Denys Ya. Khusainov

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

The purpose of this contribution is to develop a controllability method for linear discrete systems with constant coefficients and with pure delay. To do this, a representation of solutions with the aid of a discrete matrix delayed exponential is used. Such an approach leads to new conditions of controllability. Except for a criterion of relative controllability, a control function is constructed as well.

Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Stabilization of Lur’e-type nonlinear control systems by Lyapunov-Krasovskii functionals

The paper deals with the stabilization problem of Lur’e-type nonlinear indirect control systems with time-delay argument. The sufficient conditions for absolute stability of the control system are established in the form of matrix algebraic inequalities and are obtained by the direct Lyapunov method.MSC:34H15, 34K20, 93C10, 93D05.

Estimates of Exponential Stability for Solutions of Stochastic Control Systems with Delay

A nonlinear stochastic differential-difference control system with delay of neutral type is considered. Sufficient conditions for the exponential stability are derived by using Lyapunov-Krasovskii functionals of quadratic form with exponential factors. Upper bound estimates for the exponential rate of decay are derived.

Strong and Mild Extrapolated

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the data, we prove a well-posedness result and give an explicit representation of solutions. Further, we prove an exponential decay rate for the energy in the dissipative case. We also show that lower order regularizations lead to ill-posedness, also for higher order equations. Finally, an application with physically relevant constants is given.

L^{2}

Sufficient conditions of interval absolute stability of nonlinear control systems described in terms of systems of the ordinary differential equations with delay argument and also neutral type are obtained. The Lyapunov-Krasovskii functional method in the form of the sum of a quadratic component and integrals from nonlinearity is used at construction of statements.

-Solutions to the Heat Equation with Constant Delay

Abstract The paper deals with the time switching hybrid systems described by nonlinear control equations with nonlinearity of a sector form. The estimation of the solution of the control system at any moment is established through the method of Lyapunov function. Both cases, stable and unstable control systems are considered. The final estimation of the solution of the hybrid system is obtained from the composition of perturbations of the separate control systems using the condition of solution continuity in the points of switching.

On the Interval Stability of Weak-Nonlinear Control Systems with Aftereffect

The paper studies the exponential stability to a linear system of difference equations with delay x(k+1)=A(k)x(k)+B(k)x(k−m(k)), k=0,1,… where A(k), B(k) are square constant matrices, m(k) ∈ ℕ and m(k) ≤ m*, m* ∈ ℕ. New sufficient conditions for exponential stability are derived and illustrated by an example.The paper studies the exponential stability to a linear system of difference equations with delay x(k+1)=A(k)x(k)+B(k)x(k−m(k)), k=0,1,… where A(k), B(k) are square constant matrices, m(k) ∈ ℕ and m(k) ≤ m*, m* ∈ ℕ. New sufficient conditions for exponential stability are derived and illustrated by an example.

The Estimation of the Dynamics of Indirect Control Switching Systems

We investigate a special case of switched system which are described by a set of nonlinear control systems with nonlinearity of sector type. Using Lyapunov function method we obtain the estimations of switched system solution at the final time moment depending of initial state of solution. The subsystems can be stable or unstable. We investigate using Lyapunov function of Lurje-Postnikov type [5], [9] and final estimation we obtain by composition of perturbations of the separate subsystems of initial switched system on the whole interval considered.