A. A. Martynyuk

On a nonlinear inequality on the time scale

In the present paper, we consider a dynamic nonlinear integral inequality with a power-law nonlinearity. We obtain a solution of this inequality for arbitrary nonlinearity exponents exceeding unity. These results can be constructively used in the analysis of stability properties (including nonclassical stability properties) of quasilinear dynamic equations.

Analysis of the set of trajectories of nonlinear dynamics: Equations with causal robust operator

We present the results of dynamic analysis of equations and sets of equations with a causal robust operator. We obtain conditions for the local and global existence of solutions of the regularized equation, bracketing estimates for the solution set, and stability conditions for the set of stationary solutions. To this end, we use Lyapunov’s direct method and the comparison principle with a matrix Lyapunov function.

Analysis of the set of trajectories of nonlinear dynamics: Stability under interval initial conditions

For the set of equations of perturbed motion whose solutions satisfy interval initial conditions, we obtain sufficient conditions for the Lyapunov stability and the practical stability of these solutions. The analysis is performed on the basis of locally large scalar Lyapunov functions. As examples, we consider quasilinear and linear nonautonomous systems.

Analysis of the set of trajectories of nonlinear dynamics: Stability and boundedness of motions

For a set of differential equations with the Hukuhara derivative, we obtain sufficient conditions for various types of boundedness of trajectories and stability of the set of stationary solutions. To this end, we use scalar and vector Lyapunov functions constructed on the basis of an auxiliary matrix-valued function.

Analysis of the trajectory set of nonlinear dynamics: Estimates of solutions and comparison principle

For the solution set of differential equations with the Hukuhara derivative, we obtain estimates for solutions and main theorems of the comparison principle on the basis of an auxiliary matrix-valued function.

Analysis of the trajectory set of nonlinear dynamics: Systems with aftereffect under impulsive disturbances

We prove a theorem of the comparison principle with the use of an auxiliary matrix function defined on the product of spaces and a general theorem that permits one to analyze the stability of a stationary solution of a set of systems of impulsive equations with aftereffect on the basis of a scalar comparison equation. We obtain sufficient stability conditions for the case in which both the continuous and the discrete component of the system may have unstable zero solution.