Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output

Consider the following inverse problem of recovering the unknown spatial load F(x) in

$$\begin{equation}\begin{cases}\rho \left(x\right){u}_{tt}+\mu {u}_{t}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}=F\left(x\right)G\left(t\right),\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ u\left(x,0\right)={u}_{t}\left(x,0\right)=0,\quad x\in \left(0,\ell \right),\quad \\ u\left(0,t\right)={u}_{xx}\left(0,t\right)=0,\quad u\left(\ell ,t\right)={u}_{xx}\left(\ell ,t\right)=0,\quad t\in \left[0,T\right],\quad \end{cases}\end{equation} \tag{ 1 }$$

from the final time measured displacement

$$\begin{equation}{u}_{T}\left(x\right){:=}u\left(x,T\right),\quad x\in \left[0,\ell \right].\end{equation} \tag{ 2 }$$

Here, ρ(x) > 0 is the mass density, r(x) = E(x)I(x) > 0 is the spatial varying flexural rigidity, while E(x) > 0 and I(x) > 0 are the elasticity modulus and moment of inertia of the cross-section. For convenience of the analysis, we assume that the damping coefficient μ > 0 is a positive constant. The functions F ≢ 0 and G(t) > 0 are the spatial and temporal components of the acting load, respectively and ${{\Omega}}_{T}=\left\{\left(x,t\right)\in {\mathbb{R}}^{2}:0{< }x{< }\ell ,\enspace 0{< }t{< }T\right\}$. The geometry of the inverse problems (1) and (2) is given in figure 1.

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The goal of this study is to answer the following questions. Can the spatial load F(x) be uniquely determined from the final time measured displacement or velocity? If yes, then what is the role of the damping term μu_t in a positive solution to this problem? And finally, what is the relationship between the basic parameters, that is, the final time T > 0, the damping coefficient μ > 0, and also the temporal load G(t), in order for the solution to be unique? Despite the popularity of the final time inverse source problems, these important questions from the point of view of the theory of inverse problems, as well as its recent applications, have not yet been investigated so far.

The initial boundary value problem (1) is a dynamic PDE model of a simply supported damped Euler–Bernoulli beam. The term μu_t and its role in the uniqueness of the solution to the inverse problems (1) and (2) is the main focus in this paper. The nature of this term is usually determined by external damping mechanisms, although internal damping governed by the term μ_i u may also contribute (see the example given in [1]). There are two separate reasons, in terms of mathematical and physical sciences, motivating this research. The first reason is related to the non-uniqueness of the final data inverse source problem for undamped wave equation. It is shown in [12] (section 3.2) that for unique determination of the unknown source F(x) in

$$\begin{align}\begin{cases}\begin{cases}_{tt}={u}_{xx}+F\left(x\right),\quad \left(x,t\right)\in {{\Omega}}_{T}{:=}\left(0,\ell \right){\times}\left(0,T\right),\quad \\ u\left(x,0\right)={u}_{t}\left(x,0\right)=0,\quad x\in \left(0,\ell \right),\quad \\ u\left(0,t\right)=u\left(\ell ,t\right)=0,\quad t\in \left[0,T\right],\quad \end{cases}\quad \\ \hspace{25.0pt}{u}_{T}\left(x\right){:=}u\left(x,T\right),\quad x\in \left(0,\ell \right),\quad \end{cases}\end{align} \tag{ 3 }$$

the final time must satisfy the following condition

$$\begin{equation}T\ne \frac{2m}{n},\quad \text{for}\enspace \text{all}\quad m,n=1,2,3,\dots \enspace .\end{equation} \tag{ 4 }$$

Otherwise, i.e. when T = 2m/n, an infinite number of singular values σ_n defined as (see formula (3.2.12) in [12])

$$\begin{equation}{\sigma }_{n}=\frac{1}{{\lambda }_{n}}\enspace \left[1-\mathrm{cos}\left(\sqrt{{\lambda }_{n}}\enspace T\right)\right],\quad \text{for}\enspace \text{all}\quad n=1,2,3,\dots ,\end{equation} \tag{ 5 }$$

in the singular value expansion (SVE)

$$\begin{equation}F\left(x\right)=\sum\limits _{n=1}^{\infty }\frac{1}{{\sigma }_{n}}\enspace {u}_{T,n}{\psi }_{n}\left(x\right),\quad x\in \left(0,\ell \right)\end{equation} \tag{ 6 }$$

vanish, where ${u}_{T,n}{:=}{\left({u}_{T},{\psi }_{m}\right)}_{{L}^{2}\left(0,\ell \right)}$ is the nth Fourier coefficient of the final time output u_T (x) and ${\left\{{\lambda }_{n},{\psi }_{n}\left(x\right)\right\}}_{n=1}^{\infty }$ is the eigensystem of the operator −u''(x) subject to the boundary conditions in [3]. As a consequence, the Picard criterion

$$\begin{equation}\sum\limits _{n=1}^{\infty }\frac{{u}_{T,n}^{2}}{{\sigma }_{n}^{2}}{< }\infty \end{equation} \tag{ 7 }$$

is not satisfied since this criterion implicitly assumes also the requirement

$$\begin{equation}{\sigma }_{n}\ne 0,\quad \text{for}\enspace \text{all}\quad n=1,2,3,\dots \enspace .\end{equation} \tag{ 8 }$$

Therefore, if σ_n = 0 for some n, then the nth Fourier coefficient ${F}_{n}{:=}{\left(F,{\psi }_{n}\right)}_{{L}^{2}\left(0,\ell \right)}$ of the unknown function F(x) can not be determined uniquely.

From condition (4) it follows that for unique determination of the unknown source F(x) the final time T > 0 cannot be a rational number. Evidently, fulfilment of this necessary condition is impossible in practice. For this reason in [12] the above final time output inverse problems for undamped wave equation were defined as infeasible inverse problems.

The same result has been obtained in [12] for the inverse problem (3) with the final time measured displacement u_T (x) := u(x, T) replaced by the final time measured velocity ν_T (x) := u_t (x, T). Moreover, it can be proved that the same result is true for the inverse problems (1) and (2) when μ = 0 and ρ(x) = const..

Furthermore, the situation does not change, if equation (3) with pure spatial load is replaced with the equation u_tt = u_xx + F(x)G(t) which contains the temporal component G(t) of the load on the right-hand side. Namely, for an arbitrary function G(t) the uniqueness of the solution to the inverse problems (1) and (2) cannot be guaranteed. For an inverse problem related to parabolic equations with a final time output, this important issue was studied in [18].

The second reason motivating this research is that damping, as the physical phenomenon responsible for the dissipation of energy, drastically changes the nature of the solution to the vibration problem, controlling also the response of the beam [1]. Furthermore, the damped natural frequency ω_n , which is of the order $\mathcal{O}\left(\sqrt{{\lambda }_{n}}\right)$, is always less than the undamped natural frequency [4]. Consequently, the singular values corresponding to the damped problem must be greater than the singular values corresponding to the undamped problem, since σ_n and λ_n are inversely proportional as formula (5) suggests. This circumstance suggests that the damping parameter can naturally play a positive role in the unique determination of a spatial load F(x) from the final state displacement or velocity, since due to this parameter the system decays more slowly towards its equilibrium configuration. The behavior of solutions of the direct problem (1) corresponding to the different values of the damping coefficient μ shown in figure 2 clearly illustrates this typical situation.

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More detailed analysis of the inverse problems (1) and (2) shows that the role of the damping coefficient μ > 0 in the Euler–Bernoulli equation $\rho {u}_{tt}+\mu {u}_{t}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}=F\left(x\right)G\left(t\right)$ is very similar to the role of the regularization parameter α > 0 in the regularized Tikhonov functional J_α (u) = (1/2)||ΦF − u_T ||² + α||F||². Namely, adding the regularization term α||F||² to the functional J(u) = (1/2)||ΦF − u_T ||² ensures the uniqueness of the quasi-solution to the inverse problem. In other words, as a result of adding this term, instead of the ill-posed normal equation Φ^*ΦF = Φ^* u_T (or, the equation of the first kind, in terms of the theory of integral equations), we are dealing with the well-posed (regularized) normal equation αF_α + Φ^*ΦF_α = Φ^* u_T which has a unique solution (here and below, (ΦF) (x) := u(x, T; F) is the input–output operator). As a consequence, the SVE for the solution F_α of the regularized normal equation is obtained from the SVE in [5], by replacing σ_n with σ_n + α/σ_n . Hence, taking the value of this sum away from zero, the regularization term α/σ_n ensures the sufficient condition for σ_n > 0, thereby for the uniqueness of the solution. In this sense, adding the additional term μu_t to the undamped dynamic Euler–Bernoulli operator ${\mathcal{E}}^{0}u{:=}\rho {u}_{tt}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}$ plays the same role in the inverse problems (1) and (2). Specifically, this ultimately leads to the fact that the factor e^{−μ(T−t)/2} appears in the formula

$$\begin{equation*}{\sigma }_{n}=\frac{1}{{\omega }_{n}}\enspace {\int }_{0}^{T}\mathrm{sin}\left({\omega }_{n}\left(T-t\right)\right)\enspace G\left(t\right)\mathrm{d}t,\quad {\omega }_{n}=\sqrt{{\lambda }_{n}},\quad \mu =0,\end{equation*}$$

for the singular values σ_n corresponding to the undamped case, and the formula

$$\begin{align*}{\sigma }_{n}& =\frac{1}{{\omega }_{n}}\enspace {\int }_{0}^{T}{\text{e}}^{-\mu \left(T-t\right)/2}\enspace \mathrm{sin}\left({\omega }_{n}\left(T-t\right)\right)G\left(t\right)\mathrm{d}t,\\ {\omega }_{n}& =\frac{1}{2}\enspace \sqrt{4{\lambda }_{n}-{\mu }^{2}},\quad 0{< }\mu {< }2\sqrt{{\lambda }_{1}},\end{align*}$$

corresponding to the damped dynamic Euler–Bernoulli operator ${\mathcal{E}}^{\mu }u{:=}\rho {u}_{tt}+\mu {u}_{t}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}$ contains the 'regularization term' e^{−μ(T−t)/2} which ensures, under certain conditions on the damping parameter, the final time and the temporal load, the positiveness of the singular values σ_n , and thus the uniqueness of the solution to the inverse problems (1) and (2).

The analysis carried out in this paper allows us to obtain a sufficient condition for the uniqueness of the solution of the inverse problem. Moreover, this sufficient condition clearly demonstrates the relationship between the three decisive inputs: the final time T > 0, the damping coefficient μ > 0 and temporal load G(t).

At the end of this brief analysis it should be stated that although there are enough studies on inverse problems related to Euler–Bernoulli or wave equations (see [2, 6–11, 14–16, 19, 20] and references therein), none of these studies focused on the role of the damped coefficient in uniqueness of the solution. Only in [17] this issue was partially addressed and sufficient conditions for the uniqueness were derived for two specific cases (G(t) = 1 and G(t) = exp(−ηt), η > 0).

The paper is organized as follows. Section 2 introduces necessary estimates for the weak and regular weak solutions of the direct problem, and also for the output. Section 3 describes properties of the input–output operator corresponding to the inverse source problems (1) and (2), including its singular values. The sufficient condition for the uniqueness of the solution to the inverse problem is given in section 4. In section 5 the Tikhonov functional is introduced, and then existence of a quasi-solution to the inverse problem is proved. Furthermore, the Fréchet gradient of this functional is derived, and the Lipschitz continuity of this gradient is proved. The inverse source problem with time harmonic loading is examined separately, in section 6, due to the important applied significance of this problem. Issues related to the numerical implementation of both methods, their comparative analysis, as well as the results of computational experiments are presented in section 7. Some concluding remarks are provided in the final section 8.

