimal control of solutions to the initial-final problem (10), (13).

The following result follows from Theorem 9.

Corollary 2. If the operator B is (A, p)-bounded for some p E {0} U N, then for arbitrary x_mE X, m = 0,n — 1, and y E H^np+n(Y) there exists a unique optimal control to solutions of the Showalter–Sidorov problem (10), (12).

2.    Optimal Control in Sobolev Type Modelswith Relatively p-Bounded Operators

Optimal control in the Barenblatt–Zheltov–Kochina model. Consider the optimal control problem (14) for mathematical model (1), (2) with the initial Cauchy condition (11). Reduce this problem to equation (10) with n = 1. For this purpose put

X = H²(Q)n H¹(Q), Y = L2M U = L2(^),             (21)

and by formulas

A = AI — A, B = aA, C = I,                     (22)

define the operators A,B E L(X, Y). Denote by {pk} the set of eigenfunctions of the Dirichlet problem for the Laplace operator A, numbered in ascending order of eigenvalues {λ_k} taking into account their multiplicity. Consider the A-resolvent of operator B in the form

-1 X^     ^^,^pk}^pk

'"      )     ^ M(A — Ak) — aAk’ where (•, •) is an inner product in L2(Q). Hence, the A-spectrum of operator B is given by

B)=    =   "   : A = лЛ .                    (24)

l ^{л — л}к         J

Lemma 3. [3] The operator B defined by (22) is (A, 0)-bounded for any Л E R.

Let the function y E H 1(Y). Then the set My (given by (18) for n = 1) takes the form

My = {x E X : —Лa{x, ^k) = (У(0), Ek), Лк = Л} and the control space

H¹(U) = {u E L2(0,t;U) : u E L2(0,t; U),u(0) = 0} .

Choose a closed convex subset Uad in it.

Theorem 10. [2] For any y E H 1(Y) and x₀E My there exists a unique solution (x, u) E H 1(X) x U_adto the optimal control problem (14) for mathematical model (1), (2) with the initial Cauchy condition (11), that minimizes functional (20).

Remark 2. The vector function x E H 1(X) defines the desired distribution of pressure in Q, and the control u means the effect on pressure by increasing (decreasing) the influence of liquid sources (drains) in Q.

Optimal control in the Hoff model on a graph. Consider the optimal control problem for mathematical model (3) - (5) with initial-final condition (13) in the case of n = 1. Reduce the problem to equation (10) with n = 1. Introduce the Hilbert space L2 (G) = {u = (u1,u2, ...,uj,...) : uj E L2(0,lj)} with inner product lj

(u, v) = £ dj / uj vj ds,

Ej∈E   ₀

and Banach space X = {x = (x1 ,x2, ...,Xj,...) : Xj E H 1(0, lj) and (3) holds } with norm lj

MX

= ^ dj / Ej∈E   ₀

(xjs + xj )ds.

Denote by Y the conjugate space to X with respect to the duality (•, •) and by formula

(Lx, v)

-

lj

E dj/

Ej ∈E   ₀

xjsvjsds^,

x, v E X

define the operator L E L(X; Y). It was shown [32] that a(L) is negative, discrete, with finite multiplicity, and condenses only to —to. Number the eigenvalues {v_k} of the operator L in nonincreasing order, taking into account multiplicity. Then the orthonormal (in the sense of Y) family of corresponding eigenfunctions {ф_к} of operator L forms the basis in X.

For A E R+ construct the operator A = AI + L. By construction operator A E L(X; Y), and its spectrum a(A) = {A + vk}. Define the operator B by formula B = al, where a E R, and C = I. Thus, we reduced problem (3) - (5) to equation (10). Construct the relative spectrum of the operator B aA (B) = ^ki= =

-j : k E N \{l : A + Vk = 0}l . A + Vk                   J

Lemma 4. [6] The operator B is (A, 0)-bounded if one of the following conditions holds:

• (i)    ker A = {0}, A = Vk, Vk E N;

• (ii)    ker A = {0}, aj = 0 for any j.

Further, consider the initial-final problem for mathematical model (3) – (5). Represent the A-spectrum of operator B in the form

■ (B) = 4(B) U f (B), 4(B) n f(B) = 0.

