OPTIMAL CONTROL OF SOLUTIONS OF A MULTIPOINT INITIAL-FINAL PROBLEM FOR NON-AUTONOMOUS EVOLUTIONARY SOBOLEV TYPE EQUATION

. The paper presents suﬃcient conditions for existence of an optimal control of solutions to a non-autonomous degenerate operator-diﬀerential evo- lution equation. We construct families of operators that solve this equation, as well as classical and strong solutions of the multipoint initial-ﬁnal problem for the equation. We show that there exists a solution of an optimal control problem for a given operator-diﬀerential equation with a multipoint initial-ﬁnal condition. The paper, in addition to the introduction and the bibliography, contains ﬁve sections. The ﬁrst three parts contain information about the solv- ability of the multipoint initial-ﬁnal problem for a non-autonomous equation. The fourth section presents the main result of the article; that is, a theorem on existence of optimal control of solutions to a multipoint initial-ﬁnal problem. In the ﬁfth part, the optimal control problem for the non-autonomous modiﬁed Chen – Gurtin model with the multipoint initial-ﬁnal condition is investigated on the basis of the obtained abstract results.

The required complex-valued function x(r, t) in (1) describes the dynamics of the process, and the complex-valued function u(r, t) describes an external effect on the system. In the special case of d = 0 equation (1) describes the process of heat conduction with "two temperatures" [6], as well as the dynamics of fluid pressure in a cracks-porous medium [5] and the process of moisture transfer in soil [12]. Also, equation (1) is a particular case of the linearized classical Ginzburg -Landau equation [2]. If d = 0, then (1) is also the linearization of the Oskolkov equation [21].
In this paper, we consider the non-autonomous modified Chen -Gurtin equation of the form which allows for taking into account the change in the parameters of the system over time. Note that (3) is a generalization of equation (1), which can be obtained from (3) for a ≡ 1. In the case d = 0, the autonomous model (1), (2) was investigated on the finite connected oriented graphs (i.e., m = 1), for example, in [28], and the non-autonomous model (2), (3) was investigated in [23]. We also note that if the operator on the left-hand side of (1) is invertible, then (1), (2) can be investigated by methods of evolutionary semigroups (see, for example, [15,13,22,4,3]). Problem (2), (3) is reduced to a non-autonomous evolutionary equation where the operators L ∈ L(X; Y) and B ∈ L(U; Y) (i.e., linear and continuous), and the operator M ∈ Cl(X; Y) (i.e., linear, closed, densely defined in X). Equations of the form (4) with kerL = {0} are called degenerate differential equations [8] or not solvable with respect to the highest-order derivative (see the historical review in [7]). Recently, the term Sobolev type equation, which was proposed by R. Showalter [25,26], has been widely used (see, for example, [29,1]). This research is in the framework of the theory of degenerate operator semigroups, which was founded by G.A. Sviridyuk [29] and is currently actively developed, for example, in [35,18,27]. Moreover, this theory was generalized for spaces of random processes [9,10]. Note that problem (2), (3) is considered in the frame of equation (4), and refers to the relatively p-radial case [29,30] (p ∈ N 0 ). Here and below N 0 ≡ {0}∪ N. In this case the homogeneous (g(t) ≡ 0, u(t) ≡ 0) autonomous (a(t) ≡ 1) equation (4) has a degenerate strongly continuous semigroup of solving operators [11,29,Capter 2]. The main feature of such semigroups is that their operators are degenerated not only on the kernel of the operator L, but also on the linear span of M -adjoint vectors of height at most p of operator L [11,29,Capter 2]. Note that the degenerate strongly continuous semigroup of solving operators was considered for the first time in [30]. A. Favini and A. Yagi in their work [8] investigated the solvability of degenerate equations using theory of the multivalued operators and differential inclusions. Their results [8] intersect Sobolev type equations theory [29] only in case p = 0. Fix x j ∈ X and τ j ∈ R + , j = 0, n, such that τ 0 = 0 and τ j−1 < τ j (1 ≤ j ≤ n) and consider the multipoint initial-final problem [33] lim for equation (4), where P j are some spectral projections. Note that the general formulation of the multipoint initial-final problem (4), (5) in the relatively p-radial case was first introduced in [34]. If n = 1, then (5) is called an initial-final condition. Initial-final problem was considered earlier for autonomous Sobolev type equations of the first (see the review in [33]) and higher [36] orders. Note that the multipoint initial-final conditions are also a generalization of the Showalter -Sidorov condition [31], which corresponds to n = 0 (for more details, see [14]). In Hilbert spaces X, Y and U consider the optimal control problem for (4), (5): to find a vector-function v ∈ U ad such that the relation holds, where all pairs (x(u), u) satisfy the multipoint initial-final problem (4), (5).
Here J(u) is a cost functional of a special form, and U ad is a closed and convex subset of admissible controls in U. The optimal control problem for the linear autonomous Sobolev-type equation with the Cauchy condition was first considered in [32]. Later this problem was studied in different cases [23,18,36,19,20]. Separately note the paper [20], where the optimal control problem for the autonomous Chen -Gurtin equation (1) with boundary (2) and initial-final condition was investigated.
