Observability and controllability analysis of blood flow network

In this paper, we consider the initial-boundary value problem of a 
binary bifurcation model of the human arterial system. Firstly, we 
obtain a new pressure coupling condition at the junction based on 
the mass and energy conservation law. Then, we prove that the 
linearization system is interior well-posed and $L^2$ well-posed by 
using the semigroup theory of bounded linear operators. Further, by 
a complete spectral analysis for the system operator, we prove the 
completeness and Riesz basis property of the (generalized) 
eigenvectors of the system operator. Finally, we present some 
results on the boundary exact controllability and the boundary exact 
observability for the system.


1.
Introduction. In recent years, many doctors and researchers have devoted themselves to the prevention and treatment of cardiovascular diseases. For the blood flow in a single vessel shown as Figure 1(a), many models were used to obtain the physical property of the blood vessel and the interaction between blood flow and vessel in the previous literatures (e.g., see [1,3]). And some mathematical issues and the well posedness analysis of these multiscale models were presented (see, [2,4,5]). In fact, the arterial circulation is often regarded as a network of compliant vessels. Wave propagation in the arterial tree is very important to the understanding of arterial circulation, especially at bifurcation (see, [6,8,10,11]). Medical experiments and technology (see, e.g. [7,12]) also were devoted to the study and found that geometry played a key role in determining the nature of haemodynamic patterns at the human carotid arterial bifurcation.
In this paper, we consider a new problem. The mathematical models previously used can be regarded as an initial and boundary value problem of partial differential equations. The solution including its numerical solution makes sense only when the initial data and boundary value are given. In the practical situation, we have no initial state. In most cases, we have some measure-datum. Therefore, how to find some property of the blood flow network from measurement, or the observability of the system briefly, is an interesting study. In the present paper, we will pay our attention to the observability and controllability of a simple vascular bifurcation system shown as Figure 1(b).

CHUN ZONG AND GEN QI XU
A(x,t) X (a) As usual, we describe every vessel by the following one-dimensional quasilinear system: that describes conservation of mass and balance of axial momentum in terms of the cross-section area A(x, t), internal pressure over the cross-section P (x, t), average velocity u(x, t) and the flow rate Q(x, t) = A(x, t)u(x, t), where η stands for the kinetic energy coefficient, x denotes the axial direction, ρ is the blood density and K is related to the viscosity assumed in the vessel (see, e.g., [4,5,6]). To simplify the model, we introduce the following constitutive law for the pressure to system (1) (see, e.g., [4,13,14]): where A 0 , E, h and σ are the cross-section constant at rest (u = 0), the Young modulus, the thickness and Poisson coefficient of the vessel wall, respectively. This expression entails that the pressure is a linear function of the vessel radius (see, e.g., [13,15]). Using the pressure law (2), we get And we choose P and Q as variables. By linearizing the system (1) around a constant state (A, u) = (A 0 , 0), we obtain a linearization model of the single vessel as follows As pointed out in [6], the linearization model is reasonable as it can reproduce most of the essential feature of the blood flow. In particular, it is used to model the whole systemic tree. For the binary vascular bifurcation (see, Figure 1(b)) under consideration, let P i and Q i be the pressure and flow rate, where i = 1 denotes the first branch and i = 2, 3 are the binary branches. Without loss of generality, we can assume that P i and Q i are the real measurable functions on interval [0, 1] and the direction of parameter increasing coincides with that of blood flow for i ∈ {1, 2, 3}. For the connective condition at the bifurcation, the mass balance condition implies that we can choose constant α ∈ (0, 1) as the mass partition coefficient (see, e.g., [6,9]). Then the flow rates satisfy conditions Q 2 (0, t) = αQ 1 (1, t), Q 3 (0, t) = (1 − α)Q 1 (1, t). Thus the model under consideration can be described as follows In [3,14], the energy functional is defined by that provides energy estimates for the linearization system as the nonlinear entropy function guarantees the stability of the solution to the nonlinear system. Taking the derivative of (5) with respect to t and using the equations (4), we get where the term P 1 (0, t)Q 1 (0, t) is the input energy, the terms P 2 (1, t)Q 2 (1, t) and P 3 (1, t)Q 3 (1, t) are the output energy, and the term is the energy dissipation rate due to the blood viscosity. Suppose that there is no energy loss at the bifurcation, we find out the pressure relation It's worth pointing out that the result (7) is different from that used before (see, e.g., [6,8,10,11]). Based on the flow rate distribution and pressure coupling contiguity condition, we can model the arteries at the bifurcation. Furthermore, supplementing the initial and boundary conditions, we get a complete description for a one-dimensional distributed parameter model of binary vascular network where u 1 (t) is input flow rate and u 2 (t), u 3 (t) are the output pressures, that are regarded as the control variables. Moreover, we have observation data about the pressure input P 1 (0, t) and the flow rate outputs Q 2 (1, t) and Q 3 (1, t), and we set So far we have obtained a complete control-observation system described by (8) and (9).
The rest of this paper is organized as follows. In Section 2, we formulate the system (8) into a Hilbert space H and then prove the interior well-posedness and L 2 well-posedness of the network system by using the semigroup theory of bounded linear operators. In Section 3, we carry out a complete spectral analysis for the system operator. And we prove that the spectrum consists of all isolated eigenvalues, that is a union of finite many separated sets located in a strip parallel to the imaginary axis. In particular, we obtain some results on the multiplicity and separability of eigenvalues. In Section 4, we present the completeness and Riesz basis property of (generalized) eigenvectors of the system operator A. Finally, in Section 5, we discuss the exact controllability and exact observability of the system and then give the main result in Theorem5.3.

