This paper investigates the robustness of an energy space approach for solving a data completion problem in linear elasticity equations. The problem is reformulated as a variational optimal control problem, with the Dirichlet control defined in the energy space. Both continuous and discrete levels incorporate Tikhonov regularization with a parameter \(\alpha \) . At the continuous level, we show that this regularization provides a robust solution of the proposed energy space method and derive conditions ensuring an error decay bound of the form \( O\left( \frac{\delta }{ \sqrt{\alpha } }+ \sqrt{\alpha }\right) \) for noisy regularized solutions, where \(\delta \) denotes the noise level. Importantly, this rate is established assuming only that \(\overline{\pmb {\eta }}\) , representing the exact missing data on the inaccessible boundary \(\varGamma _i\) , belongs to \(\widehat{\mathbb {H}}^{\frac{1}{2}}(\varGamma _i)\) , without invoking additional source-type conditions of the classical Tikhonov theory. At the discrete level, Tikhonov regularization is integrated with a finite element method of order \(k\) . This combination leads to an optimal decay rate of the corresponding error bound for discrete noisy regularized solutions, achieving at least \(O(h^{k/3})\) for sufficiently regular data, where \(h\) represents the mesh size. Numerical experiments are included to validate the theoretical findings.
