IDENTIFICATION PROBLEMS OF RETARDED DIFFERENTIAL SYSTEMS IN HILBERT SPACES

. This paper deals with the identiﬁcation problem for the L 1 -valued retarded functional diﬀerential equation. The unknowns are parameters and operators appearing in the given systems. In order to identify the parameters, we introduce the solution semigroup and the structural operators in the initial data space, and provide the representations of spectral projections and the completeness of generalized eigenspaces. The suﬃcient condition for the identiﬁcation problem is given as the so called rank condition in terms of the initial values and eigenvectors of adjoint operator.


Introduction.
Let Ω be a bounded domain in R n with smooth boundary ∂Ω. Let A(x, D x ) be an elliptic differential operator of second order in L 1 (Ω): In this paper, we consider the inverse problem for the following retarded functional differential equation defined as A 0 u = −A(x, D x )u: −h a(s)A 0 u(t + s)ds, u(0) = g 0 , u(s) = g 1 (s), s ∈ [−h, 0). (1.1) Here A 0 , γ, and a(·) are unknown quantities to be identified and the initial condition g = (g 0 , g 1 ) is known.
In the field of control engineering, identifiability as a kind of inverse problems, or the parameter estimations of systems has attracted much interest and has been investigated in many references, for example, as for one dimensional heat equation with an unknown spatially-varying conductivity in [1][2][3][4], an abstract linear first order evolution equation within the framework of operator theory in [15], and linear retarded functional differential systems in reflexive Banach spaces in [6][7][8]. In [8,9] 78 JIN-MUN JEONG AND SEONG-HO CHO the author discussed the control problem for the following retarded system with L 1 (Ω)-valued controller: where A i (i = 1, 2) are second order linear differential operators with real coefficients, and the controller Φ 0 is a bounded linear operator from a control Banach space to L 1 (Ω). In [6], they established some results concerning identification problems for (1.1) of specific form by taking the observation. Furthermore, Yamamoto and Nakagiri [26] studied the identifiability problem for evolution equations in Banach spaces with unknown operators and initial values by means of spectral theory for linear operators.
In view of Sobolev's embedding theorem we may also consider L 1 (Ω) ⊂ W −1,p (Ω) if 1 ≤ p < n/(n − 1) as is seen in [8]. Hence, we can investigate the system (1.1) in the space W −1,p (Ω) considering Φ 0 as an operator into W −1,p (Ω). Here, we note that the space W −1,p (Ω) is ζ-convex(as for the definition and fundamental facts of a ζ-convex see [2,3]). Consequently, in view of Dore and Venni [7] the maximal regularity for the linear initial value problem: Furthermore, with the aid of a result by Seeley [21] and [8], we can obtain the maximal regularity for solutions of the retarded linear initial value problem (1.1) in the space W −1,p (Ω). In view of these results, we deal with an identification problem of (1.1) in W −1,p (Ω).
The paper is organized as follows. Section 2 presents some notations. In Section 3 from the definitions of operator A 0 and the interpolation theory as in Theorem 3.5.3 of Butzer and Berens [4], we can apply Theorem 3.2 of Dore and Venni [7] to general linear Cauchy problem in the space W −1,p (Ω). Thereafter, by using the method of Di Blasio et al. [5] to the system (1.2) with the forcing term f in place of the control term Φ 0 w, Section 4 is devoted to studying the wellposedness and regularity for solutions of (1.1) by using a solution semigroup S(t) in the initial data space Z p,q = H p,q ×L q (−h, 0; W 1,p 0 (Ω)), where H p,q = (W 1,p 0 (Ω), W −1,p ) 1/q,q (Ω) for 1 < q < ∞.
In Section 5, in order to identify the parameters, we investigate the spectrum of the infinitesimal generator Λ of S(t). We will give that the spectrum of Λ is composed of two parts of cluster points and discrete eigenvalues. Moreover, we are concerned with the representations of spectral projections and the problem of completeness of generalized eigenspaces. Based on this result, we establish a sufficient condition for the identification problem is given as the so called rank condition in terms of the initial values and eigenvectors of adjoint operator.
Finally we give a simple example to which our main result can be applied.