Consider the direct problem (1) assuming that the initial conditions are non-homogeneous:

$$\begin{equation}\begin{cases}\rho \left(x\right){u}_{tt}+\mu {u}_{t}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}=F\left(x\right)G\left(t\right),\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ u\left(x,0\right)={u}_{0}\left(x\right),\hspace{25.0pt}{u}_{t}\left(x,0\right)={u}_{1}\left(x\right),\quad x\in \left(0,\ell \right),\quad \\ u\left(0,t\right)={u}_{xx}\left(0,t\right)=0,\hspace{25.0pt}u\left(\ell ,t\right)={u}_{xx}\left(\ell ,t\right)=0,\quad t\in \left[0,T\right],\quad \end{cases}\end{equation} \tag{ 9 }$$

Although some results and a priori estimates for the weak and regular weak solutions of the initial boundary value problem (9) have been discussed in [13], we derive these estimates in a form convenient for our purpose.

We assume that the inputs in (9) satisfy the following basic conditions:

$$\begin{equation}\begin{cases}\rho ,r\in {L}^{\infty }\left(0,\ell \right),\quad 0{< }\mu {\leqslant}{\mu }^{{\ast}},\quad \\ 0{< }{r}_{0}{\leqslant}r\left(x\right){\leqslant}{r}_{1},\hspace{25.0pt}0{< }{\rho }_{0}{\leqslant}\rho \left(x\right){\leqslant}{\rho }_{1},\quad x\in \left(0,\ell \right),\quad \\ {u}_{0}\in {H}^{2}\left(0,\ell \right),\hspace{25.0pt}{u}_{0}\left(0\right)={u}_{0}\left(\ell \right)=0,\hspace{25.0pt}{u}_{1}\in {L}^{2}\left(0,\ell \right),\quad \\ F\in {L}^{2}\left(0,\ell \right),\hspace{25.0pt}F\left(x\right)\enspace \not\equiv \enspace 0,\hspace{25.0pt}G\in {L}^{2}\left(0,T\right),\quad G\left(t\right){ >}0.\quad \end{cases}\end{equation} \tag{ 10 }$$

Theorem 1. Let conditions (10) hold. Then for the weak solution $u\in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ with u_t ∈ L²(0, T; L²(0, ℓ)) and u_tt ∈ L²(0, T; H⁻²(0, ℓ)) of the initial boundary value problem (9) the following estimates hold:

$$\begin{align}\begin{aligned}{\Vert}{u}_{t}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}& {\leqslant}{C}_{1}^{2}\enspace \left[{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}\right.\\ & \quad +\left.{C}_{u}^{2}\left({\Vert}{u}_{0}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\right)\right],\\ {\Vert}{u}_{xx}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}& {\leqslant}{C}_{2}^{2}\enspace \left[{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}\right.\\ & \quad +\left.{C}_{u}^{2}\left({\Vert}{u}_{0}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\right)\right],\end{aligned}\end{align} \tag{ 11 }$$

where ${\mathcal{V}}^{2}\left(0,\ell \right){:=}\left\{v\in {H}^{2}\left(0,\ell \right):\enspace v\left(0\right)=v\left(\ell \right)=0\right\}$,

$$\begin{equation}{C}_{1}^{2}=\mathrm{exp}\left(T/{\rho }_{0}\right)-1,\hspace{25.0pt}{C}_{2}^{2}={\rho }_{0}\enspace {C}_{1}^{2}/{r}_{0},\hspace{25.0pt}{C}_{u}^{2}=\mathrm{max}\left({r}_{1},\enspace {\rho }_{1}\right),\end{equation} \tag{ 12 }$$

and ρ₀, r₀ > 0 are the constants introduced in (10).

Proof. Multiply both sides of equation (9) by 2u_t (x, t), integrate it over Ω_t := (0, ℓ) × (0, t), use the identity

$$\begin{equation}2{\left(r\left(x\right){u}_{xx}\right)}_{xx}{u}_{t}\equiv 2{\left[{\left(r\left(x\right){u}_{xx}\right)}_{x}{u}_{t}-r\left(x\right){u}_{xx}{u}_{xt}\right]}_{x}+{\left(r\left(x\right){u}_{xx}^{2}\right)}_{t},\end{equation} \tag{ 13 }$$

and then apply the integration by parts formula. Taking into account the initial and boundary conditions in [8] we obtain the following energy identity:

$$\begin{align*}& {\int }_{0}^{\ell }\left[\rho \left(x\right){u}_{t}^{2}+r\left(x\right){u}_{xx}^{2}\right]\mathrm{d}x+2\mu {\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau }^{2}\enspace \mathrm{d}x\enspace \mathrm{d}\tau \\ & \hspace{25.0pt}=2{\int }_{0}^{t}{\int }_{0}^{\ell }F\left(x\right)G\left(\tau \right){u}_{\tau }\left(x,\tau \right)\mathrm{d}x\enspace \mathrm{d}\tau +{\int }_{0}^{\ell }\rho \left(x\right){u}_{t}^{2}\left(x,{0}^{+}\right)\mathrm{d}x\\ & \quad \hspace{25.0pt}+{\int }_{0}^{\ell }r\left(x\right){u}_{xx}^{2}\left(x,{0}^{+}\right)\mathrm{d}x,\quad t\in \left[0,T\right].\end{align*}$$

We deduce from this identity the main integral inequality:

$$\begin{align}& {\rho }_{0}{\int }_{0}^{\ell }{u}_{t}^{2}\enspace \mathrm{d}x+{r}_{0}{\int }_{0}^{\ell }{u}_{xx}^{2}\enspace \mathrm{d}x+2\mu {\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau }^{2}\enspace \mathrm{d}x\enspace \mathrm{d}\tau \\ & \hspace{25.0pt}{\leqslant}{\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau }^{2}\enspace \mathrm{d}x\enspace \mathrm{d}\tau +{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}\\ & \quad \hspace{25.0pt}+{\rho }_{1}{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+{r}_{1}{\Vert}{u}_{0}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2},\end{align} \tag{ 14 }$$

for all t ∈ [0, T]. Estimates (11) are easily derived from this inequality. □

Corollary 1. Assume that conditions of theorem 1 hold. Then for the final time output u(x, T) the following estimate holds:

$$\begin{align}{\Vert}u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}& {\leqslant}2T{C}_{1}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}+2\enspace \mathrm{max}\left(1,T\enspace {C}_{1}^{2}{C}_{u}^{2}\right)\\ & \quad \enspace {\times}\left[{\Vert}{u}_{0}{{\Vert}}_{{H}^{2}\left(0,\ell \right)}^{2}+{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\right],\end{align} \tag{ 15 }$$

where C₁, C_u > 0 are the constants introduced in (12).

Proof. The identity

$$\begin{equation}u\left(x,T\right)={u}_{0}\left(x\right)+{\int }_{0}^{T}{u}_{t}\left(x,t\right)\mathrm{d}t,\quad x\in \left[0,\ell \right],\end{equation} \tag{ 16 }$$

implies that

$$\begin{equation*}{\Vert}u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\leqslant}2{\Vert}{u}_{0}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+2T{\Vert}{u}_{t}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}.\end{equation*}$$

With the first estimate in [10] this leads to the required result. □

Theorem 2. Assume that in addition to the basic conditions (10), the inputs in (9) satisfy the following regularity conditions:

$$\begin{equation}\begin{cases}r\in {H}^{3}\left(0,\ell \right),\enspace G\in {H}^{1}\left(0,T\right),\quad \\ {u}_{0}\in {H}^{4}\left(0,\ell \right),\enspace {u}_{1}\in {H}^{2}\left(0,\ell \right).\quad \end{cases}\end{equation} \tag{ 17 }$$

Then for the regular weak solution $u\in {L}^{2}\left(0,T;{H}^{4}\left(0,\ell \right)\cap {\mathcal{V}}^{2}\left(0,\ell \right)\right)$, with u_t ∈ L²(0, T; H²(0, ℓ)), u_tt ∈ L²(0, T; L²(0, ℓ)) and u_ttt ∈ L²(0, T; H⁻²(0, ℓ)) of the initial boundary value problem (9) the following estimates hold:

$$\begin{equation}\begin{aligned}{\Vert}{u}_{tt}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}& {\leqslant}{C}_{1}^{2}\enspace \left[{C}_{3}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}\right.\\ & \quad +\left.{C}_{4}^{2}\left({\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2}+{\Vert}{u}_{1}{{\Vert}}_{{H}^{2}\left(0,\ell \right)}^{2}\right)\right],\\ {\Vert}{u}_{\mathit{\text{xxt}}}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}& {\leqslant}{C}_{2}^{2}\enspace \left[{C}_{3}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}\right.\\ & \quad +\left.{C}_{4}^{2}\left({\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2}+{\Vert}{u}_{1}{{\Vert}}_{{H}^{2}\left(0,\ell \right)}^{2}\right)\right],\enspace \end{aligned}\end{equation} \tag{ 18 }$$

where

$$\begin{equation}\left.\begin{matrix}\hfill {C}_{3}^{2}=1+3{C}_{T}^{2}/{\rho }_{0},\hspace{25.0pt}\enspace {C}_{r}={C}_{r}\left({\Vert}r{{\Vert}}_{{H}^{3}\left(0,\ell \right)}\right){ >}0,\hfill \\ \hfill {C}_{4}^{2}=\mathrm{max}\left({r}_{1},3{\mu }^{2}/{\rho }_{0},3{C}_{r}^{2}/{\rho }_{0}\right),\hspace{25.0pt}{C}_{T}^{2}=2\enspace \mathrm{max}\left(T/3,\enspace 1/T\right)\hfill \end{matrix}\right.\end{equation} \tag{ 19 }$$

and C₁, C₂ > 0 are the constants introduced in theorem 1.

Proof. Differentiate equation (9) with respect to the time variable t ∈ (0, T) and multiply both sides by 2u_tt (x, t). Integrating then it over Ω_t := (0, ℓ) × (0, t) and using the identity (13) with u(x, t), replaced by u_t (x, t), we obtain the following integral identity:

$$\begin{align}& {\int }_{0}^{\ell }\left[\rho \left(x\right){u}_{tt}^{2}+r\left(x\right){u}_{\mathit{\text{xxt}}}^{2}\right]\mathrm{d}x+2\mu {\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau \tau }^{2}\enspace \mathrm{d}x\enspace \;\mathrm{d}\tau \\ & \hspace{25.0pt} =2{\int }_{0}^{t}{\int }_{0}^{\ell }F\left(x\right){G}^{\prime }\left(\tau \right){u}_{\tau \tau }\left(x,\tau \right)\mathrm{d}x\enspace \;\mathrm{d}\tau +{\int }_{0}^{\ell }\rho \left(x\right){u}_{tt}^{2}\left(x,{0}^{+}\right)\mathrm{d}x\\ & \hspace{25.0pt} \quad +\enspace {\int }_{0}^{\ell }r\left(x\right){u}_{\mathit{\text{xxt}}}^{2}\left(x,{0}^{+}\right)\mathrm{d}x,\quad \enspace t\in \left[0,T\right].\end{align} \tag{ 20 }$$

We use the limit equation $\rho \left(x\right){u}_{tt}\left(x,{0}^{+}\right)+\mu {u}_{t}\left(x,{0}^{+}\right)+{\left(r\left(x\right){u}_{xx}\left(x,{0}^{+}\right)\right)}_{xx}=F\left(x\right)G\left({0}^{+}\right)$ at t = 0⁺ to estimate the second right-hand-side integral in the above identity. This yields:

$$\begin{align*}& {\int }_{0}^{\ell }\rho \left(x\right){u}_{tt}^{2}\left(x,{0}^{+}\right)\mathrm{d}x{\leqslant}\frac{3}{{\rho }_{0}}\enspace \left[{\left(G\left({0}^{+}\right)\right)}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+{\mu }^{2}{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+{\Vert}{\left(r\left(\cdot \right){u}_{0}^{{\prime\prime}}\right)}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\right].\end{align*}$$

Substituting this in (20) with the estimates

$$\begin{align*}{\Vert}{\left(r\left(\cdot \right){u}_{0}^{{\prime\prime}}\right)}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}& {\leqslant}{C}_{r}^{2}\enspace {\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2},\\ {\left(G\left({0}^{+}\right)\right)}^{2}& {\leqslant}{C}_{T}^{2}\enspace {\Vert}G{{\Vert}}_{{H}^{1}\left(0,\ell \right)}^{2},\end{align*}$$

with the constants C_r , C_T > 0 introduced in (19), we arrive at the following integral inequality:

$$\begin{align*}& {\rho }_{0}{\int }_{0}^{\ell }{u}_{tt}^{2}\mathrm{d}x+{r}_{0}{\int }_{0}^{\ell }{u}_{\mathit{\text{xxt}}}^{2}\mathrm{d}x+2\mu {\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau \tau }^{2}\enspace \mathrm{d}x\enspace \mathrm{d}\tau \\ & \hspace{25.0pt} {\leqslant}{\int }_{0}^{t}{\int }_{0}^{\ell }{u}_{\tau \tau }^{2}\enspace \mathrm{d}x+\left({\Vert}{G}^{\prime }{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}+\frac{3}{{\rho }_{0}}\enspace {C}_{T}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}\right){\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\\ & \hspace{25.0pt} \quad +\frac{3{\mu }^{2}}{{\rho }_{0}}{\Vert}{u}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+\frac{3}{{\rho }_{0}}{C}_{r}^{2}{\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2}+{r}_{1}{\Vert}{u}_{1}^{{\prime\prime}}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2},\end{align*}$$

for all t ∈ [0, T].

