Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations

New control functions are derived for linear difference equations with delay, which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions. Later, a special case of invariant linear subspace is considered and corresponding control functions are constructed. Results for weakly nonlinear problems are also discussed.

On the New Control Functions for Linear Discrete Delay Systems

Sufficient conditions for the exponential stability of the trivial solution of nonlinear differential equations with delay and with linear parts of the form A x ( t ) + B x ( t − τ ) , τ > 0 , where A B = B A , are proved. A result on the nonexistence of blowing-up solutions is also proved.

Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices

We represent a solution of an inhomogeneous second-order differential equation with two delays by using matrix functions under the assumption that the linear parts are given by permutable matrices.

Representation of a solution of the cauchy problem for an oscillating system with two delays and permutable matrices

Conditions are derived for the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay. Utilizing a delayed matrix exponential and a method of pseudo-inverse by Moore–Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay.

Fredholm’s boundary-value problems for differential systems with a single 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.

M. Ru ring

Papers

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay