ON BOUNDARY OPTIMAL CONTROL PROBLEM FOR AN ARTERIAL SYSTEM: FIRST-ORDER OPTIMALITY CONDITIONS

. We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for 1 D system of pseudo-parabolic nonlinear equations with an unbounded coeﬃcient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and ﬁrst-order optimality conditions.

1. Introduction. The main goal of this paper is to study one class of optimal control problems (OCPs) for a viscous Boussinesq system arising in the study of the dynamics of cardiovascular networks. We consider the boundary control problem for a 1D system of coupled PDEs with the Robin-type boundary conditions, describing the dynamics of pressure and flow in the arterial segment. We discuss in this part of paper the existence of optimal solutions and provide a substantial analysis of the first-order optimality conditions. Namely, we deal with the following minimization problem: Minimize J(g, h, η, u) : η(0, ·) = η 0 in Ω, and (g, h) ∈ G ad × H ad ⊂ L 2 (0, T ) × L 2 (0, T ).
Here, β g , β h , and η * are positive constants, and G ad and H ad are the sets of admissible boundary controls. These sets and the rest of notations will be specified in the next section. Optimal control problem (1)-(5) comes from the fluid dynamic models of blood flows in arterial systems. It is well known that the cardiovascular system consists of a pump that propels a viscous liquid (the blood) through a network of flexible tubes. The heart is one key component in the complex control mechanism of maintaining pressure in the vascular system. The aorta is the main artery originating from the left ventricle and then bifurcates to other arteries, and it is identified by several segments (ascending, thoracic, abdominal). The functionality of the aorta, considered as a single segment, is worth exploring from a modeling perspective, in particular in relationship to the presence of the aortic valve.
In the first part of our investigation (see [5]) we make use of the standard viscous hyperbolic system (see [2,21]) which models cross-section area S(x, t) and average velocity u(x, t) in the spatial domain: where (t, x) ∈ Q = (0, L) × (0, T ), f = f (x, t) is a friction force, usually taken to be f = −22µπu/S, µ is the fluid viscosity, P (x, t) is the hydrodynamic pressure, L is the length of an arterial segment, and T = T pulse = 60/(HartRate) is the duration of an entire heartbeat. Here we include the inertial effects of the wall motion, described by the wall displacement η = η(x, t): where r(x, t) is the radius, r 0 = r(x, 0), S 0 = S(x, 0). The fluid structure interaction is modeled using inertial forces, which gives the pressure law Here, P ext is the external pressure, β = E 1−σ 2 h, σ is the Poisson ratio (usually σ 2 = 1 2 ), E is Young modulus, h is the wall thickness, m = ρωh 2 √ πS0 , ρ ω is the density of the wall.
This leads to the following Boussinesq system: where ρ is the blood density. Considering the relation η t = − 1 2 r 0 u x and rearranging terms in u we get the system in the form (2)- (3). It remains to furnish the system by corresponding initial and boundary conditions which we propose to take in the form (3)-(4).
As for the OCP that is related with the arterial system, we are interested in finding the optimal heart rate (HR) which leads to the minimization of the following cost functional The systolic period is taken to be consistently one quarter of T pulse , and P ref = 100 mmHg.
It is easy to note that relations (8)-(9) lead to the following representation for the cost functional (10) Since η t ≈ − 1 2 r 0 u x (see [3]) and we suppose that νη xx should be small enough, it easily follows from (11) that the given cost functional (10) can be reduced to the tracking type (1).
The research in the field of the cardiovascular system is very active (see, for instance the literature describing the dynamics of the vascular network coupled with a heart model, [2,9,10,12,15,16,17,18,19,20,21]). However, there seems to be no studies that focus on both aspects at the same time: a detailed description of the four chambers of the heart and on the spatial dynamics in the aorta. Some numerical aspects of optimizing the dynamics of the pressure and flow in the aorta as well as the heart rate variability, taking into account the elasticity of the aorta together with an aortic valve model at the inflow and a peripheral resistance model at the outflow, based on the discontinuous Galerkin method and a two-step time integration scheme of Adam-Bashfort, were recently treated in [3] for the Boussinesq system like (2). More broadly, theory and applications of optimization and control in spatial networks, basing on the different types of conservation laws have been extensively developed in literature, have been successfully applied to telecommunications, transportation or supply networks ( [6,7]).
From mathematical point of view, the characteristic feature of the Boussinesq system (2) is the fact that it involves a pseudo-parabolic operator with unbounded coefficient in its principle part. In the first part of this paper it was shown that for any pair of boundary controls g ∈ G ad and h ∈ H ad , and for given f ∈ L ∞ (0, T ; L 2 (Ω)), , and δ ∈ L 1 (Ω), the set of feasible solutions to optimal control problem (1)-(5) is non-empty and the corresponding weak solution (η(t), u(t)) of the viscous Boussinesq system (2)-(4) possesses the extra regularity properties η xx , u xt ∈ L 2 (0, T ; L 2 (Ω)), which play a crucial role in the proof of solvability of OCP (1)- (5). In this paper we deal with the existence of optimal solutions and derive the corresponding optimality conditions for the problem (1)- (5). It should be mentioned, that application of Lagrange principle requires even higher smoothness of solutions to the initial Boussinesq system (2)-(4). In order to avoid such limitations, we deal with a simplified version of the initial optimal control problem (2)-(4) (see (39), argumentation above and [3,5] for physical description of the considered model). Also, in the second part of the paper, in order to provide the thorough substantiation of the first-order optimality conditions to the considered OCP, we make the special assumption for δ to be an element of the class H 1 (Ω). Since the coefficient δ depends on such indicators as wall thickness, density of the wall and blood density, i.e. indicators varying slowly and smoothly, such assumption seems justified.
We also make use of the weighted Sobolev space V δ as the set of functions u ∈ V for which the norm is finite. Note that due to the following estimate, V δ is complete with respect to the norm · V,δ : Recall that V 0 , V , and, hence, V δ are continuously embedded into C(Ω), see [1,14] for instance. Since δ, δ −1 ∈ L 1 (Ω), it follows that V δ is a uniformly convex separable Banach space [14]. Moreover, in view of the estimate (12), the embedding V δ → H is continuous and dense. Hence, H = H * is densely and continuously embedded in V * δ , and, therefore, V δ → H → V * δ is a Hilbert triplet (see [11] for the details). Let us recall some well-known inequalities, that will be useful in the sequel (see [5]). • By L 2 (0, T ; V 0 ) we denote the space of measurable abstract functions (equivalence classes) u : By analogy we can define the spaces L 2 (0, T ; V δ ), L ∞ (0, T ; H), L ∞ (0, T ; V δ ), and C([0, T ]; H) (for the details, we refer to [8]). In what follows, when t is fixed, the expression u(t) stands for the function u(t, ·) considered as a function in Ω with values into a suitable functional space. When we adopt this convention, we write u(t) instead of u(t, x) andu instead of u t for the weak derivative of u in the sense of distribution where ·, · V * ;V denotes the pairing between V * and V . We also make use of the following Hilbert spaces , supplied with their common inner product, see [8, p. 473], for instance.

