1. Control in deterministic discrete-continuous and hybrid systems.

Dynamic systems with discrete-continuous properties (discrete-continuous systems (DCS) or hybrid systems) arise in various areas of applied sciences such as: mechanics, economics, telecommunications etc. The research performed during the last twenty years included the extension of classical methodology to a new class of DCS. The powerful method, which can be refereed as the method of discontinuous time change, was developed for various control problems in the area of DCS. By using the special transformation of the time scale, this method gives the possibility to reduce the complicated optimal control problem with impulsive controls and discontinuous trajectories to a standard one, which can be stated for some auxiliary system described by ordinary differential equations with bounded controls. The key point was the introduction of the concept of {generalized solution} in optimal control problems with discontinuous paths and their description with aid of the discontinuous time transformation method.

With aid of this method the following problems were successively solved:

• representation of discontinuous paths with aid of solutions of some auxiliary dynamic system with bounded controls;
• derivation of dynamic equations as differential equations with measures;
• proof of the existence theorems in optimal control problems with unbounded controls and discontinuous paths;
• derivation of necessary and sufficient optimality conditions within the class generalized solutions in optimal control problems with discontinuous paths.

2. Stochastic discrete-continuous systems.

The extension of the discontinuous time change method to a class of stochastic systems gave the possibility to solve the following problems:

• to derive the filtering equations of Kalman type in linear and nonlinear discrete-continuous systems, described by stochastic differential equations with measures;
• to prove the existence theorems in optimal control problems for stochastic systems with impulsive controls and discontinuous paths;
• to derive the filtering equations for stochastic systems described by differential equations with measures and driven by hidden Markov chains;
• to create the theory of observation control in complicated stochastic systems with various constraints, imposed on the composition and localization of the observations, prove the existence theorems of the optimal generalized solution, derive the necessary and sufficient optimality conditions.

3. Applications.

These theoretical methods were successively applied in the following areas:

• development of the sub-optimal algorithms of the motion estimation in dynamical systems with jumping parameters;
• modeling and optimization of computer-vision systems in visual and infra-red band;
• image processing, description of the image motion in airborne and spacecraft based optical-electronic systems, development of the image motion compensation methods;
• control of mechanical systems with unilateral constraints.