Notations.
Let Ω be a region in an n-dimensional Euclidean space R n and closure Ω. C m (Ω) is the set of all m-times continuously differential functions on Ω. C m 0 (Ω) will denote the subspace of C m (Ω) consisting of these functions which have compact support in Ω. W m,p (Ω) is the set of all functions f = f (x) whose derivative D α f up to degree m in distribution sense belong to L p (Ω) . As usual, the norm is then given by where (Ω) whose norm is denoted by || · || −1,p,∞ . If X is a Banach space and 1 < p < ∞, L p (0, T ; X) is the collection of all strongly measurable functions from (0, T ) into X the p-th powers of norms are integrable. C m ([0, T ]; X) will denote the set of all m-times continuously differentiable functions from [0, T ] into X. If X and Y are two Banach spaces, B(X, Y ) is the collection of all bounded linear operators from X into Y , and B(X, X) is simply written as B(X). For an interpolation couple of Banach spaces X 0 and X 1 , (X 0 , X 1 ) θ,p for any θ ∈ (0, 1) and 1 ≤ p ≤ ∞ and [X 0 , X 1 ] θ denote the real and complex interpolation spaces between X 0 and X 1 , respectively(see [25]).
Let 3. Cauchy problems on ζ-convex spaces. Let Ω be a bounded domain in R n with smooth boundary ∂Ω. Consider an elliptic differential operator of second order where (a i,j (x) : i, j = 1, · · · , n) is a positive definite symmetric matrix for each x ∈ Ω, a i,j ∈ C 1 (Ω), b i ∈ C 1 (Ω) and c ∈ L ∞ (Ω). The operator is the formal adjoint of A.
For 1 < p < ∞ we denote the realization of A in L p (Ω) under the Dirichlet boundary condition by A p : For p = p/(p − 1), we can also define the realization A in L p (Ω) under Dirichlet boundary condition by A p :

JIN-MUN JEONG AND SEONG-HO CHO
It is known that −A p and −A p generate analytic semigroups in L p (Ω) and L p (Ω), respectively, and A * p = A p . For brevity, we assume that 0 ∈ ρ(A p ). From the result of Seeley [22] (see also Triebel [25,p. 321]) we obtain that [D(A p ), L p (Ω)] 1 2 = W 1,p 0 (Ω), and hence, may consider that Let (A p ) be the adjoint of A p considered as a bounded linear operator from D(A p ) to L p (Ω). Let A be the restriction of (A p ) to W 1,p 0 (Ω). Then by the interpolation theory, the operator A is an isomorphism from W 1,p 0 (Ω) to W −1,p (Ω). Similarly, we consider that the restriction A of ( (Ω) to W −1,p (Ω). Furthermore, as seen in Proposition 3.1 in Jeong [8], we obtain the following result. Proposition 1. The operators A and A generate analytic semigroups in W −1,p (Ω) and W −1,p (Ω), respectively, and the inequality holds for some constants C > 0 and γ ∈ (0, π/2).

We set
Since A is an isomorphism from W 1,p 0 (Ω) onto W −1,p (Ω) and W 1,p 0 (Ω) and W −1,p (Ω) are ζ-convex spaces, it is easily seen that H p,q is also ζ-convex. From the definitions of operator A and the interpolation space H p,q as in Theorem 3.5.3 of Butzer and Berens [4], we can apply Theorem 3.2 of Dore and Venni [7] to general linear Cauchy problem as the following result.
Then the Cauchy problem . The last inclusion relation is well known and is an easy consequence of the definition of real interpolation spaces by the trace method. 4. Retarded equations and lemmas. In this section, we apply Propositions 3.1 and 3.2 to the retarded functional differential equation in the space W −1,p (Ω). Consider the following retarded equation in W −1,p (Ω): Here, A 0 = − A, and A ι u (ι = 1, 2) are the restrictions W 1,p 0 (Ω) of the linear differential operators A ι (ι = 1, 2) with real coefficients: ). Using Proposition 2 we can follow the argument as in [5] term by term to deduce the following result(see Proposition 4.1 of [8]).
The equation (4.1) can be transformed into an abstract equation in Z p,q as follows.
where G(t) = (f (t), 0), z(t) = (u(t; g), u t (·; g)) ∈ Z p,q and g = (g 0 , g 1 ) ∈ Z p,q . The mild solution of initial value problem (4.2) is the following form: We introduce the transposed problem of (4.1): Here, we remark that . We can also define the solution semigroup S T (t) of (4.3) by where y(t; φ) is the solution of (4.3). Let Λ T be the infinitesimal generator of S T (t) associated with the system (4.3).
For λ ∈ C we define a densely defined closed linear operator by The operators ∆(λ) and ∆ T (λ) are bounded in B(W 1,p 0 (Ω), W −1,p (Ω)) and B(W 1,p 0 (Ω), W −1,p (Ω)), respectively. Noting that if λ ∈ ρ(A 0 ) The structural operator F is defined by for φ ∈ Z p ,q . As in [8,18] we have that Let λ be a pole of (λ − Λ) −1 whose order we denote by k λ and P λ be the spectral projection associated with λ: where Γ λ is a small circle centered at λ such that it surrounds no point of σ(Λ) except λ. And we know that λ ∈ σ(A T ) is a pole of (λ − Λ T ) −1 and the spectral projection is given by As is well known λ is an eigenvalue of A and the generalized eigenspace corresponding to λ is given by Let us set Then we remark that It is also well known that Q k λ j λj = 0 (nilpotent) and (Λ − λ)P λj = Q λj (cf. [18,27]). The following subset of σ(Λ) are especially of use: The proof of 1) and 2) is from Proposition 7.2 and Theorem 6.1 of Nakagiri [18,19], respectively.
where Cl is denotes the closure in Z p,q .
(1) Let λ ∈ σ d (Λ T ). Then for any g ∈ Z p ,q , the spectral projection has the following representation (2) Let λ ∈ σ d (Λ). Then the spectral projection has the following representation Proof. We prove only (1) since the proof of (2) is similar. For any g ∈ Z p ,q , P T λ g is written as m λ i=1 c i ψ λi for c i ∈ C and then by (4.6) From the Laplace transform of the second equality in (4.5) we have Therefore, we have c j = (F * P T λ g, φ λj ) = ((P λ ) * F * g, φ λj ) = (F * g, P λ φ λj ) = (F * g, φ λj ). The proof of (1) is completed.