This inequality leads to the required estimates (18). □

Corollary 2. Assume that conditions of theorem 2 hold. Then the following estimate holds:

$$\begin{align}{\Vert}{u}_{x}\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}& {\leqslant}T{\ell }^{2}{C}_{2}^{2}{C}_{3}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}\\ & \quad +\enspace \left(2+T{\ell }^{2}{C}_{2}^{2}{C}_{4}^{2}\right){\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2}+T{\ell }^{2}{C}_{2}^{2}{C}_{4}^{2}{\Vert}{u}_{1}{{\Vert}}_{{H}^{2}\left(0,\ell \right)}^{2},\enspace \end{align} \tag{ 21 }$$

where C₂, C₃, C₄ > 0 are the constants introduced in theorems 1 and 2.

Proof. We use an analogue of the identity (16) with u(x, t) replaced by u_x (x, t) to deduce that

$$\begin{equation*}{\Vert}{u}_{x}\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\leqslant}2{\Vert}{u}_{0}^{\prime }{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}+2T{\Vert}{u}_{xt}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}.\end{equation*}$$

In view of the inequality ${\Vert}{u}_{xt}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}{\leqslant}\left({\ell }^{2}/2\right)\enspace {\Vert}{u}_{\mathit{\text{xxt}}}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}$ and the second estimate in (18), we arrive at the required result. □

Corollary 3. Assume that the conditions of theorem 2 are satisfied. Then for the final time output u_t (x, T) the following estimate holds:

$$\begin{align}{\Vert}{u}_{t}\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}& {\leqslant}2T{C}_{1}^{2}{C}_{3}^{2}{\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}\\ & \quad +\enspace 2T{C}_{1}^{2}{C}_{4}^{2}{\Vert}{u}_{0}{{\Vert}}_{{H}^{4}\left(0,\ell \right)}^{2}+2\left(1+T{C}_{1}^{2}{C}_{4}^{2}\right){\Vert}{u}_{1}{{\Vert}}_{{H}^{2}\left(0,\ell \right)}^{2},\enspace \end{align} \tag{ 22 }$$

with the constants introduced in theorems 1 and 2.

Proof of this corollary is similar to the proof of corollary 1. □

In the case of u₀(x) = u₁(x) = 0, the above estimates (11), (15), (17), (21) and (22) apply equally to the direct problem (1).

For convenience of the SVE analysis, here we assume that the mass density is a constant: ρ(x) ≡ ρ > 0.

Let us define the set of admissible spatial loads:

$$\begin{equation}\mathcal{F}=\left\{F\in {L}^{2}\left(0,\ell \right):\enspace {\Vert}F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{\gamma }_{F},\enspace {\gamma }_{F}{ >}0\right\}.\end{equation} \tag{ 23 }$$

Denote by u(x, t; F) the weak solution of the direct problem (1) corresponding to the given $F\in \mathcal{F}$. Introduce the input–output operator:

$$\begin{equation}\left({\Phi}F\right)\left(x\right){:=}u\left(x,T;F\right),\hspace{25.0pt}{\Phi}:\mathcal{F}\subset {L}^{2}\left(0,\ell \right){\mapsto}{L}^{2}\left(0,\ell \right).\end{equation} \tag{ 24 }$$

Then we can reformulate the inverse problems (1) and (2) in terms of this operator as follows:

$$\begin{equation}{\Phi}F={u}_{T},\quad F\in \mathcal{F},\quad {u}_{T}\in {L}^{2}\left(0,\ell \right).\end{equation} \tag{ 25 }$$

Lemma 1. Assume that conditions of theorem 2 hold. Then the input–output operator introduced in (25) is a linear compact operator.

Proof. Let $\left\{{F}^{\left(m\right)}\right\}\subset \mathcal{F}$, $m=\bar{1,\infty }$ be a sequence of inputs. Denote by {u^(m) (x, t)}, u^(m) (x, t) := u(x, t; F^(m)), the sequence of corresponding regular weak solutions of the direct problem (1). Then

$$\begin{equation*}\left\{\left({\Phi}{F}^{\left(m\right)}\right)\left(x\right)\right\},\enspace \left({\Phi}{F}^{\left(m\right)}\right)\left(t\right)={u}^{\left(m\right)}\left(x,T\right),\end{equation*}$$

is the sequence of outputs. We need to prove that the sequence {u^(m) (x, T)} is a relatively compact subset of L²(0, T) or, equivalently, the sequence $\left\{{u}_{x}^{\left(m\right)}\left(x,T\right)\right\}$ is bounded in the norm of the space H¹(0, ℓ). The last assertion follows from the estimate (21) applied to the solution of direct problem (1) with the input F^(m) (x). Namely,

$$\begin{equation*}{\Vert}{u}_{x}^{\left(m\right)}\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\leqslant}{C}_{T}^{2}{C}_{2}^{2}\left(1+{C}_{3}^{2}\right){\Vert}{F}^{\left(m\right)}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{H}^{1}\left(0,T\right)}^{2}.\end{equation*}$$

The norms ${\Vert}{F}^{\left(m\right)}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}$ are uniformly bounded by the definition (23). Hence, the sequence of norms {u^(m) (x, T)} are uniformly bounded in the norm of H¹(0, ℓ), that is, the input–output map ${\Phi}\left[\cdot \right]:\mathcal{F}{\mapsto}{L}^{2}\left(0,\ell \right)$ transforms a bounded in $\mathcal{F}\subset {L}^{2}\left(0,\ell \right)$ set to a relatively compact set in L²(0, ℓ). This implies the desired assertion. □

Compactness of the input–output operator implies that the inverse problems (1) and (2) is ill-posed.

Lemma 2. Assume that conditions of theorem 1 hold. Then the input–output operator introduced in (25) is self-adjoint. Furthermore,

$$\begin{equation}\left({\Phi}{\psi }_{n}\right)\left(x\right)={\sigma }_{n}{\psi }_{n}\left(x\right),\quad n=1,2,3,\dots ,\end{equation} \tag{ 26 }$$

where ${\left\{{\sigma }_{n},{\psi }_{n}\right\}}_{n=1}^{\infty }$ is the eigensystem of the input–output operator Φ, and

$$\begin{equation}\begin{gathered}{\sigma }_{n}=\frac{1}{{\omega }_{n}}\enspace {\int }_{0}^{T}{\text{e}}^{-\mu \left(T-t\right)/2}\enspace \mathrm{sin}\left({\omega }_{n}\left(T-t\right)\right)\enspace G\left(t\right)\mathrm{d}t,\\ {\omega }_{n}=\frac{1}{2}\sqrt{4{\lambda }_{n}-{\mu }^{2}},\quad \mu {< }2\sqrt{{\lambda }_{n}},\end{gathered}\end{equation} \tag{ 27 }$$

$$\begin{equation}{\sigma }_{n}={\int }_{0}^{T}\left(T-t\right){\text{e}}^{-\mu \left(T-t\right)/2}G\left(t\right)\mathrm{d}t,\quad \mu =2\sqrt{{\lambda }_{n}},\hspace{25.0pt}\hspace{25.0pt}\hspace{25.0pt}\enspace \end{equation} \tag{ 28 }$$

$$\begin{equation}\begin{gathered}{\sigma }_{n}=\frac{1}{2{\hat{\omega }}_{n}}{\int }_{0}^{T}{\text{e}}^{-\mu \left(T-t\right)/2}\left[{\text{e}}^{{\hat{\omega }}_{n}\left(T-t\right)}-{\text{e}}^{-{\hat{\omega }}_{n}\left(T-t\right)}\right]G\left(t\right)\mathrm{d}t,\\ \mu { >}2\sqrt{{\lambda }_{n}},\quad {\hat{\omega }}_{n}=\frac{1}{2}\sqrt{{\mu }^{2}-4{\lambda }_{n}},\end{gathered}\end{equation} \tag{ 29 }$$

while λ_n is the eigenvalue of the Bernoulli operator $\left(\mathcal{B}w\right)\left(x\right){:=}{\left(r\left(x\right){w}^{{\prime\prime}}\left(x\right)\right)}^{{\prime\prime}}$ defined on $\mathcal{D}\left(B\right){:=}\left\{w\in {\mathcal{V}}^{2}\left(0,\ell \right)\cap {H}^{4}\left(0,\ell \right):\enspace {w}^{{\prime\prime}}\left(0\right)={w}^{{\prime\prime}}\left(\ell \right)=0\right\}$, provided that there exists a function ψ_n , not identically equal to zero, solving the problem

$$\begin{equation}\begin{cases}{\left(r\left(x\right){\psi }_{n}^{{\prime\prime}}\left(x\right)\right)}^{\prime \prime }={\lambda }_{n}{\psi }_{n}\left(x\right),\quad x\in \left(0,\ell \right),\quad \\ {\psi }_{n}\left(0\right)={\psi }_{n}^{{\prime\prime}}\left(0\right)={\psi }_{n}\left(\ell \right)={\psi }_{n}^{{\prime\prime}}\left(\ell \right)=0.\quad \end{cases}\end{equation} \tag{ 30 }$$

Moreover, the input–output operator possesses the following SVE:

$$\begin{equation}\left({\Phi}F\right)\left(x\right)=\sum\limits _{n=1}^{\infty }{\sigma }_{n}{F}_{n}{\psi }_{n}\left(x\right),\end{equation} \tag{ 31 }$$

where ${F}_{n}{:=}{\left(F,{\psi }_{n}\right)}_{{L}^{2}\left(0,\ell \right)}$ is the nth Fourier coefficient of F(x) and σ_n is defined by expressions (27) to (29).