2.
Well-posedness. In this section, we study the well-posedness of the linearization blood flow network (8). At first we reformulate system (8) into an appropriate Hilbert state space. For the sake of simplicity, we introduce some notations used later. Set 3 × 3 diagonal matrices equipped with an inner product Obviously, H is a Hilbert space.
Define the system operator A in H by It is easy to check that A is a closed and densely defined linear operator in H.
A simple calculation gives that the dual operator A * of A is We take the control functions space as L 2 loc (R + , U) = L 2 loc (R + , C 3 ). For any T > 0, and for U (t) = (u 1 (t), u 2 (t), u 3 (t)) ∈ L 2 loc (R + , U), its local norm is defined by Furthermore, we define control input operator B : U = C 3 → H −1 by where H −1 = (D(A), R(λ, A) · ) with λ being a resolvent point of the system operator A, and δ(x), δ(x − 1) are the Dirac Delta function.
With the help of notations above, we can rewrite (8) into an abstract evolutionary equation in H: where U (t) = (u 1 (t), u 2 (t), u 3 (t)), X(t) = (P (x, t), Q(x, t)) T , and We take the observation space as . For each T > 0, and for Y (t) = (y 1 (t), y 2 (t), y 3 (t)) ∈ L 2 loc (R + , Y), the local norm is defined by The observation operator C : D(A) → C 3 is defined by Clearly, C has an extension on the continuous function space Thus the abstract form of the control-observation system (8) and (9) in Hilbert 2.1. Interior well-posedness of system (15). The interior well-posedness of a system means that when U (t) ≡ 0, the Cauchy problem has unique a solution for each initial X 0 ∈ D(A) and the solution depends continuously on the initial data. In other words, A generates a strongly continuous semigroup of bounded linear operators (briefly, C 0 semigroup). In this subsection, we mainly discuss the interior well-posedness of the system (13) or (15).
Firstly, we have the following lemmas.
Lemma 2.1. Let H and A be defined by (10) and (11) respectively. Then A is dissipative in H.

