In many dynamical systems, the identification of unknown parameters is essential for understanding system behavior and ensuring effective control [1]-[8]. This process becomes even more critical in distributed parameter systems, where parameters vary across space and time, and accurate identification is the key to reliable modeling and monitoring [9]-[12]. In such systems, especially for wave equations, intrinsic physical parameters (e.g., wave speed), initial states, and internal disturbances are generally not directly observable and must be inferred from indirect measurements [13]. Therefore, reconstructing these unknown parameters through boundary measurements has received extensive attention in inverse problems associated with wave-type distributed parameter systems. For hyperbolic systems, particularly wave equations, inverse problems play a crucial role in understanding wave propagation dynamics and enabling effective monitoring and control [14]. Typical inverse problems for wave equations include the identification of wave speed, source terms, initial conditions, and boundary disturbances from partial observations [2]-[4][8]-[10].

Early studies in this field mainly focused on inverse problems involving a single unknown for wave systems. Over the past few decades, extensive investigations have been conducted on inverse problems for one-dimensional wave equations from multiple perspectives, including the estimation of medium parameters such as wave speed [15][16], dispersion coefficients [17], and disturbance parameters [18][19], as well as the reconstruction of initial state distributions [20] and spatial source terms [16][21]. Provided that suitable observability conditions are satisfied, these works have obtained essential results concerning uniqueness and stability, while various analytical and computational approaches have been proposed, such as the boundary control method [22], spectral methods [23], and adaptive identification strategies [16][24]. For a more comprehensive discussion of numerical techniques for inverse problems arising from partial differential equations, interested readers are referred to the monographs [25][26].

In recent years, research attention has gradually shifted toward inverse problems involving multiple unknowns, also known as hybrid or coupled inverse problems. For wave equations, numerous works have investigated the simultaneous identification of multiple unknown quantities, either of the same type or different types, such as multiple boundary disturbances [27], two distinct source terms [28], combinations of wave speed and source terms [29], as well as measurement biases and initial states [30]. Nevertheless, most existing results rely on additional internal measurements or multiple boundary observations, which may be infeasible in practical engineering applications [19][27]. Furthermore, the simultaneous reconstruction of the wave speed together with two initial states has been rarely addressed, especially under the constraint of a single boundary observation. The difficulties outlined above, together with the recent progress reported in [31] for inverse heat-conduction models, in which the diffusion coefficient and the initial state are recovered within the same framework, provide the main motivation for the present study. Our objective is to determine the wave speed and both unknown initial data in a unified manner by using measurements acquired at only one boundary.

This study investigates an inverse problem for wave propagation in a homogeneous bar of unit length. The unknown wave velocity, initial displacement, and initial velocity are uniquely identifiable simultaneously from the available measurement data. The dynamics of the system are governed by

Here, x and t are the spatial and temporal variables, respectively. The propagation velocity c is an unknown positive constant satisfying c ≥ c 0 > 0 , while q > 0 denotes the elastic coefficient at the boundary. The two unknown initial profiles are given by u 0 ∈ H 1 ( 0 , 1 ) and u 1 ∈ L 2 ( 0 , 1 ) , corresponding to the displacement and velocity at t = 0 , respectively. Apart from boundedness, no additional restriction is imposed on either initial datum. The signal U ( t ) serves as a Neumann-type boundary input and describes the stress flux applied at the left endpoint of the bar. The measured output y ( t ) is the displacement recorded at the boundary. To explicitly indicate the dependence of the solution on the control input and the initial data, the solution of (1.1) may be denoted by u = u ( x , t ; U , u 0 , u 1 ) .

Inverse Problem. Consider system (1.1), where the wave velocity c and the initial data ( u 0 , u 1 ) ⊤ are unknown. The objective is to construct a boundary input U ∈ L 2 ( 0 , T ) such that the observation y ( t ) = u ( 0 , t ) , t ∈ ( 0 , T ) , uniquely determines c , u 0 ( x ) , and u 1 ( x ) .

The inverse problem considered here has two notable features. First, the wave velocity and both initial profiles must be determined simultaneously, whereas the available information consists solely of the displacement trace recorded at the left endpoint x = 0 . Recovering several unknown quantities from such limited data is therefore particularly difficult. Second, an appropriately chosen boundary excitation U ( t ) is essential for distinguishing the wave velocity from the two initial states. Its purpose is to enrich the system dynamics so that the resulting output y ( t ) carries enough information to identify all three unknown quantities uniquely. In this respect, the boundary input serves a role analogous to that of persistent excitation in adaptive control. To demonstrate this fact, consider the following two different collections of unknown quantities, where ω k and ω m ( k ≠ m ) denote two distinct positive roots of the characteristic equation ω tan ω = q of the Robin boundary-value problem. The corresponding eigenfunctions cos ( ω n x ) satisfy both the Neumann condition at x = 0 and the Robin condition at x = 1 :

Case 1: Let c 1 > 0 be arbitrary, u 01 ( x ) = cos ( ω k x ) , and u 11 ( x ) = 0 .

