卷 55, 编号 11 (2019)

We use the vector Lyapunov function method and the comparison method in conjunction with methods of convex geometry and the theory of mixed volumes to obtain analogs of the Liouville formula for some classes of linear differential equations with the Hukuhara derivative defined on the space of convex compact subsets of the plane. A continual generalization of the construction of a vector Lyapunov function is proposed.

The method of localization of invariant compact sets is used to study the properties of solutions of the Levinson-Smith equation with or without bounded disturbances. Necessary and sufficient conditions for the existence of localizing functions with a bounded universal section are obtained. Conditions for the existence of a bounded localizing set are established. The results are used to describe the solution behavior.

We consider a linear time-varying system with an initial state and disturbance that are known imprecisely and satisfy a common constraint. The constraint is the sum of a quadratic form of the initial state and the time integral of a quadratic form of the disturbance, and these quadratic forms are allowed to be degenerate. We obtain a linear matrix differential Lyapunov equation describing the evolution of the ellipsoidal reachability set. In the problem of estimating the state based on output observations, this result is used to find the minimum-size ellipsoidal set of admissible system states, which is determined by the optimal observer and by the reachability set of the corresponding observation error equation. A method for control law synthesis ensuring that the system state reaches the target set or the system trajectory remains in a given ellipsoidal tube is proposed. Illustrative examples are given for the Mathieu equation, which describes parametric oscillations of a linear oscillator.

We consider the problem of dynamic reconstruction of unknown boundary disturbances in a parabolic equation. The disturbances are assumed to occur in the Neumann boundary condition. Based on the Osipov-Kryazhimskii dynamic regularization method, the problem is reduced to that of constructing a feedback control law for some auxiliary differential equation. We indicate an information-noise- and roundoff-error-robust algorithm for solving this problem.

We consider the problem of constructing a bounded error observer for a multivariate system with an unknown input. The input is assumed to be bounded uniformly in time by a known constant. We study the case in which the relative degree for the system is not defined. A generalization of this concept—a degenerate relative degree—is used to solve the problem. Based on this concept, we indicate a method for reducing system to a generalized form with distinguished zero dynamics (an analog of the normal form). Further, we construct an observer for a system in this form using a hierarchical high-gain feedback.

We consider a target control problem for a system of differential equations of a special form in which the nonlinear terms depend on a single state variable. The solution is based on the passage from the original problem to an auxiliary problem for a piecewise linear system as well as on an application of the comparison principle and “smoothed” piecewise quadratic value functions. We also propose a feedback control and derive an a priori estimate for the deviation from the target set in the closed-loop original system. In addition, we present a numerical method whose efficiency is confirmed by two sample computations.