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Stabilization and motion planning problems for multi-dimensional dynamical systems with uncontrollable linearization

Work number - M 38

Nominated by the Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine

Author: Grushkovska V.V.

This work is devoted to the fundamental problem of mathematical control theory related to the development of effective control algorithms for stabilization and motion planning for multi-dimensional dynamical systems described by essentially nonlinear differential equations. The interest to such systems is motivated by their unique analytical properties and numerous applications in robotics. In particular, such systems describe the motion of many important classes of mechanical systems, from simple pendulum mechanisms to spacecrafts, wheeled mobile robots, multibody car systems, autonomous manipulators, underwater and air vehicles.

With the novel approached developed in this work, asymptotic properties of the trajectories of nonlinear systems are studied in critical cases of stability theory, the description of attractors of abstract dynamical systems is obtained, and stabilizability conditions are proposed in terms of feedback invariants of a control system. An important application of the obtained results is the solution of the optimal stabilization problem with minimax cost function for systems with uncontrollable linearization. Besides, a class of motion planning problems is solved for general nonholonomic systems satisfying the controllability condition with Lie brackets.

The obtained results have theoretical importance and make a significant contribution to mathematical control theory, and may also be of interest for scientists in related research areas. They form the scientific basis for solving such important problems as the navigation of systems with static or dynamic obstacles, optimal oscillation damping for systems with resonant frequencies, stabilization of systems with unknown parameters, etc. The main advantages of this work compared with known results in the literature are the universality and constructiveness of the developed control approaches and possibility for an effective computer implementation.

Number of publications:26, including11 papers(5 – by international publishers).

According to the Scopus database, the total citation number of author’s publications presented in the work is 5, h-index(related to work) = 1; according to theGoogleScholar database, the total citation number is 12, h-index(related to work) = 2.

On this topic, the author has defended her PhD thesis. The results of the work were presented at many international scientific conferences, and were discussed at leading Ukrainian and foreign scientific institutions.