Proof of this lemma is given in [17]. □

Evidently, the Bernoulli operator defined on $\mathcal{D}\left(B\right)$ is a self-adjoint and positive definite operator. Hence the eigenvalues are positive, and 0 < λ₁ ⩽ λ₂ ⩽ λ₁ ⩽ ⋯. Moreover, they have the asymptotic property $\mathcal{O}\left({n}^{4}\right)$ [2]. In addition, the system ${\left\{{\psi }_{n}\right\}}_{n=1}^{\infty }$, forms an orthonormal basis for L²(0, ℓ), and hence the Fourier series expansion

$$\begin{equation}w\left(x\right)=\sum\limits _{n=1}^{\infty }{w}_{n}{\psi }_{n}\left(x\right),\hspace{25.0pt}{w}_{n}{:=}{\left(w,{\psi }_{n}\right)}_{{L}^{2}\left(0,\ell \right)}\end{equation} \tag{ 32 }$$

for the weak solution of the eigenvalue problem (30) converges in L²(0, ℓ). We prove that in fact this series also converges in ${\mathcal{V}}^{2}\left(0,\ell \right)\subset {H}^{2}\left(0,\ell \right)$. Indeed, denote by

$$\begin{equation}B\left[w,v\right]{:=}{\int }_{0}^{\ell }r\left(x\right){w}^{{\prime\prime}}\left(x\right){v}^{{\prime\prime}}\left(x\right)\mathrm{d}x,\quad w,v\in {\mathcal{V}}^{2}\left(0,\ell \right)\end{equation} \tag{ 33 }$$

the symmetric bilinear form associated by the Bernoulli operator $\left(\mathcal{B}w\right)\left(x\right){:=}{\left(r\left(x\right){w}^{{\prime\prime}}\left(x\right)\right)}^{{\prime\prime}}$ defined on $\mathcal{D}\left(B\right)$. Remark that we may define the bilinear form (33) also through the regular weak solution as $w,v\in \mathcal{D}\left(B\right)={\mathcal{V}}^{2}\left(0,\ell \right)\cap {H}^{4}\left(0,\ell \right)$. However, the statements proved below show that they are true precisely for a weak solution, that is, under weaker regularity requirement $w,v\in {\mathcal{V}}^{2}\left(0,\ell \right)$.

Evidently, the energy norm ${\left(B\left[w,w\right]\right)}^{1/2}$ is equivalent to the norm ${\Vert}w{{\Vert}}_{{\mathcal{V}}^{2}\left(0,\ell \right)}$. Furthermore, from (30) it follows that

$$\begin{equation}B\left[{\psi }_{n},{\psi }_{m}\right]={\lambda }_{n}{\left({\psi }_{n},{\psi }_{m}\right)}_{{L}^{2}\left(0,\ell \right)}={\lambda }_{n}{\delta }_{n,m},\quad n,m=\bar{1,\infty },\end{equation} \tag{ 34 }$$

or

$$\begin{equation*}B\left[\frac{{\psi }_{n}}{\sqrt{{\lambda }_{n}}}\enspace ,\frac{{\psi }_{m}}{\sqrt{{\lambda }_{n}}}\right]={\delta }_{n,m},\quad n,m=\bar{1,\infty },\end{equation*}$$

where δ_n,m is the Kronecker symbol. This implies that the system ${\left\{{\psi }_{n}/\sqrt{{\lambda }_{n}}\enspace \right\}}_{n=0}^{\infty }$ is an orthonormal subset of ${\mathcal{V}}^{2}\left(0,\ell \right)$ endowed with the new inner product (33). We prove that ${\left\{{\psi }_{n}/\sqrt{{\lambda }_{n}}\enspace \right\}}_{n=0}^{\infty }$ is in fact an orthonormal basis of ${\mathcal{V}}^{2}\left(0,\ell \right)$. To this end, we need to show that $B\left[{\psi }_{n}/\sqrt{{\lambda }_{n}},w\right]=0$, for all n = 1, 2, 3, ⋯, implies w ≡ 0. But this assertion is evidently holds since $B\left[{\psi }_{n}/\sqrt{{\lambda }_{n}},w\right]=\sqrt{{\lambda }_{n}}{\left({\psi }_{n},w\right)}_{{L}^{2}\left(0,\ell \right)}$ by (34), and the conditions

$$\begin{equation*}{\left({\psi }_{n},w\right)}_{{L}^{2}\left(0,\ell \right)}=0,\quad \text{for}\enspace \text{all}\quad n=1,2,3,\cdots \end{equation*}$$

imply w(x) ≡ 0, as ${\left\{{\psi }_{n}\right\}}_{n=0}^{\infty }$ is a basis for L²(0, ℓ). Thus, ${\left\{{\psi }_{n}/\sqrt{{\lambda }_{n}}\enspace \right\}}_{n=0}^{\infty }$ is an orthonormal basis of ${\mathcal{V}}^{2}\left(0,\ell \right)$ and, as a consequence, the series

$$\begin{equation*}\sum\limits _{n=1}^{\infty }{\hat{w}}_{n}\frac{{\psi }_{n}}{\sqrt{{\lambda }_{n}}},\quad {\hat{w}}_{n}{:=}B\left[w,\enspace \frac{{\psi }_{n}}{\sqrt{{\lambda }_{n}}}\right].\end{equation*}$$

converges in ${\mathcal{V}}^{2}\left(0,\ell \right)$. Comparing this series with the series (32) we find that ${\hat{w}}_{n}=\sqrt{{\lambda }_{n}}\enspace {w}_{n}$. This means that the series (32) in fact converges also in ${\mathcal{V}}^{2}\left(0,\ell \right)$.

Remark 1. The cases $\mu {< }2\sqrt{{\lambda }_{n}}\enspace$, $\mu =2\sqrt{{\lambda }_{n}}$ and $\mu { >}2\sqrt{{\lambda }_{n}}$ correspond to underdamped, critically damped and overdamped vibrating systems, according to the commonly accepted classification [21]. It should be emphasized that only one term ${\sigma }_{{n}_{{\ast}}}$ associated with the case $\mu =2\sqrt{{\lambda }_{{n}_{{\ast}}}}$, introduced in (28), can appear in the expansion (31). If $\mu =2\sqrt{{\lambda }_{{n}_{{\ast}}}}$ and n_* > 1, then the terms σ_n , n = 1, 2, ⋯, n_* − 1 associated with the case $\mu { >}2\sqrt{{\lambda }_{n}}$ and defined by (29), appear in the expansion (31), due to the fact that the sequence of eigenvalues ${\left\{{\lambda }_{n}\right\}}_{n=1}^{\infty }$ increases monotonically as n → ∞. Finally, the case $\mu \in \left(2\sqrt{{\lambda }_{m}},2\sqrt{{\lambda }_{m+1}}\right)$ means that the terms σ₁, ⋯, σ_m in the expansion (31) are defined by the integral in (29). □

Remark 2. In the case where the flexural rigidity in (1) is a constant, r(x) ≡ r = constant > 0, one can prove that the eigensystem ⟨λ_n , ψ_n (x)⟩, n = 1, 2, ⋯ of the Bernoulli operator $\mathcal{B}$ defined on $\mathcal{D}\left(B\right)$ are given by (see, for instance [21])

$$\begin{equation*}{\lambda }_{n}=\frac{{\pi }^{4}\enspace r}{{\ell }^{4}\rho }\enspace {n}^{4},\hspace{25.0pt}{\psi }_{n}\left(x\right)=\sqrt{2/\ell }\enspace \mathrm{sin}\enspace \frac{\pi nx}{\ell },\quad x\in \left(0,\ell \right),\quad n=1,2,\cdots \enspace .\end{equation*}$$

The left formula also shows the dependence of the eigenvalues λ_n on the physical and geometrical parameters of the Euler–Bernoulli beam. Thus, λ_n (r₁, ρ) < λ_n (r₂, ρ), if r₁ < r₂, and λ_n (r, ρ₁) > λ_n (r, ρ₂), if ρ₁ < ρ₂, that is increase of the flexural rigidity r > 0 of a homogeneous beam leads to increase of the values of the eigenvalues, and, conversely, increase of the density ρ > 0 of the beam leads to decrease of the values of the eigenvalues. □

Formula (27) shows that even the positivity G(t) > 0 of the temporal load cannot guarantee the positivity σ_n > 0 of the singular values for all n = 1, 2, ⋯, which means that the expansion (31) of the input–output operator is only formal. Furthermore, if σ_n = 0 for some n, then the nth Fourier coefficient of the function F(x) cannot be taken into account in the expansion (31). Note that Picard criterion (7) also requires the positivity of the singular values.

These considerations show that before proceeding with the solution of the inverse problems (1) and (2), by any method, one needs first obtain a sufficient condition for the positivity of the singular values defined in (27) to (29).

When G(t) > 0, the positivity of the singular values, corresponding to the critically damped $\mu =2\sqrt{{\lambda }_{{n}_{{\ast}}}}$ and overdamped $\mu { >}2\sqrt{{\lambda }_{n}}$ cases, is obvious, as it follows from expressions (28) and (29). Hence, in these cases no additional condition needs to be imposed on the final time for the positivity of the singular values.

Consequently, we consider the underdamped case $\mu {< }2\sqrt{{\lambda }_{1}}$, which is the most common case. We recall that by virtue of the monotonicity 0 < λ₁ ⩽ λ₂ ⩽ λ₃ < ⋯, the condition $\mu {< }2\sqrt{{\lambda }_{1}}$ guarantees the fulfilment of the condition $\mu {< }2\sqrt{{\lambda }_{n}}$ for all n > 1.

Theorem 3. Let conditions (10) hold. Assume that the damping coefficient satisfies the condition $\mu {< }2\sqrt{{\lambda }_{1}}$ and the temporal load G(t) belongs to H¹(0, T). Suppose that the damping coefficient, final time and the temporal load satisfy the following inequality:

$$\begin{align}G\left(T\right)& { >}\left(G\left(0\right){\text{e}}^{-\mu T/2}+{\left(\left(1-{\text{e}}^{-\mu T}\right)/\mu \right)}^{1/2}{\Vert}{G}^{\prime }{{\Vert}}_{{L}^{2}\left(0,T\right)}\right)\\ & \quad {\times}{\left(1-{\left(\mu /\left(2\sqrt{{\lambda }_{1}}\right)\right)}^{2}\right)}^{-1/2}.\end{align} \tag{ 35 }$$

Then the SVE (6) of the solution of the inverse problems (1) and (2) is unique.

Proof. We transform the integral in (27) as follows:

$$\begin{align*}{\sigma }_{n}& =\frac{1}{{\omega }_{n}}\enspace {\int }_{0}^{T}{\text{e}}^{-\mu t/2}\enspace \mathrm{sin}\left({\omega }_{n}t\right)G\left(T-t\right)\mathrm{d}t\\ & =\frac{1}{{\omega }_{n}^{2}}\left[G\left(T\right)-G\left(0\right){\text{e}}^{-\mu T/2}\enspace \mathrm{cos}\left({\omega }_{n}T\right)\right.\\ & \quad -\left.{\int }_{0}^{T}{\text{e}}^{-\mu t/2}\enspace \mathrm{cos}\left({\omega }_{n}t\right)\left(\frac{\mu }{2}G\left(T-t\right)+{G}^{\prime }\left(T-t\right)\right)\enspace \mathrm{d}t\right]\\ & =\frac{1}{{\omega }_{n}^{2}}\left\{G\left(T\right)-G\left(0\right){\text{e}}^{-\mu T/2}\left[\mathrm{cos}\left({\omega }_{n}T\right)+\frac{\mu }{2{\omega }_{n}}\enspace \mathrm{sin}\left({\omega }_{n}T\right)\right]\right.\\ & \quad -{\int }_{0}^{T}{\text{e}}^{-\mu t/2}\left[\mathrm{sin}\left({\omega }_{n}t\right)\left(\frac{{\mu }^{2}}{4{\omega }_{n}}G\left(T-t\right)+\frac{\mu }{2{\omega }_{n}}{G}^{\prime }\left(T-t\right)\right)\right.\\ & \quad \left.+\left.\mathrm{cos}\left({\omega }_{n}t\right){G}^{\prime }\left(T-t\right)\right]\enspace \mathrm{d}t\right\}.\end{align*}$$

After rearranging the terms, this gives

$$\begin{align*}{\sigma }_{n}& =\frac{1}{{\omega }_{n}^{2}+{\left(\mu /2\right)}^{2}}\left\{G\left(T\right)-{\text{e}}^{-\mu T/2}G\left(0\right)\left[\mathrm{cos}\left({\omega }_{n}T\right)+\frac{\mu }{2{\omega }_{n}}\enspace \mathrm{sin}\left({\omega }_{n}T\right)\right]\right.\\ & \quad -\left.{\int }_{0}^{T}{\text{e}}^{-\mu t/2}\left[\frac{\mu }{2{\omega }_{n}}\enspace \mathrm{sin}\left({\omega }_{n}t\right)+\mathrm{cos}\left({\omega }_{n}t\right)\right]{G}^{\prime }\left(T-t\right)\enspace \mathrm{d}t\right\}.\end{align*}$$

