Research Interests

Research interests deal with control theory. The object under consideration is a linear differential-algebraic time-delay systems, with some equations being differential, the other equations being difference. This systems refer to the difference-differential hybrid class (DAD systems). The subjects of research are reachability analysis and relative controllability of DAD systems.

The following new results are obtained:

- solution representations of differential-algebraic systems with retarded argument are obtained by using the corresponding adjoint system solutions;

- an exponential increasing solution estimate is proved for the stationary system solutions;

- algebraic properties of determining equation solutions of the linear stationary differential-algebraic time-delay systems are studied, in particular, it is shown that there is a finite set of generators in the span of determining equation solutions;

- determining equation solution series representations are given for linear stationary difference-differential hybrid systems under action of both continuous and piecewise continuous control functions;

- effective parametric criteria of multi-point Euclidean controllability are proved for linear differential-algebraic systems with retarded argument.

The results obtained can be applied to solving several theoretical and practical problems of the theory of differential-algebraic time-delay systems and in the theory of optimal control that appear in various fields of science and technologies. The results in whole can be qualified as the theoretical basis for analysis and synthesis of real systems of control.

Selected Recent Publications

Poddubnaya O.N. "Some Criteria of Hybrid System Relative Controllability," Proceedings of the National Academy of Sciences of Belarus//Physics and Mathematics Series, Vol. 4.,2005.

Marchenko V.M., Poddubnaya O.N."Solution Representations and Relative Controllability of Linear Differential Algebraic Systems with Several Delays," Doklady Mathematics, Vol.72, No 2, 2005.

Marchenko V.M., Poddubnaya O.N." Representations of Solutions for Controlled Hybrid Systems," Jornal of Automation and Information Sciences, Vol.34(11), 2002.

V. M. Marchenko, O. N. Poddubnaya, and Z. Zaczkiewicz

"On the Observability of Linear Differential-Algebraic

Systems With Delays," IEEE Transactions on Automatic Control, Vol. 51, ą 8, August, 2006.

V. M. Marchenko and O. N. Poddubnaya "Representation of Solutions of Hybrid Difference-Differential Systems,"Jornal of Differential Equations,Vol. 42, No. 6, 2006.

V. M. Marchenko and O. N. Poddubnaya "Linear Stationary Differential–Algebraic Systems:I. Solution Representation," Journal of Computer and Systems Sciences International, Vol. 45, ą 5, 2006.

V.M.Marchenko and O.N.Poddubnaya. Linear Stationary Differential-Algebraic Systems: II. Relative controllability // Journal of Computer and Systems Sciences, Vol. 45, No 6, 2006.

Research activities

Marchenko V.M., Poddubnaya O.N. Relative controllability of stationary hybrid systems // Methods and Models in Automation and Robotics: Proceedings of the 10th IEEE Intern. Conf., Miedzyzdroje, Poland, on 30 August-2 September 2004/ IEEE - Miedzyzdroje, 2004. - Vol. 1.

Marchenko V.M., Poddubnaya O.N., Zaczkiewicz Z. Hybrid control and observation systems in symmetric form // RoMoCo'05: Proceedings of the 5th Intern. Workshop on Robot motion and Control, Dymaczewo, Poland, on 23-25 June 2005/ Poznan,- 2005.

Research projects

"Qualitative Investigation of Differential Equation Properties and Solving of Some Problems of Control and Mechanics", that is the separate part of the scientific direction "Mathematical Structures-21" (one of the branches of research work in Belarus) 1999-2000.

"Construction of Qualitative Theory of Control and Observation for Differential-algebraic Systems with Impulse Influence", Grant of Education Ministry of Belarus 2001.

"Analysis of the Properties of Hybrid Sistem Solutions and Their Applications”, as a part of the state program of the fundamental structures and problems of mathematical modelling 2002-2005.