The paper, in addition to the introduction and the bibliography, contains five parts. The first part provides the necessary information about the theory of relatively p-radial operators. In the second and the third parts, the solvability of the homogeneous non-autonomous equation (4) and the multipoint initial-final problem (5) for equation (4) is considered. The fourth part presents the main result of the article that is the theorem on existence of optimal control of solutions to the multipoint initial-final problem (4), (5). In the fifth part the obtained abstract results are applied to study of the optimal control problem for the non-autonomous modified Chen -Gurtin equation (3) with boundary conditions (2) and multipoint initial-final conditions (5).
1. Relatively p-radial operators. Let us recall the standard definitions and notations of the theory of relatively p-radial operators [29,30,11]. The proofs of the statements of this part can be found in [11,29,Chapter 2].
Let X and Y be Banach spaces, L(X; Y) be the space of continuous linear operators, and Cl(X; Y) be the space of closed densely defined linear operators. The operators L ∈ L(X; Y) and M ∈ Cl(X; Y) are given. Following [29,30,11], the sets ρ L (M ) = {µ ∈ C : (µL − M ) −1 ∈ L(Y; X)} and σ L (M ) = C \ ρ L (M ) are called L-resolvent set and L-spectrum of the operator M , respectively. By the results of [29,30,11] We call the largest index of a vector in a chain (starting from 0) its height, thus the eigenvector is an M -adjoint vector of height 0. Using this lemma, we obtain the statement of the following lemma.
Remark 1.4. The concept of (L, p)-radial operator M is a generalization of the conditions of the Hille -Yosida theorem [13, Chapter 12, § 3]. More details can be found in [11].
By virtue of Lemma 1.2, introduce the notation (ii) X t is strongly continuous for t > 0 and there exists lim Definition 1.6. A semigroup {X t ∈ L(X) : t ∈ R + } is called exponentially bounded with constants C and α, if the following condition holds: . Moreover, this semigroup is exponentially bounded with the constants K, α from Definition 1.3.
Here and below the notation s-lim means the limit in the strong topology of the corresponding space of operators.
Example. To illustrate these concepts, we consider a degenerate system of ordinary differential equations For such operators L and M , the L-resolvent, the right L-resolvent and the left L-resolvent of the operator M have the form Note that ker L = span Then the projection P from Remark 1.9 has the form P = Introduce the condition Let L 1 (M 1 ) be the restriction of the operator L (M ) to X 1 (dom M ∩ X 1 ). Also, introduce one more additional condition: there exists an operator L −1 Theorem 1.10. [29, Chapter 2] Let the operator M be (L, p)-radial (p ∈ N 0 ) and conditions (7), (8) be satisfied. Then 2. C 0 -semiflows of solving operators for non-autonomous evolution equations. Let X, Y be Banach spaces, L ∈ L(X; Y) and M ∈ Cl(X; Y) be given operators.
is called a degenerate strongly continuous semiflow (briefly, a degenerate C 0 -semiflow) of operators, if the following conditions are satisfied: (ii) X(t, ς) are strongly continuous for all t, ς > 0 and for all ς ≥ 0 there exists lim t→ς+ X(t, ς)x = x for all x from a dense in linear subspace X 1 .
Proof. Show that X(t, ς) is a semiflow of operators. Let us show that condition (i) of Definition 2.1 holds. Let 0 ≤ ς < τ < t, then where the notations t 1 = t τ a(ζ)dζ, t 2 = τ ς a(ζ)dζ and the form of semigroup operators (6) are used. By the conditions of the theorem t 1 , t 2 > 0, and, consequently, from condition (i) of Definition 1.5, we obtain Let us show that condition (ii) holds. For ς < t we replace t 3 = t ς a(ζ)dζ > 0 in (9) and in view of (6), we obtain and, consequently, the operator X(t, ς) is strongly continuous by the condition (ii) of Definition 1.5. From (10) for ς ≥ 0 and x ∈ X we obtain in view of Remark 1.9.
Remark 2.3. In view of Theorem 1.10, we obtain By Remark 1.9, from this formula we get X(t, ς) where S = L −1 1 M 1 ∈ Cl(X 1 ). Remark 2.4. By analogy with (9), the semiflow of operators in the space L(Y) can be also given by the following formula: The proof of the semiflow properties for operators (11) is analogous to the proof of Theorem 2.2.
On the interval (t 0 , T ] ⊂ R + consider the weakened Cauchy problem (in sense of S.G. Krein [16, 15, for a homogeneous non-autonomous equation where the function a : [t 0 , T ] → R + will be defined later. Definition 2.6. A closed set P ⊂ X is called a phase space of (13), if (i) any solution x(t) of (13) lies in P, i.e. x(t) ∈ P ∀t ∈ (t 0 , T ]; (ii) for any x t0 from P there exists a unique solution of the weakened Cauchy problem (12) for equation (13).