Identification problem in case
In this section we deal with the identification problem in the case where A 1 = γA 0 with some constant γ, A 2 = A 0 as follows.
Here A 0 , γ, and a(·) are unknown quantities to be identified and the initial conditions g i = (g 0 i , g 1 i ) ∈ Z p,q , i = 1, . . . , l are known We denote by the model system (5.1) m by the equation (5.1) with A 0 , γ, a replaced by A m 0 , γ m , a m respectively. The solutions of (5.1) and the model system (5.1) m are denoted by u(t; g) and u m (t; g), respectively, and the solution semigroup for model system by S m (t). We assume that A m 0 and a m satisfy the same type of assumptions as A 0 and a.
The identifiability for (5.1) is to find conditions such that if . . , l, is a finite set of initial values, then At first we investigate the spectral properties of the infinitesimal generator Λ m of solution semigroup S m (t) for the equation (5.1) m . Since Ω is bounded, the imbedding of W 1,p 0 (Ω) to H p,q is compact. From [1,Theorem 3.4], it follows that the system of generalized eigenspaces of A 0 is complete in H p,q . According to Riesz-Schauder theorem A m 0 has discrete spectrum σ(A m 0 ) = {µ j : j = 1, · · · } which has no point of accumulation except possibly λ = ∞. For

JIN-MUN JEONG AND SEONG-HO CHO
The structural operator F defined by (4.4) is written as for g = (g 1 , g 1 ) ∈ Z p,q . The m m and F m are the structural operators of the model system (5.1) m in place of m in (5.2) and F , respectively. Let λ ∈ σ p (Λ m ), and {φ λ k : k = 1, · · · , d λ } denote the basis of P m λ Z p,q . Let Λ m T be the infinitesimal generator of transposed solution semigroup associated with (5.1). Then λ ∈ σ p (Λ m T ). Let {ψ λ k : k = 1, · · · , d λ } be a basis of (P m ) T λ Z Z p ,q , where (P m ) T µ denotes the projection of Λ m T at µ. As shown in [21, Theorem 8.1] the projection (P m ) T λ has the following eqivalent representation Throughout this section we shall assume following: • RANK CONDITION: For set of the initial values {g 1 , . . . , g l } is said to be satisfy the Rank condition for the model system (5.1) m if and only if for n = 1, 2, · · · and j = 1, 2, · · · .
Then the system (6.1) can be written in the same form as of (5.1) on the space H p,q . It is well known that {e n :, n = 1, · · · } is an orthogonal base for H p,q , and so {sin(nx), n = 1, · · · } is complete In H p,q . Thus, we can solve the identification problem of the system (6.1) for parameters α, β, and the function a 1 (·) in the terminology of Theorem 5.3.