The zero-input solution takes the form u 1 ( x , t ) = cos ( ω k c 1 t ) cos ( ω k x ) , which yields the boundary output y ( t ) = cos ( ω k c 1 t ) at x = 0 .

Case 2: Let c 2 = ω k ω m c 1 , u 02 ( x ) = cos ( ω m x ) , and u 12 ( x ) = 0 .

The zero-input solution becomes

u 2 ( x , t ) = cos ( ω m c 2 t ) cos ( ω m x ) = cos ( ω k c 1 t ) cos ( ω m x ) , which produces the

identical boundary output y ( t ) = cos ( ω k c 1 t ) at x = 0 .

By construction, both initial profiles are eigenfunctions of the underlying Sturm-Liouville problem and thus satisfy the boundary conditions of system (1.1). Nevertheless, the two distinct sets of unknown parameters yield exactly the same boundary displacement trace. Hence, the measured trace cannot be used to discriminate between the two cases. Equivalently, when no boundary excitation is applied, the input-output map is not injective. Therefore, the observation y ( t ) alone is insufficient to uniquely determine the wave velocity c together with the initial data u 0 and u 1 .

The contributions of this work can be summarized in three points. First, motivated by the switched boundary-input strategy proposed in [31] for the joint identification of the diffusion coefficient and the initial profile of a one-dimensional heat equation, we adapt the underlying identification idea to a hyperbolic wave equation. Second, in view of the oscillatory nature of wave propagation, which is fundamentally different from the dissipative behavior of heat conduction, we establish a reconstruction framework for the simultaneous identification of the constant wave speed c and two initial states u 0 ( x ) and u 1 ( x ) from a single boundary observation. Third, in contrast to many identifiability results based on inverse spectral theory, which assume that the initial data have a nonzero projection onto every eigenfunction, our method eliminates this requirement through the designed boundary control and therefore permits some modal coefficients of the initial states to vanish.

The remainder of this article is arranged as follows. Section 2 studies the joint identifiability of the wave speed c and the two initial states by means of a Fourier-series argument. Section 3 develops an identification procedure that integrates the matrix pencil technique, controlled-residual fitting, and regularized least-squares reconstruction. Section 4 examines the errors associated with applying the matrix pencil technique to the corresponding infinite-dimensional spectral estimation problem. Finally, Section 5 reports numerical experiments that assess the performance of the procedure developed in Section 3.

System (1.1) is formulated on the Hilbert space ℋ = H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) . We are equipped ℋ with the following inner product 〈 ⋅ , ⋅ 〉 ℋ defined by

and the norm induced by this inner product will be written as ‖   ⋅   ‖ .

Let the operator A : D ( A ) ( ⊂ ℋ ) → ℋ be specified by

where its domain is given by

With respect to the inner product defined in (2.1), the operator A is skew-adjoint on the separable Hilbert space ℋ . It follows that all its eigenvalues lie on the imaginary axis and appear in complex-conjugate pairs; namely,

Here, the sequence { ω n } n = 1 ∞ consists of all positive solutions to

For each eigenvalue λ n ± , an associated eigenvector can be chosen as

It is not immediate from the unboundedness and skew-adjointness of A that its eigenvectors form an orthogonal basis for ℋ . This property can instead be established through the inverse operator. Indeed, the Rellich-Kondrachov compact embedding theorem ensures that A − 1 is compact. Moreover, the skew-adjointness of A yields ( A − 1 ) * = − A − 1 , and hence A − 1 is also skew-adjoint. Since ℋ is separable, the spectral theorem for compact normal operators guarantees that the eigenvectors associated with A − 1 constitute a complete orthogonal system in ℋ . Furthermore, A and A − 1 possess the same eigenvectors, with their corresponding eigenvalues being reciprocal. It therefore follows that { ϕ n ± ( x ) } n = 1 ∞ forms an orthogonal basis of ℋ .

Normalizing these eigenvectors yields the orthonormal basis { ϕ ˜ n ± ( x ) } n = 1 ∞ , i.e.,

where the normalization constant is

satisfying ‖ ϕ ˜ n ± ‖ ℋ = 1 .