Together with equation (13) we consider the equivalent equation Theorem 2.7. Let M be an (L, p)-radial operator (p ∈ N 0 ), conditions (7), (8) be satisfied and a ∈ C(R, R + ). Then the set X 1 is the phase space of equation (13), and the set Y 1 is the phase space of equation (14).
Proof. In view of Theorem 1.10, equation (13) is equivalent to the systeṁ where x 1 = P x ∈ X 1 , x 0 = (I X − P )x ∈ X 0 and P is taken from Remark 1.9. From the second equation in (15), we obtain Differentiate (16) and apply the operator 1 a H, in view of (16), we obtain 1 a Repeating this procedure p times we obtain Therefore, since the operator H is nilpotent, we get that x 0 = 0. Now consider the first equation of (15). The solution of this equation has the form [22, Theorem 6.1.5] where S = L −1 1 M 1 ∈ Cl(X 1 ). Hence, for x t0 ∈ X 1 , in view of Remark 2.3, the solution of (13) has the form x(t) = X(t, t 0 )x t0 ∈ X 1 . Definition 2.8. A semiflow of operators X(·, ·) : R + ×R + → L(X) is called the semiflow of solving operators of (13), if for any x t0 ∈ X the vector-function x(t) = X(t, t 0 )x t0 is a solution of (13) (in the sense of Definition 2.5).
Remark 2.10. Similar results for the relatively p-bounded case were obtained in [24].
3. Strong solutions of the multipoint initial-final problem for the nonautonomous evolution equation. Let X and Y be Hilbert spaces, L ∈ L(X; Y), M ∈ Cl(X; Y) be given operators, M be an (L, p)-radial operator (p ∈ N 0 ) and conditions (7), (8) be satisfied. On the interval (τ 0 , τ n ) ⊂ R + consider the inhomogeneous equation Additionally we assume the conditions: In view of holomorphy of the relative resolvents, there exist projections P j ∈ L(X) and Q j ∈ L(Y) (j = 0, n), which have the form, [34], Q j , where P and Q are defined in Remark 1.9.
Introduce the subspaces X 1 j = im P j , Y 1 j = im Q j , j = 0, n. By construction, Let L 1j be the restriction of the operator L to X 1 j , j = 0, n, and M 1j be the restriction of the operator M to dom M ∩ X 1 j , j = 0, n. Since it is easy to show that P j ϕ ∈ dom M , if ϕ ∈ dom M , then dom M 1j = dom M ∩ X 1 j is dense in X 1 j , j = 0, n. Theorem 3.1. (The generalized splitting theorem) [34]. Let L ∈ L(X; Y) and M ∈ Cl(X; Y), the operator M be (L, p)-radial (p ∈ N 0 ) and conditions (7), (8) and (18) be satisfied. Then Fix x j ∈ X and τ j ∈ R + (j = 0, n) such that τ 0 = 0 and τ j−1 < τ j (1 ≤ j ≤ n). For such data, consider the multipoint initial-final problem [33] lim for equation (17).  (17) is called a classical solution of the multipoint initial-final problem (17), (19), if it satisfies (19).
The solution of (21) for j = 0 has the form which is directly verified, similarly to [22,Theorem 6.1.5]. Now consider problem (22). From (22) we obtain Using (23) we get the equality Hẋ and, in view of (22), we obtain Repeating these transformations and using the nilpotency of the operator H, we obtain Therefore, the solution of (17), (19) has the form x(t) = x 0 (t) + n j=0 x 1 j (t).
Proof. In view of Theorem 3.3 it is clear that the function, given by (20), satisfies (17), (19). We verify that the function x(t) belongs to the required class. Suppose that g(t) satisfies the conditions of the theorem; then the third term of (20) belongs to H 1 (X), and functions belong to L 2 (τ 0 , τ n ; Y) in view of the properties of the semiflow X(t, s), the fact that the operator M is closed and a(t) is continuous for t ∈ [τ 0 , τ n ]. It also follows from the last equality that x(t) satisfies (17), (19). In view of Theorem 3.3, the obtained solution is unique.
Remark 3.6. It is clear that if vector-function g ∈ H p+1 (Y), then condition (24) is fulfilled automatically.
Let Z be a Hilbert space and the operator C ∈ L(X; Z). Consider the optimal control problem for (19), (25) in the form where z = Cx, N q ∈ L(U) (q = 0, 1, . . . , p + 1) are self-adjoint and positively defined operators, z d = z d (t) ∈ Z is a target observation. Note that if x ∈ H 1 (X), then z ∈ H 1 (Z). Similarly to H p+1 (Y), consider the space H p+1 (U), which is the Hilbert space in view of the Hilbert property of U. Let U ad be a non-empty, closed and convex subset of the space H p+1 (U), which is the set of admissible controls.
Proof. Using the mapping D from Lemma 4.1, rewrite the cost functional (28) in the form Denotex = Dû forû ≡ 0. In other words,x is given by (20). Hence, is a bilinear continuous coercive form on H p+1 (U) and is a linear and continuous on H p+1 (U) form. Therefore, the statement of the theorem follows from [17, Chapter 1].

5.
Optimal control of solutions of the Chen -Gurtin equation with complex coefficients.