Use here the inequality $\vert a\enspace \mathrm{cos}\enspace \alpha +b\enspace \mathrm{sin}\enspace \alpha \vert {\leqslant}\sqrt{{a}^{2}+{b}^{2}}$ with the estimate

$$\begin{align*}& \left\vert {\int }_{0}^{T}{\text{e}}^{-\mu \tau /2}\left[\mathrm{sin}\left({\omega }_{n}\tau \right)\frac{\mu }{2{\omega }_{n}}+\mathrm{cos}\left({\omega }_{n}\tau \right)\right]{G}^{\prime }\left(T-\tau \right)\mathrm{d}\tau \right\vert \\ & \hspace{25.0pt} {\leqslant}{\left(\frac{1-{\text{e}}^{-\mu T}}{\mu }\right)}^{1/2}{\Vert}{G}^{\prime }{{\Vert}}_{{L}^{2}\left(0,T\right)}{\left[1+{\left(\frac{\mu }{2{\omega }_{n}}\right)}^{2}\right]}^{1/2}.\end{align*}$$

Taking into account also the relations

$$\begin{align*}\begin{gathered}{\omega }_{n}^{2}+{\left(\mu /2\right)}^{2}={\lambda }_{n},\\ 1+{\left(\frac{\mu }{2{\omega }_{n}}\right)}^{2}={\left(1-{\left(\mu /\left(2\sqrt{{\lambda }_{n}}\right)\right)}^{2}\right)}^{-1},\end{gathered}\end{align*}$$

we obtain

$$\begin{align*}{\sigma }_{n}& {\geqslant}\frac{1}{{\lambda }_{n}}\left\{G\left(T\right)-\left(G\left(0\right)\enspace {\text{e}}^{-\mu T/2}+{\left(\left(1-{\text{e}}^{-\mu T}\right)/\mu \right)}^{1/2}\enspace {\Vert}{G}^{\prime }{{\Vert}}_{{L}^{2}\left(0,T\right)}\right)\right.\\ & \quad {\times}\left.{\left(1-{\left(\mu /\left(2\sqrt{{\lambda }_{1}}\right)\right)}^{2}\right)}^{-1/2}\right\}.\end{align*}$$

This inequality shows that under condition (35), all singular values σ_n defined in (27) are positive. This implies the required assertion of the theorem. □

Remark 3. Consider the case of pure spatial load, that is G(t) = 1 in (1). In this case, the inequality (35) holds for all large enough values of the final time T > 0. □

Remark 4. Consider the case $\mu =\sqrt{2{\lambda }_{1}}$. Then

$$\begin{equation*}{\left[1+\frac{{\mu }^{2}}{4{\lambda }_{1}-{\mu }^{2}}\right]}^{1/2}=\sqrt{2},\hspace{25.0pt}{\text{e}}^{-\mu T/2}={\text{e}}^{-T\sqrt{{\lambda }_{1}/2}},\hspace{25.0pt}{\left(\frac{1-{\text{e}}^{-\mu T}}{\mu }\right)}^{1/2}{< }\frac{1}{\sqrt{\mu }}.\end{equation*}$$

In this case the inequality (35) is valid for all large enough values of T > 0, and $G\left(T\right){ >}{\Vert}{G}^{\prime }{{\Vert}}_{{L}^{2}\left(0,T\right)}{\left(2/{\lambda }_{1}\right)}^{1/4}$. The latter inequality holds if, for example, G(t) = exp(αt) with large enough α > 0. □

Evidently, only the positivity condition (35) does not guarantee the convergence of the series (6), as the Picard criterion (7) shows. Indeed, from formula (27) it follows that

$$\begin{align*}0{< }{\sigma }_{n}& {< }\frac{1}{{\omega }_{n}}\enspace {\left({\int }_{0}^{T}{\mathrm{sin}}^{2}\left({\omega }_{n}t\right)\mathrm{d}t\right)}^{1/2}{\left({\int }_{0}^{T}{G}^{2}\left(T-t\right)\mathrm{d}t\right)}^{1/2}\\ & {\leqslant}\frac{\sqrt{T}}{\sqrt{{\lambda }_{n}-{\mu }^{2}/\left(2{\lambda }_{n}\right)}}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}.\end{align*}$$

With the asymptotic property ${\lambda }_{n}\sim \mathcal{O}\left({n}^{4}\right)$ this implies that the singular values σ_n , n = 1, 2, 3, ⋯ have the asymptotic property $\mathcal{O}\left({n}^{-2}\right)$. As a consequence of this and the Picard criterion (7) we deduce that the series (6) converges if and only if

$$\begin{equation}\sum\limits _{n=1}^{\infty }{n}^{4}{u}_{T,n}^{2}{< }\infty .\end{equation} \tag{ 36 }$$

Based on characterization of Sobolev spaces by Fourier transform [5], we conclude that (36) holds, if the measured output u_T (x) satisfies the following regularity and consistency conditions:

$$\begin{equation}{u}_{T}\in {H}^{2}\left(0,\ell \right),\hspace{25.0pt}{u}_{T}\left(0\right)={u}_{T}\left(\ell \right)=0.\end{equation} \tag{ 37 }$$

Theorem 4. Assume that conditions of theorem 3 hold. Suppose that the measured output u_T (x) defined in (2) satisfies the conditions (37). Then the inverse problems (1) and (2) has a unique solution. Furthermore, this solution possesses the convergent SVE given by (6), with the singular values defined in (27).

From the above analysis it follows that the main merit of the approach based on the singular value decomposition of the input–output operator, is that it allows us to find the relationship between the main inputs, namely, the final time, the damping coefficient and the temporal load, in which the inverse problem has a unique solution. Of course, it provides a simple computational algorithm for recovering the unknown spatial load according to the singular value decomposition (6) and formulas (27) to (29). However, due to the rapid decay $\mathcal{O}\left({n}^{-2}\right)$ of the singular values, in fact, only a few Fourier coefficients u_T,n of the final time output u_T (x) can be used in the series (6) to recover F(x). Moreover, if the output u_T (x) contains a low random noise of, say, 10⁻², in the 3rd or 4th terms of the series (6) this error is amplified by the factor of the order 10! The second drawback of this method is that it requires very smooth final time output as the conditions (37) show, which is not always the case in applications. In the case of non-smooth data u_T ∈ L²(0, T), of course, the convergence of the functional series (6) cannot be guaranteed. The advantage of the adjoint problem approach based on minimization the Tikhonov functional is, in particular, that this method does not require such a restriction.

The measured output u_T (x) always contains a random noise and, as a result, exact equality in equation (25) is not possible in practice. Hence one needs to introduce the Tikhonov functional

$$\begin{equation}J\left(F\right)=\frac{1}{2}\enspace {\Vert}{\Phi}F-{u}_{T}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2},\quad F\in \mathcal{F}\end{equation} \tag{ 38 }$$

and reformulate the inverse problems (1) and (2) as the minimization problem

$$\begin{equation}J\left({F}_{{\ast}}\right)=\underset{F\in \mathcal{F}}{\mathrm{inf}}\enspace J\left(F\right)\end{equation} \tag{ 39 }$$

for this functional. A solution of this problem is defined as a quasi-solution of the inverse problems (1) and (2).

Lemma 3. Let conditions (10) hold and u_T ∈ L²(0, ℓ). Then the Tikhonov functional is Lipschitz continuous, that is

$$\begin{equation}\vert J\left({F}_{1}\right)-J\left({F}_{2}\right)\vert {\leqslant}{L}_{J}{\Vert}{F}_{1}-{F}_{2}{{\Vert}}_{{L}^{2}\left(0,\ell \right)},\quad \forall \enspace {F}_{1},{F}_{2}\in \mathcal{F},\end{equation} \tag{ 40 }$$

where

$$\begin{equation*}{L}_{J}={C}_{3}\left[{C}_{3}{\gamma }_{F}+{\Vert}{u}_{T}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}\right]{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}\end{equation*}$$

and C₃ > 0 is the constant defined in corollary 1.

Proof. We estimate the difference |J(F₁) − J(F₂)| as follows:

$$\begin{align}\left\vert J\left({F}_{1}\right)-J\left({F}_{2}\right)\right\vert & {\leqslant}\frac{1}{2}\left[\enspace {\Vert}{\Phi}{F}_{1}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}+{\Vert}{\Phi}{F}_{2}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}+2{\Vert}{u}_{T}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}\right]\enspace {\Vert}{\Phi}{F}_{1}-{\Phi}{F}_{2}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}.\end{align} \tag{ 41 }$$

We use the definition ${\Vert}{\Phi}{F}_{1}-{\Phi}{F}_{2}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}={\Vert}\delta u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}$, where δu(x, t) := u(x, t; F₁) − u(x, t; F₂) solves the following problem

$$\begin{equation}\begin{cases}\rho \left(x\right)\delta {u}_{tt}+\mu \delta {u}_{t}+{\left(r\left(x\right)\delta {u}_{xx}\right)}_{xx}=\delta F\left(x\right)G\left(t\right),\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ \delta u\left(x,0\right)=\delta {u}_{t}\left(x,0\right)=0,\quad x\in \left(0,\ell \right),\quad \\ \delta u\left(0,t\right)=\delta {u}_{xx}\left(0,t\right)=0,\hspace{25.0pt}\delta u\left(\ell ,t\right)=\delta {u}_{xx}\left(\ell ,t\right)=0,\quad t\in \left[0,T\right],\quad \end{cases}\end{equation} \tag{ 42 }$$

and estimate right-hand-side norm ${\Vert}\delta u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}$ in (41). From estimate (15) in corollary 1 applied to the solution of problem (42) it follows that

$$\begin{equation}{\Vert}\delta u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{C}_{3}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}{\Vert}\delta F{{\Vert}}_{{L}^{2}\left(0,\ell \right)},\end{equation} \tag{ 43 }$$

that is, the input–output operator defined in (24) is Lipschitz continuous. Substituting this estimate in (41) with the estimate

$$\begin{equation*}{\Vert}{\Phi}{F}_{i}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{C}_{3}{\Vert}{F}_{i}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)},\quad \forall \enspace {F}_{i}\in \mathcal{F},\end{equation*}$$

i = 1, 2, and taking into account that ${\Vert}{F}_{i}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{\gamma }_{F}$, we obtain:

$$\begin{equation*}\left\vert J\left({F}_{1}\right)-J\left({F}_{2}\right)\right\vert {\leqslant}{C}_{3}\left[{C}_{3}{\gamma }_{F}+{\Vert}{u}_{T}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}\right]{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}{\Vert}\delta F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}.\end{equation*}$$

□

Theorem 5. Under the conditions (10) and u_T ∈ L²(0, ℓ), there exists a quasi-solution of the inverse problems (1) and (2).