Remark 1.
The following result is fundamental (see [8] 3. On solvability of optimal control problem (1)- (5). Let ν > 0 be a viscosity parameter, and let be given distributions. In particular, f stands for a fixed forcing term, u Ω and η Q are given desired states for the wall displacement and average velocity, respectively, α Ω and α Q are non-negative weights (without loss of generality we suppose that α Q is a nonnegative constant function on [0, T ] × [0, L]), u 0 and η 0 are given initial states, and δ is a singular (possibly locally unbounded) weight function such that δ(x) ≥ δ 0 > 0 for a.e. x ∈ Ω. We assume that the sets of admissible boundary controls G ad and H ad are given as follows The optimal control problem we consider in this paper is to minimize the discrepancy between the given distributions (u Ω , η Q ) ∈ L 2 (Ω) × L 2 (Q) and the pair of distributions (u(T ), η(t) + η tt (t)) (see, for instance, [5] for the physical interpretation), where (η(t), u(t)) is the solution of a viscous Boussinesq system, by an appropriate choice of boundary controls g ∈ G ad and h ∈ H ad . Namely, we deal with the minimization problem (1)-(5).
Definition 3.1. We say that, for given boundary controls g ∈ G ad and h ∈ H ad , a couple of functions (η(t), u(t)) is a weak solution to the initial-boundary value problem (2)-(4) if and the following relations hold true for all ϕ ∈ V 0 and ψ ∈ V δ and a.e. t ∈ [0, T ].