Then the initial-final condition (13) takes the form

^2 <(x(0) - Xo),Vk) Vk = 0, ^2 ((x(t) - Xt),Vk) Vk = 0.

^k ^еоА(^в)                               ^k ^еа^п(^в)

Introduce the control space

{ Nq Uq

lj и^ы> = E d,[ Ej eE 0

Kjq (ufq^ds,

where κjq are positive numbers.

Theorem 11. [8] For any y E H 1(Y) and A E R₊, aj E R, satisfying condition (i) or (ii) of Lemma 4, for arbitrary x₀,x_TE X there exists a unique solution (X,u) E H 1(X) x Uad of optimal control problem (14) for mathematical model (3) – (5) with initial-final conditions (26), that minimizes functional (20).

Optimal control in the Boussinesq–L¨ove model. Consider the optimal control problem for mathematical model (1), (6) with the initial Showalter–Sidorov condition (12). Reduce the problem to equation (10) with n = 2. To do this, put

X = {x E H^l+2(Q) : x(s) = 0,s E dQ}, Y = H^l(Q), U = H^l(Q), l E {0} U N.

The operators A,B E L(X; Y) are given by formulas A = AI — A, B = в(A — A’), C = I. Then the A-resolvent of operator B takes the form

∞

(m2a — B)-1 = £ k=1

<•, Vk)Vk

^{(A — A}k)v²+ ^{в (A}‘^{— A}k),

where (•, •) is an inner product in L2(Q), Xk,pk are the same as in Barenblatt-Zheltov-Kochina model. Then for the A-spectrum of operator B we get aA(B) = {мк = -^7---r^ : X = Xk| •                  (28)

^{X — X}k

Lemma 5. [10] The operator B is (A, 0)-bounded.

Construct the projector:

( I, if X = Xk;

P = ( I -E •■ ■ ^k’ if Xk = X.

I      Ak =A

The Showalter–Sidorov condition (12) is given by

^2 < n, x(s, 0) - xo(s) > ^k = 0, ^2 < ^k, Xt(s, 0) - xi(s) > ^k = 0.    (29)

Ak=A                                      Ak=A

Let us proceed to the optimal control problem. Introduce the control space

H²(U) = {u g L₂(0,t;U) : u G L2(0,t;U)}.

In the space H²(U) single out a closed convex subset U_ad, which will be the set of admissible controls.

Theorem 12. [10] For any а, в G R\ {0} , X G R, and т G R₊,xm G X,m = 0,1, there exists a unique solution (x,u) G H²(X) x U_adto the optimal control problem (14) for the Boussinesq–L¨ove model (1), (6) with the Showalter–Sidorov condition (29), that minimizes functional (20).

3.    Relatively p-Sectorial Operators

In this section, definitions and results of the theory of relatively sectorial operators are given [3, Ch. 3]. Let X, Y,U be Banach spaces, operators A G L(X; Y), B G Cl(X; Y), C G L(U; Y), functions y : (0,т) C R+ G Y (т < то) to be determined later. Let further Mk G pA(B), k = 0,1,... ,p. Operator-functions pp

Cp) = ПС(B), CPB) = nLA.(B) k=o                        k=o are called, respectively, a right and a left (A, p)-resolvents of operator B.

Definition 8. [3, Ch. 3] The operator B is called p-sectorial with respect to the operator A with the number p G {0} U N (or simply (A, p)-sectorial for some p G {0} U N) if

• ( i) there are constants a G R and 0 G (ПП ,n) such that

^SAe⁽B) = ^{M ^{G C}: I arg⁽p ^-a)| < ^0,p = a^{} C}p^A(B),

• ( ii) there is a constant K E R₊such that

max {HRL^B)^l(x), ^L^p)(b)^l(y)}<

K

p

П |Mq - al q=0

for any p,q E SA₀(B), q = 0,1,...,p.