Proof. The proof follows from the generalized Weierstrass existence theorem (section 2.5, theorem 2D) since the Tikhonov functional is lower semicontinuous, as a Lipschitz continuous functional, and is defined on the closed convex set $\mathcal{F}$. □

Now, assume that $F,F+\delta F\in \mathcal{F}$ and $u\in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ is the solution of the direct problem (1), corresponding to the given spatial load $F\in \mathcal{F}$. Obviously the function δu(x, t) := u(x, t; F + δF) − u(x, t; F) solves the problem (42). Then the first variation δJ(F) := J(F + δF) − J(F) of the Tikhonov functional J(F) is

$$\begin{equation}\delta J\left(F\right)={\int }_{0}^{\ell }\left[u\left(x,T,F\right)-{u}_{T}\left(x\right)\right]\delta u\left(x,T\right)\mathrm{d}x+\frac{1}{2}{\int }_{0}^{l}{\left[\delta u\left(x,T\right)\right]}^{2}\mathrm{d}x,\quad \delta F\in \mathcal{F}.\end{equation} \tag{ 44 }$$

Lemma 4. Let conditions (10) hold and u_T ∈ L²(0, ℓ). Then between the inputs and outputs the following integral relationship holds:

$$\begin{equation}-{\int }_{0}^{\ell }\rho \left(x\right)q\left(x\right)\delta u\left(x,T\right)\mathrm{d}x={\int }_{0}^{\ell }\left({\int }_{0}^{T}G\left(t\right)\phi \left(x,t;q\right)\mathrm{d}t\right)\delta F\left(x\right)\mathrm{d}x,\end{equation} \tag{ 45 }$$

for all F ∈ L²(0, ℓ), where $\phi \in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ is the weak solution of the following backward problem

$$\begin{equation}\begin{cases}\rho \left(x\right){\phi }_{tt}-\mu \left(x\right){\phi }_{tt}+{\left(r\left(x\right){\phi }_{xx}\right)}_{xx}=0,\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ \phi \left(x,T\right)=0,\hspace{25.0pt}{\phi }_{t}\left(x,T\right)=q\left(x\right),\quad x\in \left(0,\ell \right),\quad \\ \phi \left(0,t\right)={\phi }_{xx}\left(0,t\right)=0,\hspace{25.0pt}\phi \left(\ell ,t\right)={\phi }_{xx}\left(\ell ,t\right)=0,\quad t\in \left(0,T\right),\quad \end{cases}\end{equation} \tag{ 46 }$$

with the input (final velocity) q(x) at t = T, and $\delta u\in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ is the weak solution of the problem (42).

Proof. Multiply both sides of equation (42) by arbitrary function ϕ(x, t), integrate over (0, Ω_T ) and apply the integration by parts formula multiple times. Using the homogeneous initial and boundary conditions in (42) we obtain:

$$\begin{align*}& {\int }_{0}^{T}{\int }_{0}^{\ell }\left[\rho \left(x\right){\phi }_{tt}-\mu \left(x\right){\phi }_{t}+{\left(r\left(x\right){\phi }_{xx}\right)}_{xx}\right]\delta u\enspace \mathrm{d}t\enspace \mathrm{d}x\\ & \hspace{25.0pt} \quad +{\int }_{0}^{\ell }{\left[\rho \left(x\right)\delta {u}_{t}\phi -\rho \left(x\right)\delta u{\phi }_{t}+\mu \delta u\phi \right]}_{t=T}\enspace \mathrm{d}x\\ & \hspace{25.0pt} \quad +{\int }_{0}^{T}{\left[{\left(r\left(x\right)\delta {u}_{xx}\right)}_{x}\phi +r\left(x\right)\delta {u}_{x}{\phi }_{xx}\right]}_{x=0}^{x=\ell }\enspace \mathrm{d}t\\ & \hspace{25.0pt} ={\int }_{0}^{T}{\int }_{0}^{\ell }\delta F\left(x\right)G\left(t\right)\phi \left(x,t\right)\mathrm{d}x\enspace \mathrm{d}t.\end{align*}$$

We require now that the function $\phi \in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ the weak solution of the backward problem (46). Then in view of the final and boundary conditions in (46) we arrive at the required relationship (45). □

Note that the backward problem (46) is a well-posed problem as the change of the variable t with τ = T − t shows. Hence all estimates derived in section 2 can also be applied to the solution of the problem (46).

Assume that the arbitrary final time input q(x) in (46) is specified as follows:

$$\begin{equation}q\left(x\right)=-\frac{1}{\rho \left(x\right)}\left[u\left(x,T;F\right)-{u}_{T}\left(x\right)\right],\quad x\in \left(0,\ell \right).\end{equation} \tag{ 47 }$$

The backward problem (46) with the final time input (47) is defined as an adjoint problem corresponding to the inverse problems (1) and (2).

Substituting the input (47) in the integral relationship (45) we get:

$$\begin{equation*}{\int }_{0}^{\ell }\left[u\left(x,T;F\right)-{u}_{T}\left(x\right)\right]\delta u\left(x,T\right)\mathrm{d}x={\int }_{0}^{T}{\int }_{0}^{\ell }\delta F\left(x\right)G\left(t\right)\phi \left(x,t;F\right)\mathrm{d}x\enspace \mathrm{d}t,\end{equation*}$$

where ϕ(x, t; F) is the solution of the backward problem (46) with the input defined in (47) which, in turn, depends on the input given $F\in \mathcal{F}$.

This integral relationship with the formula (44) for the first variation of the Tikhonov functional leads to

$$\begin{equation}\delta J\left(F\right)={\int }_{0}^{\ell }\left({\int }_{0}^{T}G\left(t\right)\phi \left(x,t;F\right)\mathrm{d}t\right)\delta F\left(x\right)\mathrm{d}x+\frac{1}{2}{\int }_{0}^{\ell }{\left[\delta u\left(x,T\right)\right]}^{2}\enspace \mathrm{d}x,\end{equation} \tag{ 48 }$$

for all $F,\delta F\in \mathcal{F}$.

Theorem 6. Assume that conditions (10) hold and u_T ∈ L²(0, ℓ). Then the Tikhonov functional is Fréchet differentiable. Furthermore, for the Fréchet gradient ∇J(F) the following formula holds:

$$\begin{equation}\nabla J\left(F\right)\left(x\right)={\int }_{0}^{T}\phi \left(x,t;F\right)G\left(t\right)\mathrm{d}t,\quad F\in \mathcal{F},\end{equation} \tag{ 49 }$$

where $\phi \in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ is the weak solution of the adjoint problem (46) with the input q(x) defined in (47).

Proof. The last right-hand-side integral in formula (48) is of the order $\mathcal{O}\left({\Vert}\delta F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}\right)$, as estimate (43) shows. Then the proof follows from the definition of the gradient. □

Remark 5. According to theorem 1 applied to the adjoint problem (46), for existence of the weak solution $\phi \in {L}^{2}\left(0,T;{\mathcal{V}}^{2}\left(0,\ell \right)\right)$ the input q(x) defined in (47) must satisfy the condition q ∈ L²(0, ℓ). From the condition u_T ∈ L²(0, ℓ) and corollary 1 it follows that q(x) actually belongs to L²(0, ℓ). This justifies the validity of the substitution (47). □

Now, we prove that the Fréchet gradient of the Tikhonov functional is Lipschitz continuous.

Theorem 7. Assume that the conditions of theorem 6 hold. Then the Fréchet gradient defined by formula (49) is Lipschitz continuous, that is

$$\begin{equation}{\Vert}\nabla J\left({F}_{1}\right)-\nabla J\left({F}_{2}\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{L}_{\nabla }{\Vert}{F}_{1}-{F}_{2}{{\Vert}}_{{L}^{2}\left(0,\ell \right)},\quad {F}_{1},{F}_{2}\in \mathcal{F},\end{equation} \tag{ 50 }$$

where ${L}_{\nabla }={T}^{3/2}\enspace {C}_{1}{C}_{u}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}$ is the Lipschitz constant and C₁, C_u > 0 are the constants introduced in theorem 1.

Proof. From the gradient formula (49) it follows that

$$\begin{align}{\Vert}{J}^{\prime }\left({F}_{1}\right)-{J}^{\prime }\left({F}_{2}\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}& ={\int }_{0}^{\ell }{\left({\int }_{0}^{T}\delta \phi \left(x,t\right)G\left(t\right)\mathrm{d}t\right)}^{2}\mathrm{d}x\\ & {\leqslant}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}{\Vert}\delta \phi {{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}\hspace{1pt},\end{align} \tag{ 51 }$$

where δϕ(x, t) is the weak solution of the problem

$$\begin{equation}\begin{cases}\rho \left(x\right)\delta {\phi }_{tt}-\mu \left(x\right)\delta {\phi }_{tt}+{\left(r\left(x\right)\delta {\phi }_{xx}\right)}_{xx}=0,\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ \delta \phi \left(x,T\right)=0,\hspace{25.0pt}\delta {\phi }_{t}\left(x,T\right)=\delta u\left(x,T\right),\quad x\in \left(0,\ell \right),\quad \\ \delta \phi \left(0,t\right)=\delta {\phi }_{xx}\left(0,t\right)=0,\hspace{25.0pt}\delta \phi \left(\ell ,t\right)=\delta {\phi }_{xx}\left(\ell ,t\right)=0,\quad t\in \left(0,T\right),\quad \end{cases}\end{equation} \tag{ 52 }$$

with the input δu(x, T) := u(x, T; F₁) − u(x, T; F₂).

We employ the inequality ${\Vert}\delta \phi {{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}{\leqslant}\left({T}^{2}/2\right)\enspace {\Vert}\delta {\phi }_{t}{{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}$ with the first estimate (11) of theorem 1 applied to the weak solution δϕ(x, t) of problem (52) to get

$$\begin{equation*}{\Vert}\delta \phi {{\Vert}}_{{L}^{2}\left(0,T;{L}^{2}\left(0,\ell \right)\right)}^{2}{\leqslant}\frac{{T}^{2}}{2}\enspace {C}_{1}^{2}{C}_{u}^{2}\enspace {\Vert}\delta u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}.\end{equation*}$$

Then we use estimate (15) in corollary 1 to estimate the right-hand side norm ${\Vert}\delta u\left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}$. We have:

$$\begin{equation*}{\Vert}\delta \phi \left(\cdot ,T\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\leqslant}{T}^{3}{C}_{1}^{2}\enspace {C}_{u}^{2}\enspace {\Vert}\delta F{{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}{\Vert}G{{\Vert}}_{{L}^{2}\left(0,T\right)}^{2}.\end{equation*}$$

With the above inequality this leads to the required assertion. □

The case G(t) = cos(ωt) is called the harmonic loading, where ω > 0 is the frequency of the applied temporal load. This leads to the inverse problem of recovering the unknown spatial load F(x) in

$$\begin{align}\begin{cases}\begin{cases}\rho {u}_{tt}+\mu {u}_{t}+{\left(r\left(x\right){u}_{xx}\right)}_{xx}=F\left(x\right)\mathrm{cos}\enspace \left(\omega t\right),\quad \left(x,t\right)\in {{\Omega}}_{T},\quad \\ u\left(x,0\right)={u}_{t}\left(x,0\right)=0,\quad x\in \left(0,\ell \right),\quad \\ u\left(0,t\right)={u}_{xx}\left(0,t\right)=u\left(\ell ,t\right)={u}_{xx}\left(\ell ,t\right)=0,\quad t\in \left[0,T\right],\quad \end{cases}\quad \\ \hspace{25.0pt}{u}_{T}\left(x\right){:=}u\left(x,T\right),\quad x\in \left(0,\ell \right).\quad \end{cases}\end{align} \tag{ 53 }$$

Harmonic excitation, in which the magnitude of the external load varies within a harmonic envelope, is one of the most encountered types of loading in applications. Determination of the harmonic response of engineering structures in which the beam-like elements are involved is of great importance especially at the design stage. From the point of view of inverse problems, an important problem here is the determination of the unknown spatial load F(x) from the final time displacement or velocity, at different admissible values of the frequency ω > 0 of the applied harmonic load G(t) = cos(ωt). Positions, as function of time t ∈ [0, T], for the undamped and two damped vibrations are shown in figure 2. This figure illustrates that in the undamped case, a mass on a beam, displaced out of its equilibrium position, will oscillate about that equilibrium for all time, while in the damped case, it will relax towards that equilibrium.