Lemma 2.2.
Let H and A be defined by (10) and (11) respectively. Then 0 ∈ ρ(A). Furthermore, A −1 is compact on H, and hence the spectra of A consist of all isolated eigenvalues of finite multiplicity, i.e., σ(A) = σ p (A).
Proof. For any fixed (u(x), v(x)) T ∈ H, we consider the solvability of the equations that is, Solving the differential equations above we find out the formal solutions Inserting above into the boundary conditions yields A direct calculation shows that (I − C) −1 = I + C and (I − C T ) −1 = I + C T . Thus, Inserting above into the formal solutions, we can get a unique solution pair to (16) The closed operator Theorem asserts that 0 ∈ ρ(A). By the expression (17), (f, g) = A −1 (u, v) is an integral operator of continuous kernel function and so A −1 is compact on H. Thus, the spectra of A consist of all isolated eigenvalues of finite multiplicity (see [16]) and σ(A) = σ p (A). The proof is then complete. Lemma 2.1 and Lemma 2.2 together with the Lumer-Phillips Theorem in [17] assert the following result. Theorem 2.3. Let H and A be defined by (10) and (11) respectively. Then A generates a C 0 semigroup of contractions on H, and hence the system (15) is interior well-posed.
2.2. L 2 well-posedness of system (15). L 2 well-posedness of a system means that if a system has local L 2 input, then its output (observation) also is L 2 locally. Sometimes, it also is called the exterior well-posedness of the system. In this section, we shall discuss the L 2 well-posedness of the system (15).
We begin with introducing the following lemma that gives a sufficient and necessary condition for L 2 well-posedness of a system(see [18, section 4, pp.281-284]). Lemma 2.4. Let X, U and Y be Hilbert spaces, L 2 loc (R + , U) and L 2 loc (R + , Y) be abstract function spaces consisting of all local square integrable functions. Suppose that A generates a C 0 semigroup T (t), t ≥ 0. Let B be a linear operator from U to X −1 and C also be a linear operator from D(A) to Y.
is L 2 well-posed if and only if, for any given t > 0, there exists a constant M t > 0, which depends on the time t, such that L 2 well-posedness implies that the control operator B and the observation operator C are admissible for C 0 semigroup T = (T (t)) t>0 . B is an admissible control operator for T means that for ∀t > 0 and ∀u ∈ L 2 loc (R + ; U), the mild solution to (13) is given by x(t) = T (t)x 0 + t 0 T (t − s)Bu(s)ds; C is an admissible observation operator for T implies that, for ∀x 0 ∈ D(A), the mapping x 0 → CT (t)x 0 can be extended to a bounded mapping from X to L 2 loc (R + ; Y) (see [19, chapter 4 , pp.128-131]).
Using the multiplier method, we can prove the L 2 well-posedness of controlobservation system (15).
Since the proof of Theorem 2.5 has long and complicated calculation, we postpone the detail in Appendix A.
3. Spectral analysis of A. In this section, we shall discuss the spectral property of the system operator and its eigenvectors. Firstly, we observe that D(A) = D(A * ) (see (11) and (12)) and for (f, g) ∈ D(A),

So we can define two operators in H by
and Obviously, the following result is true.
Theorem 3.1. Let A 0 and A 1 be defined by (18) and (19) respectively. Then A 0 is a skew adjoint operator and A 1 is a bounded linear operator on H. In particular, Since A 0 is a skew adjoint operator, for any λ ∈ iR, we have ||R(λ, A 0 )|| ≤ 1 | λ| . So when | λ| > ||A 1 ||, we have λ ∈ ρ(A) and From Lemma 2.1 we know that ρ(A) ⊂ {λ ∈ C λ ≤ 0}. Therefore, we have the following result. 3.1. Eigenvalues of A. In this subsection, we shall determine the distribution of spectra σ(A). In Lemma 2.2, we have proved that σ(A) = σ p (A). Therefore, we only need to discuss the eigenvalue problem of A.
Let λ ∈ C be an eigenvalue of A and (P, Q) T be the corresponding eigenvector.
In what follows, we calculate the det(D(λ)). Since and e xB is a diagonal matrix, so . Thus we can write D(λ) as a product of two matrices Obviously, det(D(λ)) = 0 if and only if det(d(λ)) = 0. Since Thus we find out Thus we get an explicate expression for det(d(λ)): Clearly, the spectra of A are determined by det(d(λ)) = 0. When λ → +∞, it holds that e −B(λ) → 0, and hence

