The research of control theory has been concentrated in optimal control of nonlinear distributed systems. Emphasis has been on the development of numerical methods and the solution of real-life applications. The research of parameter identification has been concentrated in identification of the coefficients in distributed systems by using the measured data. The identification has several applications, for example, in computer tomography, geology and environmental sciences. Emphasis has been on the analysis and development of numerical methods.
The purpose is to study optimal control problems, especially, systems arising from free boundary problems. The main interest has been in optimal control problems governed by nonlinear parabolic systems, including, among others, parabolic variational inequalities and systems with phase transitions. The aim is twofold: firstly, to give a theoretical approach to the subject and, secondly, to present detailed algorithms (with convergence proofs) that are necessary in computerizing the optimal control processes. Several practical examples are being worked out in detail in order to demonstrate the usefulness of the proposed methods.
Collaboration with researchers from France, Germany, USA, Japan and Romania.
In this project, hemivariational inequalities, generalized variational inequalities involving nonmonotone, multivalued inclusion, are studied. The existence results have been shown for a constrained stationary hemivariational inequality and for a parabolic hemivariational inequality. In addition, a stable and convergent FEM approximation has been developed for these equations. This approximation has been used in the numerical solutions of nonmonotone contact problems of linear elasticity by nonsmooth, nonconvex optimization methods.
Collaboration with Aristotle University, Greece (prof. P.D. Panagiotopoulos).
The aim of the project is to develop methods for the identification of functional coefficients. It is assumed that one has a distributed observation of the solution of an elliptic or parabolic partial differential equation. These observations are used to determine the unknown coefficients in the equation. In physical systems, unknown parameters can represent, for example, the heat conductivity or the diffusion coefficient. The main emphasis lies on derivation and analysis of new efficient numerical algorithms.
Collaboration with Karl-Franzens-University of Graz, Austria (prof. K. Kunisch).