Vol 54, No 11 (2018)

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ℝⁿ {0}.

Finitely many embedded localizing sets are constructed for invariant compact sets of a time-invariant differential system. These localizing sets are used to divide the state space into three subsets, the least localizing set and two sets called sets of the first kind and the second kind. We prove that the trajectory passing through a point of the set of the first kind remains in this set and tends to infinity. For a trajectory passing through a point of the set of the second kind, there are three possible types of behavior: it either goes to infinity or, at some finite time, enters the least localizing set, or has a nonempty ω-limit set contained in the intersection of the boundary of one of the constructed localizing sets with the universal section of the corresponding localizing function.

The problem of positional boundary control is considered for the spatially one-dimensional wave equation. The objective of control is to transfer the system from an unknown initial state into the state of rest in finite time. A specific feature of the statement of the problem is the weakening of the requirements on the regularity of generalized solutions, observations, and control. The smoothing procedure is used to transfer the problem into the class of strong generalized solutions, where the method previously developed by the authors can be used. The procedure is mathematically justified, and the corresponding results of numerical experiments are given.

For a distributed second-order differential equation, we consider the problem of constructing a control law ensuring that the solution of this equation tracks the solution of a standard equation subjected to an unknown disturbance. A control design algorithm based on constructions of feedback control theory is proposed. The algorithm is stable under information noise and computational errors.

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.

The problem of transformation of an affine system into a linear controllable system is considered. For affine systems with a single control, the notion of A-orbital linearizability is introduced, which generalizes the notion (well known for affine systems) of orbital linearizability to the case where the control-dependent changes of independent variable are used. A necessary and sufficient condition for the A-orbital linearizability is proved, and an algorithm for determining linearizable transformations is proposed based on the construction of the derived series of the codistribution associated with the original system.

The problem of stabilization of multiple-input switched linear systems operating under the conditions of bounded coordinate disturbances is considered. It is assumed that the operation modes can have different dynamical orders. To solve this problem, an algorithm for constructing a variable-structure controller is proposed based on the dynamical order extension method.