3.2.
Eigenvectors. In this subsection, we shall discuss the eigenvectors of A, that is equivalent to determine the vectors P (0) and Q(0) from equation The equations (22) If λ ∈ C satisfies det d(λ) = 0, then (26) have nonzero solutions. In what follows, we classify four cases to determine the solution according to whether or not λ makes d 6 (λ) = 0 and d 7 (λ) = 0 respectively.
3.2.1. Case 1: d 6 (λ) = 0 and d 7 (λ) = 0. In this case, according to (26) we have Substituting above into the first equation in (26) leads to = 0, which means that P 1 (0) is an arbitrary constant. Thus the nonzero solution of (26) is and the vector (Q(0), P (0)) T is given by Since there is only one arbitrary parameter ξ, so the eigen-space of A corresponding to λ is one-dimensional (see, Eqs. (20)).
If A 10 = A 20 , by a complex calculation we see that only if all parameters satisfy the following equality the function equations have a solution. Otherwise, the function equations have no solution.
From above we get
Hence m and n satisfy condition If this condition is fulfilled, then it holds that According to the discussions of case 2 and case 3, we can assume d 4 (λ) = 0, d 5 (λ) = 0. From (26) we get The above discussions show that the following result is true.
Theorem 3.4. Let H and A be defined by (10) and (11) respectively. Then for each λ ∈ σ(A), the corresponding eigen-space N (λI − A) is one dimensional.
Note that Therefore, the eigenvector (P (x, λ), Q(x, λ)) T can be rewritten by 3.3. Multiplicity and separability of eigenvalues. In this subsection, we shall discuss the algebraic multiplicity and separability of eigenvalues of A. In the sequel, we simply denote G(λ) = det(d(λ)). At first, we have the following result.
Theorem 3.5. Let H and A be defined by (10) and (11) respectively. The for any λ ∈ ρ(A), and (f, g) ∈ H, we have the resolvent expression where E(λ)(f, g) is an exponential-type H-valued entire function with respect to λ.
This result is a direct verification(we omit the details). As a direct consequence of Theorem 3.5, we have the following corollary. Corollary 1. Let H and A be defined by (10) and (11) respectively. Then for λ ∈ σ(A) = σ p (A), the multiplicity of the λ is equal to the multiplicity of zeros of G(λ).
In what follows, we shall discuss the multiplicity and separability of eigenvalues of A based on Corollary 1. We classify three cases according to whether the three vessels are the same or not.
Theorem 3.6. Let H and A be defined by (10) and (11) respectively, if three vessels are the same for system (15), all eigenvalues of A are simple and separable .

3.3.2.
Two sub-branches are the same. We consider the case that A 20 = A 30 and β 2 = β 3 . Thus b 2 (λ) = b 3 (λ) and whose zeros are given by Clearly, if e 2b2(λ) = −1, the zeros of G(λ) are separable and simple(at most finite number non-simple). We consider the second case. Set we have asymptotic expression for sufficiently large |λ|

CHUN ZONG AND GEN QI XU
When λ ∈ C satisfies | λ| ≤ r, cosh( i λ + δ i ) and sinh( i λ + δ i ) are bounded functions, so we have So the asymptotic zeros of G(λ) are given by To show the separability of eigenvalues, we consider function equations: Solving the equations yields Thus λ ∈ C is of the form Comparing the imaginary parts of λ leads to Therefore, only when the above two equations are satisfied simultaneously, the function equations have a solution λ. Otherwise, the function equations have no solution. Thus, the zeros are simple and separable.
Theorem 3.7. Let H and A be defined by (10) and (11) respectively, when the two sub-branch vessels are the same for system (15) but (29) does not hold, all eigenvalues of A are separable and only at most finite number of eigenvalues are not simple.

