Introduced a notion of category with approximate morphisms. Approximate decomposition is defined as morphism in such category.

Derived backpropagation equations from maximum principal. Introduced a new class of "symmetric" neural networks with learning cycle twice faster than classical.

Reflexion is studied in a neural networks models using methods of group analysis. Group invariants of the model are treated in terms of reflexive behavior.

Detailed discussion with proofs and examples of results published in some of the previous papers. In addition, new numerical method of decomposition in optimal control is discussed.

Defined almost controllability for affine systems. Criterion of almost controllability and its connections to weak controllability and controllability were obtained using non-standard analysis and differential geometry.

Almost controllability is applied for order reduction in optimal control problem.

Numerical analysis of the flow with free surface and axial symmetry based on Tikhonov regularization methods. Presented a new approach to a classical problem using integral equations.

Artificial small parameter allows order reduction for all smooth optimal control system. This is a new behavior since order reduction is rare and unstable property of non-optimal dynamic systems.

Linear quadratic control problem without symmetries form a set of zero measure. For each linear-quadratic problem exists arbitrary small perturbation which allows to reduce its order.

New method of order reduction in classical Lagrange problem was found. New types of first integrals of Hamiltonian systems allow to reduce order of Hamiltonian systems in a new way. New categories of Hamiltonian and Lagrangian systems were build. Criterion of decomposition in this categories was obtained.

New method of transition from 2D model to 3D model of flow with axial symmetry was introduced. Computer modeling showed good correspondence to experimental data (5 percents precision).