Let the operator B be (A,p)-sectorial for some p E {0} U N, then there are degenerate analytic semigroups of operators (see [3, Ch. 3])

X^tMr B'

Г

and

^Y t = ^il LAB ^d*

Г

where t E R₊, and the contour Г C SA0(B) is such that | arg^| ч 0 for p Ч те, p E Г, 0 E (2,n). Denote by X• = {Xt : t E R₊}. Put X⁰= ker X•, X¹= imX•, Y⁰= ker Y•, Y¹= imY• and denote by Ak the restriction of operator A to Xk, and by Bk the restriction of operator B to Xk П dom B, k = 0,1.

There are two approaches to splitting of spaces X and Y. The first approach is outlined in [3], where a stronger condition on operator B (strong (A,p)-sectoriality of operator B) is set. We follow the equivalent approach firstly proposed by G.A. Sviridyuk. Further, we need two conditions:

X⁰Ф X¹= X and Y⁰ф Y¹= Y,                    (A1)

there exists an operator A^-1E L(Y1; X¹).                    (A2)

The equivalence of these approaches was shown in [33]. Note that condition (A1) occurs when the operator B is strongly (A,p)-sectorial on the right (left). Condition (A2) is met either when the operator B is strongly (A, p)-sectorial, or when it is (A, p)-sectorial, (A1) holds and Y¹= im A1. If conditions (A1), (A2) are satisfied and operator B is (A,p) sectorial, then there are projectors P = s- lii0i Xt, Q = s- lim Yt, operators H = B-A₀E L(X⁰) and S = A-1B1 E Cl(X1). Moreover, the operator H is nilpotent of degree p, and the operator S is sectorial.

Further we consider the case n = 1 and equation (10) in the form

Ai: = Bx + y.

Proceed to the study of solvability of the Cauchy problem (11) for equation (30). Similarly to Section 1 for the solvability of the Cauchy problem, it is necessary to construct a condition connecting the initial value and y in the right-hand side of (30). Introduce the set

p

My = {x E X : (I - P)x + ^H^qB^-1(I - Q)y^(q)(0) = 0}.          (31)

q=0

The solvability of the Cauchy problem (11), (30) was studied in [3, 13].

Theorem 13. [3,13] Let the operator B be (A, p)-sectorial for some p E {0}UN, conditions (A1), (A2) be fulfilled and y⁰= (I - Q)y E C^p([0,t]; Y⁰) П C^p+1((0,T); Y⁰), y¹= Qy E

C([0,т]; Y¹). Then for any x₀E My there exists a unique classical solution to (11), (30)

given by

x(t) = X tx₀

-

£ HqB-(I - Q)y(q)(t) + q=0

t

У X ^t-sy(s)ds.

Proceed to more general initial-final condition (13) and the Showalter–Sidorov condition (12). If we use these conditions, we can omit conditions on the initial value x₀, and take it from the whole space X. Suppose that conditions (A1), (B) are satisfied. Construct a relatively spectral projectors [4]

Pfin = Л [ RA(B^,  Qfin =    [ LA(B)d".

2ni J                      2ni J

γ0                                    γ0

It turns out that under (A,p)-sectorial for some p E {0} U N and conditions (A1), (B) being fulfilled PfinP = PPfin = Pfin, QfinQ = QQfin = Qfin. So there exist projectors

Pin = P - Pfin,  Qin = Q - Qfin.

Then Yin = imQin, Y^fin = imQ_fin.

Theorem 14. [14] Let the operator B be (A, p)-sectorial for some p E {0} U N and conditions (A1), (A2), (B) be satisfied. Then for any x0,xT E X and vector function y such that y0 = (I - Q)y E Cp([0,T]; Y0) П Cр+1((0,т); Y0), yin = Qiny E C([0,t]; Yin), yfin = Qfiny E C([0,t]; Yfin) there exists a unique classical solution to (13), (30), given by pt

x(t) = - E H^qB-Py⁰(t) + Xinxo + Zt—syyin(s)ds+ q=0                               0

τ

'■'■ Xt - Zfny y'"№, t where

^Xi^t

•“ = ini I/ RA(B^d" -1 RA(B^d"j Γ                     γ0

,

t ^Zin

= 2—i I У ("A - B) 1^d" - j ("A - B')¹e^dp, j , Γ                          γ0

^Xf^t

'■" = 2П7 у ^RA b"^{1 d}",

Z'in = 77- [("A - B ■

'     2ni J

γ