CHUN ZONG AND GEN QI XU
Obviously, the zeros set of Γ(λ) is and λ ∈ C\σ Γ makes F (λ) = 0 if and only if G(λ) = 0. We rewrite F (λ) as For any η ∈ C, we consider the function equations Obviously, for some η ∈ C, λ ∈ C\σ Γ is a solution of the above equations if and only if F (λ) = 0 and hence λ makes G(λ) = 0. We assume that η ∈ C makes the function equations have at least one solution. For such an η, λ is a corresponding solution, then , .
Theorem 3.8. Let H and A be defined by (10) and (11) respectively and the three vessels be different for system (15). If ∀λ ∈ C with G(λ) = 0 makes (31) do not hold, then all eigenvalues of A are simple and separable.

Completeness and Riesz basis property of (generalized) eigenvectors of A.
In this section, we shall discuss the completeness and Riesz basis property of (generalized) eigenvectors of A. Firstly, we establish the completeness of the (generalized) eigenvectors of A.
Lemma 4.1. Let H be a Hilbert space. Suppose that A 0 is a skew adjoint operator in H with compact resolvent, and there exist two positive constants M and β such that its resolvent satisfies ||R(λ, A 0 )|| ≤ M e β|λ| for all λ satisfying dist(λ, σ(A 0 )) ≥ δ, and A 1 is a bounded linear operator on H. Then the root vectors of operator Proof. Denote the spectral subspace of A by where E(λ k , A) is the Riesz projector corresponding to λ k . We shall prove Sp(A) = H. Let u 0 ∈ H such that u 0 ⊥ Sp(A). Then R * (λ; A)u 0 is an H-valued entire function on C. For any u ∈ H, we denote a scalar function on C by
Based on the above properties, the Phragmén-Lindelöf Theorem (see [23]) asserts that F (λ) is a bounded function on complex plane C, and then the Liouville's Theorem says that F (λ) ≡ 0. So R * (λ, A)u 0 ≡ 0, which implies u 0 = 0. Hence Sp(A) = H. This completes the proof.
As a direct consequence of Lemma 4.1, we have the following result. In what follows, we shall discuss the Riesz basis property of eigenvector and generalized eigenvectors of A. At first we introduce the conception of general divide difference of exponentials.

CHUN ZONG AND GEN QI XU
If µ 1 = µ 2 , the first-order generalized divide difference In general, the k-order Generalized divide difference is defined by The following result gives the subspace Riesz basis of the eigenvectors and the family of exponentials (see e.g. [26,27,29]). ( is the Riesz projector associated with λ k ; (3) There is a constant α such that   Remark 2. In Theorem 4.5, we do not use the exact multiplicity of eigenvalues of A. Set σ(A) = {λ n , n ∈ Z + }. Under certain conditions, the eigenvalues of A are simple and separable. In this case, the exponentials {e λnt , n ∈ Z + } forms a Riesz basis sequence in L 2 (0, T ) for some sufficiently large T , and there is a sequence of eigenvectors of A that is a Riesz basis for H. These facts will be used in the next section.

