Extension principle and degenerate control problems
On the basis of the extension principle, a series of new nontraditional
optimal control methods were elaborated and effectively applied to
theoretically complicated degenerate problems with turnpike solutions.
Results are presented in books (Gurman, 1997; Gurman, 1998; Gurman, Ukhin, 2005).
Geometric control theory
Geometric methods are fruitfully applied to many controllability and
optimal control problems. In particular, complete controllability conditions
for bilinear systems and invariant control systems were obtained.
Results are systematically presented in books
(Agrachev, Sachkov, 2004; Agrachev, Sachkov, 2005).
Control of oscillations
Control problems for the wave equation with boundary conditions of
the first, second, third kind, and of mixed type, are extensively studied.
There was also stated and studied the problem on oscillations control for
multichain elastic system with a bounded control resource oriented towards
applications to large constructions in the space. Results are presented
in the monograph (Znamenskaya, 2004).
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Algorithms and program systems for modeling and control
New first-order and second-order iteration methods for the search and
locally optimal synthesis of control are developed. The methods are based
on sufficient local optimality conditions and local description of
attainable sets, and on sequential improvement of the model with
wide possibilities of adaptation to external conditions.
There are elaborated new methods of approximate synthesis of optimal
control on the basis of polynomial approximation of solutions of
Bellman equation and trajectory recovery of Bellman-Krotov function.
The concept and experimental version of the Intelligent System for
Control Optimization is developed and equipped with a complex of
programs and tutorial supporting multimethod procedures of search
of optimal control.
The web page describing the project and containing description of
the multimethod intelligent system, algorithms, and test problems,
was created (http://www.botik.ru/~cprc/ISOU).
Results are described in book (Gurman, Ukhin, 2004).
Research on regional problems (with the example of Pereslavl region)
- Analysis of innovative activity at large enterprises of the region,
including innovations of ecological and management direction.
- Investigations on the effect of innovations on economics and environment.
- Development of mathematical socio-ecological-economical model of region
with account of innovation processes and the study of optimal strategies
and scenarios of sustainable development.
Results are presented in monograph: (Gurman, Ryumina, 2001, 2003).
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