Define the state vector

where u ( t ) = u ( ⋅ , t ) and u t ( t ) = ∂ ∂ t u ( ⋅ , t ) , and the initial state

encodes the initial displacement and velocity prescribed by u ( x , 0 ) = u 0 ( x ) and u t ( x , 0 ) = u 1 ( x ) .

Accordingly, system (1.1) admits the following first-order representation in the dual space [ D ( A ) ] ′ :

Here, B = ( 0 c 2 δ ( x ) ) , where δ ( x ) denotes the Dirac delta distribution.

Theorem 2.1 (Well-posedness of the nonhomogeneous abstract evolution equation)Let ℋ = H 1 ( 0 , 1 ) × L 2 ( 0 , 1 ) be equipped with the inner product introduced in(2.1), and consider the operator A : D ( A ) ( ⊂ ℋ ) → ℋ specified by (2.2) - (2.3).Assume that A is the generator of an isometric C 0 -semigroup { S ( t ) } t ≥ 0 on ℋ ,and that B is admissible as a control operator for A .Equivalently, B * is an admissible observation operator for A * .Then,for everyinitial datum z 0 = ( u 0 , u 1 ) ⊤ ∈ ℋ and every input U ∈ L loc 2 ( [ 0 , ∞ ) ; ℝ ) ,the abstractsystem(2.10)possesses exactly one mild solution z ∈ C ( [ 0 , ∞ ) ; ℋ ) . This solution is represented by the variation-of-constants formula

Moreover, for each T > 0 , there exists a constant C T > 0 such that

Consequently, the solution varies continuously with respect to both the initial datum z 0 and the control input U .

Proof. The proof proceeds in two steps.

Step1. A generates an isometric C 0 -semigroup on ℋ .

According to the Lumer-Phillips theorem, it is enough to establish that A is a densely defined dissipative operator and that the range of λ I − A covers ℋ for some λ > 0 . The density of D ( A ) in ℋ follows directly from the definition of the domain. Moreover, for any z ∈ D ( A ) , we have

which implies that A is dissipative. Therefore, it remains only to prove the surjectivity of λ I − A . Let ( h 1 , h 2 ) ⊤ ∈ ℋ be arbitrary and consider the equation.

which is equivalent to

Substituting v = λ u − h 1 into the second equation yields the elliptic boundary value problem

By the well-posedness theory for second-order boundary value problems, the above equation possesses a unique solution u ∈ H 2 ( 0 , 1 ) . Consequently, v = λ u − h 1 ∈ H 1 ( 0 , 1 ) , which implies that ( u , v ) ⊤ ∈ D ( A ) is a solution of the resolvent equation. Therefore, λ I − A is onto. The Lumer-Phillips theorem then guarantees that A is the generator of an isometric C 0 -semigroup { S ( t ) } t ≥ 0 on ℋ .

Step 2. The control operator B is admissible for A .

Using the duality principle, the admissibility of B is equivalent to that of the observation operator B * for A * . Since A is skew-adjoint, one has A * = − A , and B * = ( 0 , c 2 δ ( x ) ) . Hence, B * ( u , u t ) ⊤ = c 2 u t ( 0 , t ) . Accordingly, it remains to establish the existence of positive constants T and K T such that every solution of the dual system satisfies

satisfy

Define the energy

Using the multiplier ( x − 1 ) v x , we derive the following identity:

Upon integrating both sides of equation (2.18) with respect to t over the interval [ 0 , T ] , one arrives at

Owing to the isometric dissipativity property, E ( t ) = E ( 0 ) holds for every t ≥ 0 . Observing that | x − 1 | ≤ 1 , it follows that

which implies that the left-hand term in equation (2.19) satisfies

Moreover, by (2.17), we obtain

Substituting (2.20) and (2.21) into the integral equality (2.19) and rearranging terms, we obtain

Upon multiplying both sides of equation (2.22) by a factor of 2, it follows that

where the constant K T = T + 1 min { 1 , c 2 } . By definition (2.17), we have

Combining this with inequality (2.23), we immediately obtain

Therefore, the admissibility of B * for A * directly leads to the admissibility of B as a control operator for A .

Since A generates an isometric C 0 -semigroup and B is admissible, the standard theory of abstract control systems ensures that the nonhomogeneous evolution equation possesses a unique mild solution z ∈ C ( [ 0 , ∞ ) ; ℋ ) , which is represented by the variation-of-constants formula (2.11). Moreover, the solution satisfies the energy estimate (2.12), guaranteeing its continuous dependence on both the initial condition and the control input.

