Inverse problems with respect to vocal tract shape, area function, articulatory parameters or control commands appear both in the theory of speech production and perception, and in technical applications like speech recognition, synthesis and compression. The inverse problems are ill-posed because of non-unique mapping from acoustical parameters to area function, to articulatory parameters, to control commands. The observation of speech pathology, especially laryngectomy and glossectomy, and artificial disturbance of speech production and perception have lead to the hypothesis of the so-called internal model used by the articulatory control system to transform motor commands in order to achieve desired acoustic or articulatory patterns. This hypothesis is supported by the theory of ill-posed inverse problems. One of the most powerful methods of ill-posed problems solving is the variational one where a mathematical model of speech production is used together with some criteria of optimality and constraints to obtain a stable solution. The measured acoustical parameters of the speech signal serve as external constraints while the geometry of the vocal tract, the mechanics of the articulation, the aerodynamics, and the phonetic properties of the language play the role of internal constraints. Optimality criteria, such as the work of articulation and the muscle force, provide good accuracy for both static and dynamic tasks, and reproduce the effects of the bite-block and motor control reorganization for different articulation rates.