Consider the underdamped case: $\mu {< }2\sqrt{{\lambda }_{1}}$. Substituting G(t) = cos(ωt) in (27) and calculating the integral, we find the following formula for the singular values corresponding to the undamped case:

$$\begin{align}{\sigma }_{n}& =\frac{1}{{\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}+{\mu }^{2}{\omega }^{2}}\enspace \left\{\left[2\mu \omega \enspace \mathrm{sin}\left(\omega T\right)+\left({\lambda }_{n}-{\omega }^{2}\right)\mathrm{cos}\left(\omega T\right)\right]\right.\\ & \quad -\left.\left[\frac{\mu \left({\lambda }_{n}+{\omega }^{2}\right)}{2{\omega }_{n}}\enspace \mathrm{sin}\left({\omega }_{n}T\right)+\left({\lambda }_{n}-{\omega }^{2}\right)\mathrm{cos}\left({\omega }_{n}T\right)\right]{\text{e}}^{-\mu T/2}\right\},\\ & \hspace{25.0pt}\quad \mu {< }2\sqrt{{\lambda }_{1}},\quad {\omega }_{n}=\frac{1}{2}\enspace \sqrt{4{\lambda }_{n}-{\mu }^{2}},\quad n=1,2,3,\cdots \enspace .\end{align} \tag{ 54 }$$

Lemma 5. Let G(t) = cos(ωt). Assume that the damping coefficient μ > 0, the frequency ω > 0 of the applied harmonic load G(t) = cos(ωt) and the final time T > 0 satisfy the following conditions

$$\begin{equation}\mu {< }2\sqrt{{\lambda }_{1}},\quad 0{< }\omega {< }\sqrt{{\lambda }_{1}},\quad \omega T{< }\pi /4,\end{equation} \tag{ 55 }$$

where λ₁ > 0 is the principal eigenvalue of the Euler–Bernoulli operator. Denote by ω* = ω*(λ₁, μ) the root of the equation

$$\begin{equation}{\text{e}}^{\mu \pi /\left(4\omega \right)}=2\left[1+\frac{{\mu }^{2}{\left({\lambda }_{1}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{1}-{\mu }^{2}\right){\left({\lambda }_{1}-{\omega }^{2}\right)}^{2}}\right]\end{equation} \tag{ 56 }$$

and let

$$\begin{equation}{T}_{{\ast}}=\frac{\pi }{4{\omega }^{{\ast}}\left({\lambda }_{1},\mu \right)},\hspace{25.0pt}{T}^{{\ast}}=\frac{\pi }{4\omega }.\end{equation} \tag{ 57 }$$

Then for all of ω ∈ (0, ω*(λ₁, μ)) and T ∈ (T_*, T*), the singular values defined by formula (54) are positive.

Proof. Rewrite formula (54) as follows:

$$\begin{align}{\sigma }_{n}& =\frac{{\lambda }_{n}-{\omega }^{2}}{{\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}+{\mu }^{2}{\omega }^{2}}\enspace \left\{\left[\frac{2\mu \omega }{{\lambda }_{n}-{\omega }^{2}}\enspace \mathrm{sin}\left(\omega T\right)+\mathrm{cos}\left(\omega T\right)\right]\right.\\ & \quad -\left.\left[\frac{\mu \left({\lambda }_{n}+{\omega }^{2}\right)}{\sqrt{4{\lambda }_{n}-{\mu }^{2}}\enspace \left({\lambda }_{n}-{\omega }^{2}\right)}\enspace \mathrm{sin}\left({\omega }_{n}T\right)+\mathrm{cos}\left({\omega }_{n}T\right)\right]{\text{e}}^{-\mu T/2}\right\}.\end{align} \tag{ 58 }$$

By the third condition of (55) the sum in the first right-hand side square bracket in formula (58) is positive. Using the inequality $\vert a\enspace \mathrm{sin}\enspace \alpha +b\enspace \mathrm{cos}\enspace \alpha \vert {\leqslant}\sqrt{{a}^{2}+{b}^{2}}$ for the sum in the second right-hand side square bracket and also the assumption (55) we deduce that

$$\begin{align}{\sigma }_{n}& {\geqslant}\frac{{\lambda }_{n}-{\omega }^{2}}{{\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}+{\mu }^{2}{\omega }^{2}}\enspace \\ & \quad {\times}\left\{\mathrm{cos}\left(\omega T\right)-{\left[1+\frac{{\mu }^{2}{\left({\lambda }_{n}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{n}-{\mu }^{2}\right){\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}}\right]}^{1/2}{\text{e}}^{-\mu T/2}\right\}.\end{align} \tag{ 59 }$$

By the third condition of (55), $\mathrm{cos}\left(\omega T\right){ >}1/\sqrt{2}$ and the right-hand side of inequality (59) is positive for all n = 1, 2, 3, ⋯, , if

$$\begin{equation*}\frac{1}{\sqrt{2}}{ >}{\left[1+\frac{{\mu }^{2}{\left({\lambda }_{n}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{n}-{\mu }^{2}\right){\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}}\right]}^{1/2}{\text{e}}^{-\mu T/2},\quad n=1,2,3,\cdots \enspace .\end{equation*}$$

This yields:

$$\begin{equation}{\text{e}}^{\mu T}{ >}2\left[1+\frac{{\mu }^{2}{\left({\lambda }_{n}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{n}-{\mu }^{2}\right){\left({\lambda }_{n}-{\omega }^{2}\right)}^{2}}\right]{=:}{\Lambda}\left({\lambda }_{n},\omega \right),\quad n=1,2,3,\cdots ,\enspace .\end{equation} \tag{ 60 }$$

Since Λ(λ_n , ω), n = 1, 2, 3, ⋯, is a monotone decreasing function of λ_n , the condition

$$\begin{equation}{\text{e}}^{\mu T}{ >}2\left[1+\frac{{\mu }^{2}{\left({\lambda }_{1}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{1}-{\mu }^{2}\right){\left({\lambda }_{1}-{\omega }^{2}\right)}^{2}}\right]{=:}{\Lambda}\left({\lambda }_{1},\omega \right)\end{equation} \tag{ 61 }$$

ensures the fulfillment of the conditions (60) for all n = 2, 3, ⋯.

Consider now the equation (56) which right-hand side is the above introduced function Λ(λ₁, ω). We prove that for all $\mu \in \left(0,2\sqrt{{\lambda }_{1}}\enspace \right)$ there exists a unique root ω^* = ω^*(λ₁, μ) of this equation in $\left(0,\sqrt{{\lambda }_{1}}\enspace \right)$. Indeed, Λ(λ₁, ω) is a monotone increasing function of $\omega \in \left(0,\sqrt{{\lambda }_{1}}\right)$, for a given λ₁ > 0, and tends to infinity as $\omega \to \sqrt{{\lambda }_{1}}$. On the other hand, the left-hand side of equation (56), i.e. the function e^μπ/(4ω), is a monotone decreasing function of $\omega \in \left(0,\sqrt{{\lambda }_{1}}\right)$ and tends to infinity as ω → 0. This implies that the equation (56) has a unique root ω* = ω*(λ₁, μ) in $\left(0,\sqrt{{\lambda }_{1}}\enspace \right)$.

From the above considerations it follows that for all ω ∈ (0, ω*(λ₁, μ)) and T ∈ (T_*, T^*], with T_*, T^* > 0 defined in (57),

$$\begin{equation*}{\text{e}}^{\mu T}\equiv {\text{e}}^{\mu \pi /\left(4\omega \right)}{ >}{\text{e}}^{\mu \pi /\left(4{\omega }^{{\ast}}\right)}\equiv {\Lambda}\left({\lambda }_{1},{\omega }_{{\ast}}\right){ >}2\left[1+\frac{{\mu }^{2}{\left({\lambda }_{1}+{\omega }^{2}\right)}^{2}}{\left(4{\lambda }_{1}-{\mu }^{2}\right){\left({\lambda }_{1}-{\omega }^{2}\right)}^{2}}\right],\end{equation*}$$

which means the fulfillment of the condition (61). In view of inequalities (59) and (60) this leads to the positivity of the singular values for all n = 1, 2, 3, ⋯. □

The version of theorem 4 for the case of the time harmonic inverse source problem (53) can be formulated as follows.

Theorem 8. Assume that conditions of lemma 5 are satisfied. Suppose that the measured output u_T (x) defined in (2) satisfies the regularity conditions (37). Then the inverse problem (53) has a unique solution. Furthermore, this solution possesses the convergent SVE given by (6), with the singular values defined in (54).

Example 1. The admissible intervals for the values of the frequency of the applied temporal load ω > 0 and final time T > 0.

Consider a simply supported Euler–Bernoulli beam with unit flexural rigidity and mass density r = ρ = 1, and with the length ℓ = π. The formula λ_n = π⁴ rn⁴/(ℓ⁴ ρ) given in remark 2 suggests that in this case the first eigenvalue is λ₁ = 1, and the underdamped case $\mu {< }2\sqrt{{\lambda }_{1}}$ means μ ∈ (0, 2).

Table 1 illustrates the admissible upper and lower limits of ω > 0 and T > 0 corresponding to some reasonable values of the damping parameter μ ∈ (0, 2) given on the first row of the table. The upper limits of ω > 0 calculated as a root of equation (56), and the final time are given on the second and third rows of the table, respectively.

For simplicity of calculations, we consider 0.5ω*, i.e. half of the upper limit, as lower limit of ω given on the fourth row of the table. The values of the lower, T_* > 0 and the upper, T* > T_* limits of the final time, given on the third and fifth rows show that they also differ by about two times. This allows us to select the value of the admissible final time from the interval (T_*, T*] with sufficient length, to generate the measured output u_T (x).

□

In this section, we illustrate the application of the truncated singular value decomposition (TSVD) and the conjugate gradient algorithm (CG-algorithm) to the inverse problem (53) related to the forced vibration under the harmonic load G(t) = cos(ωt). For the numerical solution of the direct problem, the method of lines approach is employed. Specifically, the finite element method with Hermite cubic basis functions is used for discretization of the spatial domain, while the second order backward finite difference scheme, are used for time discretizations, to obtain full discrete problem. Uniform element lengths are performed for discretization of both spatial and temporal intervals. Additional details of this two stage numerical technique are explained in (12).

In the first part of this subsection, we provide numerical examples to demonstrate the ability of the TSVD algorithm to recover the unknown spatial load from noise free as well as noisy final time output. This algorithm is especially convenient when the eigenvalues of the Euler–Bernoulli operator are known. As noted in remark 2, when the flexural rigidity r(x) is constant, then the eigensystem is given by formulas given in remark 2. In the numerical examples below, we assume that

$$\begin{equation}\rho \left(x\right)=1,\quad \mu =1/2,\quad \ell =\pi ,\quad T=3/2,\quad \omega =1/2.\end{equation} \tag{ 62 }$$

Notice that the values of the frequency ω > 0 of the applied temporal load and the final time T > 0 given in (62) are defined according to the table 1.

In view of (62), the eigensystem of the Euler–Bernoulli operator with r ≡ 1 subject to the clamped boundary conditions is

$$\begin{equation}{\lambda }_{n}={n}^{4},\hspace{25.0pt}{\psi }_{n}\left(x\right)=\sqrt{\frac{2}{\pi }}\enspace \mathrm{sin}\left(nx\right),\quad x\in \left[0,\pi \right].\end{equation} \tag{ 63 }$$

Since λ₁ = 1, for the above value of the damping coefficient the condition $\mu {< }2\sqrt{{\lambda }_{n}}$ for the underdamped case holds for all $n\in {\mathbb{N}}^{+}$. Substituting λ_n = n⁴ in (54) we obtain the following formula for the singular values corresponding to the underdamped case:

$$\begin{align}{\sigma }_{n}& =\frac{1}{{\left({n}^{4}-{\omega }^{2}\right)}^{2}+{\mu }^{2}{\omega }^{2}}\enspace \left\{\left[2\mu \omega \enspace \mathrm{sin}\left(\omega T\right)+\left({n}^{4}-{\omega }^{2}\right)\mathrm{cos}\left(\omega T\right)\right]\right.\hspace{25.0pt}\\ & \quad \left.-\left[\frac{\mu \left({n}^{4}+{\omega }^{2}\right)}{2{\omega }_{n}}\enspace \mathrm{sin}\left({\omega }_{n}T\right)+\left({n}^{4}-{\omega }^{2}\right)\mathrm{cos}\left({\omega }_{n}T\right)\right]{\text{e}}^{-\mu T/2}\right\},\\ & \hspace{25.0pt}\quad \mu {< }2,\quad {\omega }_{n}=\frac{1}{2}\sqrt{4{n}^{4}-{\mu }^{2}},\quad n=1,2,3,\cdots \enspace .\end{align} \tag{ 64 }$$

Then the TSVD solution of the inverse problem (53) is

$$\begin{equation}{F}_{\alpha ,N}\left(x\right)=\sum\limits _{n=1}^{N}\frac{q\left(\alpha ;{\sigma }_{n}\right)}{{\sigma }_{n}}\enspace {u}_{T,n}\enspace {\psi }_{n}\left(x\right),\quad \enspace x\in \left(0,\pi \right),\end{equation} \tag{ 65 }$$

where N > 1 is the truncation parameter,

$$\begin{equation}{u}_{T,n}=\sqrt{\frac{2}{\pi }}{\int }_{0}^{\pi }{u}_{T}\left(x\right)\mathrm{sin}\left(nx\right)\mathrm{d}x,\quad n=1,2,3,\cdots \enspace \end{equation} \tag{ 66 }$$

in the Fourier coefficients of the measured output u_T (x),

$$\begin{equation}q\left(\alpha ;\sigma \right)=\frac{{\sigma }^{2}}{{\sigma }^{2}+\alpha }\end{equation} \tag{ 67 }$$

is the filter function and α > 0 is the parameter of regularization.