We expand the initial state z 0 = ( u 0 , u 1 ) ⊤ ∈ ℋ with respect to the orthonormal eigenbasis { ϕ ˜ n ± } n ≥ 1 as

where a n ± = 〈 z 0 , ϕ ˜ n ± 〉 ℋ denote the initial modal coefficients.

We rely on a fundamental property satisfied by the isometric C 0 -semigroup { S ( t ) } t ≥ 0 associated with the operator A , that is,

Combining the variation-of-constants formula (2.11) for the system solution, we compute the observations contributed by the initial state term and the control term, respectively, and finally obtain the expression for the boundary observation y ( t ) at x = 0 as

where the Green’s function is expressed as

Assume that T 2 > T 1 > 0 and that the boundary control vanishes identically, i.e., U ( t ) = 0 for every t ∈ [ 0 , T 2 ] . Under this assumption, the boundary observation takes the form

in which the coefficients are given by C n ± = a n ± α n for each n ∈ ℕ * .

Because the initial data u 0 and u 1 are not known a priori, one cannot determine in advance which coefficients C n ± are nonzero. We define an unknown set K ⊂ ℕ * such that C k ± ≠ 0 for k ∈ K and C k ± = 0 for k ∉ K .

Theorem 2.2Assume that 0 < T 1 < T 2 < ∞ , u 0 ∈ H 1 ( 0 , 1 ) , and u 1 ∈ L 2 ( 0 , 1 ) . Then the eigenvalues and corresponding coefficients { ( λ k + , C k + ) , ( λ k − , C k − ) } k ∈ K canbe uniquely determined from the observation y ( t ) on the interval [ T 1 , T 2 ] .

Proof. The zero-input boundary response admits the nonharmonic Fourier series representation

where the frequencies ± ω n c are purely imaginary and derived from the characteristic equation ω tan ω = q . By the properties of the Sturm-Liouville problem, the sequence { ω n } n ≥ 1 is uniformly separated: there exists a constant γ > 0 depending only on q such that

Consequently, the frequency set Λ = { ± ω n c } n ≥ 1 is also uniformly separated with minimal spacing c γ > 0 .

The regularity assumptions u 0 ∈ H 1 ( 0 , 1 ) and u 1 ∈ L 2 ( 0 , 1 ) imply that the coefficients satisfy

which guarantees absolute and uniform convergence of the series on any bounded time interval.

We now invoke the Ingham inequality for uniformly separated frequencies. If the observation interval length satisfies

then there exists a constant C Ing > 0 such that

Suppose there exist two sets of spectral parameters { λ k ± , C k ± } and { λ ˜ k ± , C ˜ k ± } that generate identical boundary observations on [ T 1 , T 2 ] . Subtracting the two representations yields a series that vanishes identically on [ T 1 , T 2 ] . Applying the Ingham inequality to the difference series forces all coefficients to be zero, which implies that the two sets of frequencies (poles) must coincide exactly, and the corresponding modal coefficients are equal.

Therefore, the eigenvalues and corresponding coefficients

{ ( λ k + , C k + ) , ( λ k − , C k − ) } k ∈ K are uniquely determined by the boundary observation y ( t ) on [ T 1 , T 2 ] , provided the observation interval is sufficiently long.

Theorem 2.3Assume that 0 < T 1 < T 2 < T 3 < ∞ , u 0 ∈ H 1 ( 0 , 1 ) ,and u 1 ∈ L 2 ( 0 , 1 ) . Let the control input U ∈ L 2 ( 0 , T 3 ) satisfy

Denote the associated boundary measurement by

Then, the wave speed c and the initial states u 0 ( x ) and u 1 ( x ) can be uniquely determined from the boundary observation y ( t ) on the interval [ T 1 , T 3 ] .

Proof. From equation (2.27), we define the difference observation

Let U ˜ ( t ) = U ( t + T 2 ) for t ∈ ( 0 , T 3 − T 2 ] , and denote the Green’s function as

Substituting these into the observation formula yields

We first justify the continuity of the Green function G ( t , 0 ) and then establish its unique identifiability from finite-time convolution data. Substituting the spectral expansion and normalization constants into the expression for G ( t , 0 ) , we obtain the trigonometric series

From the Sturm-Liouville eigenvalue condition ω n tan ω n = q , we have ω n ~ ( n − 1 2 ) π as n → ∞ , which yields coefficient decay

The sequence { A n } is positive and monotonically decreasing toward zero. By the Dirichlet test for trigonometric series, the series converges uniformly on every compact time interval [ 0 , T ] . Consequently,

and term-by-term evaluation gives G ( 0 , 0 ) = 0 .

