Vol 53, No 11 (2017)

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered.

We study the problem on the construction of coverings by a given system of differential equations and the description of systems covered by it. This problem is of interest in view of its relationship with the computation of nonlocal symmetries, recursion operators, B¨acklund transformations, and decompositions of systems. We show that the distribution specified by the fibers of the covering is determined by a pseudosymmetry of the system and is integrable in the infinite-dimensional sense. Conversely, every integrable pseudosymmetry of a system defines a covering by this system. The vertical component of the pseudosymmetry is a matrix analog of the evolution differentiation, and the corresponding generating matrix satisfies a matrix analog of the linearization of an equation.

We prove a theorem on the coincidence points of two mappings acting on spaces equipped with a vector metric. By way of application, we obtain sufficient conditions for the existence of a solution of an ordinary differential equation unsolved for the derivative of the unknown function and local solvability conditions for a control system with mixed constraints.

We study the statements and solvability of the modal control problem (the pole assignment problem) for linear time-invariant hybrid difference-differential systems in symmetric form and for the associated delay systems of neutral type. We obtain constructive necessary and sufficient parametric conditions for the modal controllability of the systems in question in various scales of difference-differential controllers. Methods for the construction of such controllers solving the corresponding modal controllability problem are indicated. The results are illustrated by examples and counterexamples.

To transform single-input affine systems into linear control systems, we suggest to use control-dependent changes of independent variable. We show that the use of such changes of variables in conjunction with feedback linearization enables one to linearize systems to which known linearization methods do not apply. We prove that a linearizing change of independent variable can be found by solving a system of partial differential equations. The approach developed in the paper is applied to the construction of solutions of terminal problems.

We consider the stabilization problem for multiple-input switched linear systems operating under bounded coordinate disturbances and arbitrary switching signals. To solve this problem, we suggest an algorithm for the construction of a variable-structure controller based on methods of simultaneous stabilization theory.

We consider the problem of finding a bounded solution of a linear finite-difference equation with continuous time. We present conditions for the existence of such a solution and algorithms for finding the solution in real time under various additional assumptions.

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.