Therefore, for the given eigensystem (63), the TSVD solution of the inverse problem (53), given by (65) and also by relations (64), (66) and (67), can be computed without the large computational effort involved in a complete TSVD computation.

The synthetic noise free output u_T (x) := u(x, T) is generated from the finite-element solution of the direct problem defined in (53), assuming that F(x) = x sin(5x) and G(t) = cos(t/2). Having the noise free output, the synthetic noisy output ${u}_{T}^{\delta }\left(x\right)$, with noise level δ > 0 is then generated by using standard MATLAB rand function. These outputs are plotted in figure 3 on the left.

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The reconstructed spatial loads from noise free and noisy outputs with the noise levels δ = 0.05 (%5) and δ = 0.1 (%10) are shown in figure 3 on the right, together with the function F(x) = x sin(5x) (exact solution). The worst numerical result is obtained when the regularization parameter is zero: α = 0 (thin dotted line), as expected. In each remaining cases, ⟨α = 3 × 10⁻⁹⟩, the results of reconstructions corresponding to noise free and noisy outputs, are fairly more accurate. Thus, if the eigensystem of the Euler–Bernoulli operator is known, then with a very small values α ∈ (10⁻⁹, 10⁻⁸) of the regularization parameter, the TSVD solution of the inverse problem can be obtained by formula (65), taking the values of the truncation parameter N between 6 and 11. This means that the TSVD solution F_N (x), corresponding to a sufficiently high accuracy of reconstruction, can be obtained by using only a few terms of the sum (65), if the optimal value of the regularization parameter α > 0 is found.

Thus, the TSVD algorithm is a simple computational tool, when the coefficients of the Euler–Bernoulli equation are constant. However, in the case of variable coefficients, additional computational work is required to find the eigensystem of the Euler–Bernoulli operator numerically, which increases the overall cost of the entire computational algorithm. Furthermore, the numerically found eigenvalues introduce additional error in the formula (65).

To compare the TSVD and the CG-algorithm, the above example with the constant flexural rigidity r ≡ 1, was solved by the CG-algorithm. The iteration scheme of the CG-algorithm used here is as follows:

• Step 1: set n = 0, choose the initial iteration as F⁽ⁿ⁾ ≡ 0 and compute the gradient ∇J(F⁽ⁿ⁾). Then set the descent direction: d⁽ⁿ⁾ = −∇J(F⁽ⁿ⁾).
• Step 2: find the parameter ${\gamma }_{{\ast}}^{\left(n\right)}=\frac{{\Vert}\nabla J\left({F}^{\left(n\right)}\right){{\Vert}}^{2}}{{\Vert}u\left(\cdot ,T;{d}^{\left(n\right)}\right){{\Vert}}^{2}}$ from the minimization problem:
  $$\begin{equation*}J\left({F}^{\left(n\right)}+{\gamma }_{{\ast}}^{\left(n\right)}{d}^{\left(n\right)}\right)=\underset{\gamma { >}0}{\mathrm{min}}\enspace J\left({F}^{\left(n\right)}+\gamma {d}^{\left(n\right)}\right);\end{equation*}$$
  then define the next iteration ${F}^{\left(n+1\right)}={F}^{\left(n\right)}+{\gamma }_{{\ast}}^{\left(n\right)}{d}^{\left(n\right)}$ and compute J(F⁽ⁿ⁺¹⁾).
• Step 3: set n = n + 1, compute ∇J(F⁽ⁿ⁾) and calculate the descent direction d⁽ⁿ⁾ according to the rule:
  $$\begin{equation*}{d}^{\left(n\right)}=\frac{{\Vert}\nabla J\left({F}^{\left(n\right)}\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}}{{\Vert}\nabla J\left({F}^{\left(n-1\right)}\right){{\Vert}}_{{L}^{2}\left(0,\ell \right)}^{2}}\enspace {d}^{\left(n-1\right)}-\nabla J\left({F}^{\left(n\right)}\right).\end{equation*}$$
• Step 4: if the following stopping condition holds,
  $$\begin{equation*}{\Vert}u\left(\cdot ,T;{F}^{\left(n\right)}\right)-{u}_{T}^{\delta }{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}\varepsilon \delta {\Vert}{u}_{T}^{\delta }{{\Vert}}_{{L}^{2}\left(0,\ell \right)}{\leqslant}{\Vert}u\left(\cdot ,T;{F}^{\left(n-1\right)}\right)-{u}_{T}^{\delta }{{\Vert}}_{{L}^{2}\left(0,\ell \right)}\end{equation*}$$
  for known parameter ɛ > 0, stop the iteration process; otherwise, return to step 2.

The reconstruction results by the CG-algorithm from noisy free and noisy, with δ = 0.05 and δ = 0.1 final time output are given in figure 4, together with the function F(x) = x sin(5x) (solid line).

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Comparison of the reconstruction results given in figure 3 on the right, obtained by the TSVD, and the results given in figure 4, obtained by the CG-algorithm, one can easily find out that the accuracy of reconstruction in both cases is approximately the same. However, the optimal value α = 3 × 10⁻⁹ of the regularization parameter in the TSVD was found as a result of computational experiments, i.e. empirically, while in the CG-algorithm, sufficiently accurate results were obtained without regularization (α = 0), that is this parameter may not even be needed to get accurate results.

The behavior of the convergence error $e\left(n;F;\delta \right)={\Vert}u\left(\cdot ,T;{F}^{\left(n\right)}\right)-{u}_{T}^{\delta }{{\Vert}}_{{L}^{2}\left(0,\ell \right)}$ and the accuracy error $E\left(n;F;\delta \right)={\Vert}F-{F}^{\left(n\right)}{{\Vert}}_{{L}^{2}\left(0,\ell \right)}$, depending on the iteration number n, is shown in figure 5, on the left and on the right, respectively.

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In order to find out how the variability of the coefficient r(x) affects the accuracy of the reconstruction, let us consider now the case where the flexural rigidity r(x) is not constant. Specifically, we assume in the above example, with the inputs given in (62), and the flexural rigidity is defined as $r\left(x\right)=\mathrm{exp}\left(-\sqrt{x}\enspace \right)$. The rest of the parameters are the same. The synthetic noise free and noisy final time outputs here is generated in the same way as above, assuming F(x) = x sin(5x) and G(t) = cos(t/2) in the direct problem. These outputs are shown in figure 6 on the left.

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The above outlined CG-algorithm is then applied to the inverse problem (53), for reconstruction of the spatial load F(x). The reconstructed spatial loads from noise free and noisy outputs with the noise levels δ = 0.05 and δ = 0.1 are shown in figure 6 on the right. The behavior of the convergence error e(n; F; δ) and the accuracy error E(n; F; δ), depending on the iteration number n, is shown on the left and on the right in figure 7, respectively. The reconstruction results presented in figure 6 on the right were obtained in about 150 iterations. Note that for very small value α = 3 × 10⁻⁹ of the regularization parameter, the reconstruction results are almost the same in accuracy with those obtained for α = 0. A further increase in the value of the regularization parameter, degrades the accuracy of the reconstruction.

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Let us compare the results presented in figure 4 and in figure 6 on the right, corresponding to the cases r(x) = 1 and $r\left(x\right)=\mathrm{exp}\left(-\sqrt{x}\enspace \right)$, respectively. In terms of accuracy of the reconstructions, they are qualitatively the same, although the results corresponding to the second case are slightly inferior to that corresponding to the first one. This shows that the effect of the coefficients in the Euler–Bernoulli equation being constant or variable on the numerical results obtained by the CG-algorithm is very low.

The reconstructions presented in figures 4 and 6 on the right, and also the accuracy and convergence errors plotted in figures 5 and 7 show that in terms of the accuracy of the results obtained, as well as the number of iterations, the CG-algorithm demonstrates high quality and performance.

Remark 6. The results presented in figures 5 and 7 on the right show that starting with some iteration n_*, the value of the accuracy error E(n; F; δ) begins to grow. This means that stopping the CG-algorithm a few iterations earlier, say at n_* − 2, will give a better reconstruction of F(x), especially for the case of noisy output. But since this function is unknown in real applications, such an artificial interference into the algorithm is impossible. As a consequence of this, a slight divergence is quite natural, and should appear as it looks in the figure 7 on the right, especially when working with the noisy measurement. □

This work is devoted to a class of inverse source problems for damped wave and Euler–Bernoulli equations with final time outputs (displacement or velocity). One of the goals of this study, is to show the feasibility of these inverse problems in the presence of the damping term. As the first main result, we proved that, in contrast to the similar inverse problems for undamped equations, these inverse problems make sense and are feasible, precisely because of the damping term. Furthermore, we have found a sufficient condition which guarantees the positivity of the singular values of the input–output operator, and therefore, the uniqueness of the solution to the inverse problem. Moreover, the same sufficient condition expresses the relationship, in the form of an inequality, between the main parameters: the damping coefficient, final time and temporal load. Thus, we have revealed a new ability of the SVE approach. At the same time, we propose the adjoint problem approach combined with the Tikhonov functional, as an alternative method, which allows to solve this class of inverse problems with non-smooth measured output, unlike the TSVD algorithm. Thus, our results demonstrate that using both approaches combined, it is possible to successfully investigate a wide class of applied inverse problems with final time outputs. In addition, the numerical results show that the CG-algorithm looks more suitable for realistic problems due to flexible applicability.

The results obtained here remain valid in the case where the final time displacement u_T (x) := u(x, T) in (2) is replaced by the final time velocity ν_T (x) := u_t (x, T).

In this article, the term μu_t determined by the external damping mechanisms has been the focus of attention. As noted above, in the physical model, the term μ_i u determined by the internal damping mechanisms can also be present in the model separately or together with the term μ u_t . In this case, the problem of uniquely determining the unknown spatial load F(x) from the final time measured displacement or velocity arises again. This problem needs to be examined separately. Another important direction in this area is the generalization of the results to the case of Kirchhoff late model. These problems will be the subject of future articles.

The research of the first and third authors have been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) through the Incentive Program for International Scientific Publications (UBYT). The second author was supported by Mathematical Center in Akademgorodok at the Novosibirsk State University (the agreement with Ministry of Science and High Education of the Russian Federation No. 075-15-2019-1613). The third author has also been supported by CAPES/PRINT, Brazil through the Post-Doctoral research fellowship program '6685-Programa Institucional de internacionalização-Edital No. 41/2017. The authors would like to thank the anonymous reviewers for their comments and suggestions that helped us to improve the paper.

All data that support the findings of this study are included within the article (and any supplementary files).