Convergence of approximate solutions for singular difference systems with maxima

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

OPTIMIZING RELIABILITY OF LINEAR FRACTIONAL DIFFERENCE SYSTEMS UNDER UNCERTAINTY AND RANDOMNESS

Some complex systems may suffer from failure processes arising from soft failures and hard failures. The existing researches have shown that the reliability of a dynamic system is not constant under uncertain random environments. First, two types of uncertain random optimization models are proposed where reliability index is quantified by chance measure based on whether soft and hard failures are independent or not. It is considered that internal degradation is driven by left Caputo fractional linear difference equation, while shocks are defined as discrete i.i.d. random variables. The shocks may generate additional uncertain degradation shifts when considering the competing dependent failure processes. Then, two proposed optimization reliability problems may be transformed into their equivalent deterministic forms on the basis of [Formula: see text]-path, and improved gradient descent method is applied to obtain optimal solutions. Finally, the numerical example of a micro-engine indicates that the optimization models are beneficial to the reliability of engineering systems.

Optimisation problem for some class of hybrid differential-difference systems with delay

In the paper, the linear differential-difference dynamic systems with delayed arguments are considered. Such systems have a lot of application areas, in particular, processes with repetitive and learning structure. We apply the method of the separation hyperplane theorem for convex sets to establish optimality conditions for the control function to drive the trajectory to zero equilibrium state in the fastest possible way. For the special case of the integral control constraints, the proposed method is detailed to establish an analytical form of the optimal control function. The illustrative example is given to demonstrate the obtained results with the step-by-step calculation of the basic elements of the optimal control.