Recall the shifted input U ˜ ( t ) = U ( t + T 2 ) for t ∈ ( 0 , T 3 − T 2 ] . From the assumption U ∈ L 2 ( 0 , T 3 ) , we obtain U ˜ ∈ L 2 ( 0 , T 3 − T 2 ) ⊂ L 1 ( 0 , T 3 − T 2 ) . Both G ( ⋅ , 0 ) and U ˜ satisfy the function-space hypotheses of the Titchmarsh convolution theorem ([24] Chap. 1): for functions f , g ∈ L 1 ( 0 , T ) , if the convolution

vanishes for almost every t ∈ ( 0 , T ) , then there exists α ∈ [ 0 , T ] such that f = 0 almost everywhere on ( 0 , α ) and g = 0 almost everywhere on ( 0 , T − α ) .

We now deduce the deconvolution uniqueness needed for our argument. Suppose two candidates G 1 , G 2 ∈ L 1 ( 0 , T 3 − T 2 ) satisfy G 1 ∗ U ˜ = G 2 ∗ U ˜ almost everywhere on ( 0 , T 3 − T 2 ] . Define f : = G 1 − G 2 , so that f ∗ U ˜ = 0 a.e. Applying the Titchmarsh result, there exists α ∈ [ 0 , T 3 − T 2 ] such that f = 0 a.e. on ( 0 , α ) and U ˜ = 0 a.e. on ( 0 , T 3 − T 2 − α ) . By hypothesis, U ˜ ( t ) ≠ 0 for almost every t ∈ ( 0 , T 3 − T 2 ] , which forces T 3 − T 2 − α = 0 , i.e., α = T 3 − T 2 . Therefore, f = 0 a.e. on ( 0 , T 3 − T 2 ] , meaning G 1 = G 2 almost everywhere. Since G 1 , G 2 are continuous functions, almost-everywhere equality implies pointwise equality for all t ∈ ( 0 , T 3 − T 2 ] . Hence, G ( t , 0 ) is uniquely determined from the convolution relation.

Moreover, according to Theorem 2.2, the spectral parameters { ( λ k + , C k + ) , ( λ k − , C k − ) } k ∈ K are determined by the boundary measurement { y ( t ) | t ∈ [ T 1 , T 2 ] } . Hence, the extended observation data { y ˜ ( t ) | t ∈ ( 0 , T 3 − T 2 ] } is uniquely determined by the original measurement { y ( t ) | t ∈ [ T 1 , T 3 ] } . Since G n ≠ 0 for all n , applying Theorem 2.2 yields that the eigenvalue sequence { λ n + } n ∈ ℕ * is identifiable from { G ( t , 0 ) | t ∈ ( 0 , T 3 − T 2 ] } . Combining this result with the relation λ n ± = ± i ω n c , the wave speed c can consequently be identified uniquely.

We now proceed to investigate whether the initial data u 0 ( x ) and u 1 ( x ) can be uniquely determined. Substituting the identified c back into equation (2.27), we rearrange it as

The identification of c from the measured output { y ( t ) | t ∈ [ T 1 , T 3 ] } enables us to evaluate the corresponding spectral relation completely. In this relation, the unknown terms appear in the form of a complex exponential expansion. The uniqueness property of such spectral representations guarantees that the expansion coefficients { 1 α n ( a n + , a n − ) } can be uniquely obtained from the available observation data. Therefore, the modal coefficients associated with the initial conditions are determined.

Here, the Fourier coefficients 〈 u 0 , ϕ n 〉 and 〈 u 1 , ϕ n 〉 can be calculated from the identified spectral quantities { λ n ± } and { a n ± } . Hence, the initial profiles u 0 ( x ) and u 1 ( x ) are uniquely determined from the boundary measurement { y ( t ) | t ∈ [ T 1 , T 3 ] } .

From the analysis in the preceding sections, it is evident that the core challenge of the identification problem lies in retrieving the spectral-coefficient pairs { ( C n + , λ n + ) , ( C n − , λ n − ) } n ∈ ℕ * from the boundary measurements y ( t ) on the interval [ T 1 , T 2 ] , based on the complex exponential series representation given in (2.27). A primary obstacle lies in the fact that an infinite number of coefficients C n ± in (2.27) may be nonzero. To address this, we adopt the matrix pencil technique in this section to extract a subset of the spectral-coefficient data { ( C n + , λ n + ) , ( C n − , λ n − ) } from the series truncated to its first M terms in (2.27), with the remaining higher-order components treated as measurement noise.