A scheme for controlling oscillations is proposed, in which, within the framework of a coupled system, the van der Pol cycle mode is imposed on a conservative system that admits a family of periodic motions. The acting control is found, the cycle is constructed, the attraction domain of the cycle is determined, and a control law and an algorithm for the asymptotic exit of the coupled system to the cycle mode are given.

A twin-rotor electromechanical system whose output variables make angular motions in the vertical and horizontal planes is explored as a control plant. The mathematical model of the mechanical subsystem, which is essentially nonlinear, includes cross-coupling between propellers, dry-friction torques, and unspecified parameters; it is also subject to uncontrolled external perturbations. A mathematical model of the electric subsystem (considered dynamics of dc-drives) is approximated by linear differential equations. Directly measured are output variables, drive armature currents, and control inputs. The block-based approach was used to develop a decomposition procedure for the synthesis of a dynamic feedback that enables output variables to track specified signals with a given accuracy. To suppress unmatched perturbations, everywhere bounded nonlinear S-shaped local couplings are formed that are ensured by discontinuous controls with constant amplitudes; this enables the design constraints on state and control variables to be taken into account at the stage of synthesis. Measurements of angular positions have been used to develop a procedure for the adjustment of a reduced dynamic observer of angular velocities with S-shaped corrective actions, which does not require the model of the plant to be defined or a dynamic model of external perturbations to be introduced. The effectiveness of the approach developed has been confirmed by simulations in the MATLAB-Simulink environment.

The existence of real solutions of a nonlinear equation in a neighborhood of an abnormal (degenerate) point is studied. We prove that if the mapping describing this equation admits a lambda-truncation that is regular in some direction, then there exist solutions for all right-hand sides in some neighborhood of the image of the abnormal point. Under the same assumption, we prove that there exist nontrivial smooth curves passing through the abnormal point and lying on the level set of the map in question.

For nonlinear mappings acting in Banach spaces, we examine inverse and implicit function theorems under various smoothness assumptions. For various regularity (normality) conditions imposed on such mappings, we prove that the corresponding equations have solutions under any sufficiently small (in the norm) completely continuous perturbations. A priori estimates for these solutions are obtained.

We present Battle–Lemari´e wavelet systems of natural orders. Our main result is a decomposition theorem in Besov and Triebel–Lizorkin spaces with local Muckenhoupt weights, which is formulated in terms of bases generated by systems of such a type. The Battle–Lemari´e wavelets are splines and suit very well the study of integration operators.