5.
The exact observability and the exact controllability. In this section, we shall discuss the exact observability and the exact controllability of the system (15) in finite time. We first recall the concepts of the exact observability and exact controllability.
Obviously, the mild solution of (33) is x 0 makes sense. The admissibility of operator C for T (t) means that the operator Ψ : can be extended to a bounded linear operator from H to L 2 loc (R + , Y). The exact observability of the system means that the operator Ψ τ is injective and has closed range in L 2 ((0, τ ), Y). Hence, we have that means that we can find out the initial state x 0 of the system from the observation Y (t), t ∈ (0, τ ) and hence for all state X(t). Therefore, the exact observability in finite time is an important property of the system.
is said to be exactly controllable in finite time τ if for any x 0 , x 1 ∈ H, there exists a control function u(t) ∈ L 2 loc (R + , U) such that Note that, the admissibility of operator B for T (t) means that, for each u(·) ∈ L 2 loc (R + , U), the operator Φ : is a bounded linear operator from L 2 loc (R + , U) to H. Thus the mild solution of (35) is The exact controllability of the system means that for any given final state x 1 , we can find out a control u(t) such that the mild solution X(t, u) of the system arrives at x 1 at finite time τ . So the exact controllability also is an important property of the system. Remark 3. (see [19, chapter 11, pp.365]) Usually it is difficult to verify (36). Note that (36) implies that the range of operator Φ τ is the whole space H. By the duality theory, we can transfer the controllability into the observability of the dual system. Then the system (35) is exactly controllable in finite time τ if and only if its dual system is exactly observable in finite time τ , or equivalently, there exists a positive constant m τ such that 5.1. The exact observability of system (15). In this subsection, we shall discuss the exact observability of system (15) in finite time. Without loss of generality, we assume that all eigenvalues of A are separable and simple(also available for at most finite number non-simple eigenvalues). As pointed in Remark 2, for sufficiently large τ , the family of exponentials {e λnt , n ∈ Z} forms a Riesz basis sequence in L 2 (0, τ ), and there exists a sequence of eigenvectors of A, saying {Φ n (x, λ n ), n ∈ Z}, which forms a Riesz basis for H. The property of the Riesz basis sequence {e λnt , n ∈ Z} in L 2 (0, τ ) for sufficiently large τ implies that there exist two positive constants m τ and M τ such that And the Riesz basis property of eigenvectors {Φ n (x, λ n ), n ∈ Z} of A means that for each F = (f, g) ∈ H, F = n∈Z a n Φ n (x), a n = (F, Ψ n (λ n )), where the sequence {Ψ n (x, λ n ), n ∈ Z} is bi-orthogonal to {Φ n (x, λ n ), n ∈ Z} and there exist m 1 and m 2 such that Now let A be defined by in (11) and T (t) be the semigroup generated by A. The C is the observation operator defined in Section 2. Then T (t) has an analytic expression as T (t)F = n∈Z a n e λnt Φ n (x), ∀F ∈ H.
In particular, for F ∈ D(A), the observation Y (t) satisfy Y (t) = CT (t)F = n∈Z a n e λnt CΦ n (x).

5.2.
The exact controllability in finite time. In this subsection, we shall discuss the exact controllability of system (15) in finite time. Thanks to the duality, we only need to discuss the exact observability of the dual system (37). We complete the proof by the following three steps.
Step 1. Calculating the conjugate operator B * . Firstly we notice that L 2 , and λ 1 , λ 2 are the resolvent point of operator A and A * , respectively. So, B * : H * −1 → U * = C 3 is given by Step 2. Calculating the eigenvectors of A * .
Recall the results of the simplicity and separability of the eigenvalues of the system operator A in Theorem 3.6, Theorem 3.7 and Theorem 3.8. Then we have obtained the following result.
Theorem 5.3. The system (15) are exactly controllable and exactly observable in finite time in the following three cases: (1) The three vessels are the same; (2) The two sub-branch vessels are the same but (29) does not hold; (3) The three vessels are different and if ∀λ ∈ C with G(λ) = 0 makes (31) do not hold.

6.
Conclusion. In the previous several sections, we have carried out a complete system analysis for a linearization blood flow network described by (8) and (9). We have proved that the system is L 2 well-posed, exactly observable and exactly controllable in finite time. Note that this system is the simplest model of blood flow network, and we can extend this method to more complicated ones. The key point in the proof of observability is to show the Riesz basis sequence property of exponential family of eigenvalues for the corresponding system. And the main difficulty we encounter is the proof of separability of eigenvalues of the system operator. In the future, we shall study the observability and the controllability of more complicated blood flow networks.
Appendix A: The proof of Theorem 2.5.