Remark 2.
Let us mention that if we multiply the left-and right-hand sides of equations (23)-(24) by function χ ∈ L 2 (0, T ) and integrate the result over the interval (0, T ), all integrals are finite. Moreover, closely following the arguments of Korpusov and Sveshnikov (see [13]), it can be shown that the weak solution to (2)-(4) in the sense of Definition 3.1 is equivalent to the following one: (η(t), u(t)) is a weak solution to the initial-boundary value problem (2)-(4) if the conditions (19)-(22) hold true and where . Assume that the conditions (15)-(17) hold true. Let g ∈ G ad and h ∈ H ad be an arbitrary pair of admissible boundary controls. Then there exists a unique solution (η(·), u(·)) of the system (2)-(4) in the sense of Definition 3.1 such that and there exists a constant D * > 0 depending only on initial data (15), (17) and control constrains h 1 , g 1 , satisfying the estimates We also define the feasible set to the problem (1)-(5), (18) as follows: We say that a tuple g 0 , h 0 , η 0 , u 0 ∈ Ξ is an optimal solution to the problem In [5] it was shown that Ξ = ∅ and Ξ λ = {(g, h, η, u) ∈ Ξ : While proving these hypotheses, the authors in [5] obtained a series of useful estimates for the weak solutions to initial-boundary value problem (2)-(4). Lemma 3.3. [5, Lemmas 6.3 and 6.5 along with Remark 6.5] Let g ∈ G ad and h ∈ H ad be an arbitrary pair of admissible boundary controls. Let (η(·), u(·)) = (w(·) + η * , u(·)) be the corresponding weak solution to the system (2)-(4) in the sense of Definition 3.1. Under assumptions (15)-(17), there exist positive constants C 1 , C 2 , C 3 depending on the initial data only such that for a.a. t ∈ [0, T ] In the context of solvability of OCP (18)- (5), the regularity of the solutions of the corresponding initial-boundary value problem (2)-(4) plays a crucial role. Now we proceed with the result concerning existence of optimal solutions to OCP (1)-(5), (18).
It is worth to mention here that, in fact, (δ(u 0 ) x ) x ∈ (H 1 (Ω)) * because the element δ(u 0 ) x belongs to L 2 (Ω). Indeed, It remains to note that the property T 0 Let us consider two operators γ 1 and γ 2 that define the restriction of the functions from V = H 1 (Ω) to the boundary ∂Ω = {x = L, x = 0}, respectively (i.e. γ 1 [u(t, ·)] = u(t, L) and γ 2 [u(t, ·)] = u(t, 0)). Also we put into consideration two operators A, B : defined on the set of vector functions p = (p, q) t ∈ L 2 (0, (37) Here, we use the fact that . Then the following result holds true. (36), satisfies the following conditions: A(t) is radially continuous, i.e. for any fixed v 1 , v 2 ∈ V 0 × V δ := V and almost each t ∈ (0, T ) the real-valued function s → A(t)(v 1 + sv 2 ), v 2 V * ; V is continuous in [0, 1]; for some constant C and some function g ∈ L 2 (0, T ) it is strictly monotone uniformly with respect to t ∈ [0, T ] in the following sense: there exists a constant m > 0, independent of t, such that Moreover, the operator B : Proof. Since the radial continuity of operator A is obvious, we begin with the proof of the second property. Let v = (v, w), z = (z, y) ∈ V be arbitrary elements. Then As for the monotonicity property, for every p 1 ,

It remains to show the Lipschitz continuity of operator B(t). With that in mind we consider three vector-valued functions
, z 2 (t)) H + (σ 1 (t) + u 0 (t, L))(v 2 (t, L) − w 2 (t, L))z 2 (t, L) Taking into account the continuous embedding V δ , V 0 → C(Ω) and the corresponding inequality we finally have where L = max{C 1 ; C 2 } and ). This concludes the proof.

Lemma 4.2. Operator
which is defined by (36), is radially continuous, strictly monotone and there exists an inverse Lipschitz-continuous operator and for all f ∈ L 2 (0, T ; Proof. It is easy to see that the action of operator A(t) on element p = (p, q) t can be also given by the rule: It is easy to see, that A 1 (t) is the identity operator. Therefore, A −1 1 (t) ≡ A 1 (t). As for the operator A 2 (t), it is strongly monotone for all t ∈ [0, T ] because

Moreover, A 2 (t) satisfies all preconditions of [11, Lemma 2.2] that establishes the existence of a Lipschitz continuous inverse operator
The proof is complete.
Before proceeding further, we make use of the following result concerning the solvability of Cauchy problems for pseudoparabolic equations (for the proof we refer to [11,Theorem 2.4]).

Theorem 4.3. For operators
defined in (36),(37), and for any admits a unique solution.
Notice that, for a fixed t, we have u ∈ V ⊂ C(Ω) and w ∈ V 0 ⊂ C(Ω), hence, the inner products (w x (t)u(t) + w(t)u x (t), p(t)) H and (u(t)u x (t), q(t)) H are correctly defined almost everywhere in [0, T ].
Further we make use of the following relation η t = − 1 2 r 0 u x that was introduced in [3]. Substituting this one to (2), we have νη xx = (ηu) x = η x u + u x η. Also, to simplify the deduction and in order to avoid the demanding of the increased smoothness on solutions of the initial Boussinesq system (2)-(5), we use (see [4] and [5]) elastic model for the hydrodynamic pressure Indeed, if we suppose the wall thickness h to be small enough, the last term in the inertial model (38) appears negligible. As a result, the cost functional J(g, h, w, u), where η = w + η * , takes the form In order to formulate the conjugate system for an optimal solution (g 0 , h 0 , η 0 , u 0 ), where η 0 = w 0 + η * , we have to find the Fréchet differentials L w z and L u v, where z ∈ W 0 (0, T ) ∩ L 2 (0, T ; H 2 (Ω) ∩ V 0 ) and v ∈ W 1,∞ (0, T ; V δ ) × L 2 (0, T ).
Taking into account the definition of the Fréchet derivative of nonlinear mappings, we get J(g, h, w + z, u) = J(g, h, w, u) + J w z + R 0 (w, z), where R 0 (w, z) stands for the remainder, which takes the form and