Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection

This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1−d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

Parametrical Synthesis of Radio Devices with the Set Quantity of Identical Cascades for Inclusion Variants of Jet Two-port Networks between a Nonlinear Part and Loading

Introduction. The ability to analytically determine some parameters of various radio devices, which are optimal according to the criterion of providing the set values of the modules and phases of transfer functions at the required number of frequencies, significantly reduces the time for numerical optimization of the rest of the parameters according to the criterion of forming the required frequency response and frequency response in the frequency band. Until now, such problems with respect to radio devices have been solved only for one stage of the "nonlinear part – matching device" or "matching device – nonlinear part" type. As a matching device, reactive, resistive, complex, or mixed quad-poles were used.Aim. Development of algorithms for parametric synthesis of radio devices with an arbitrary number of identical cascades of the "nonlinear part – matching reactive quadrupole" type according to the criterion of ensuring the specified frequency characteristics. Non-linear parts are represented as a non-linear element and parallel or serial current or voltage feedback.Materials and methods. Four-pole theory, matrix algebra, decomposition method, method of synthesis of microwave control devices, numerical optimization methods.Results. Systems of algebraic equations are formed and solved. Models of optimal quadrupole conductors are obtained in the form of mathematical expressions for determining the relationships between the elements of their classical transmission matrix and for finding the frequency dependences of the resistances of two-pole conductors.Conclusion. It is shown that the frequency characteristics of the studied radio devices from the same stages are identical or similar to the frequency characteristics of radio devices from the same stage, but with the signal source and load resistances changed in a certain way. Such schemes are called equivalent. A comparative analysis of the theoretical results (frequency response and frequency response of radio devices) obtained by mathematical modeling in the "MathCad" system, and the experimental results obtained by circuit modeling in the "OrCAD" and "MicroCap" systems, shows their satisfactory agreement.

Parametrical synthesis of radio devices with the set quantity of unequal cascades for variants of inclusion of jet two-port networks between a nonlinear part and loading

The algorithm of parametrical synthesis of various radio devices with any quantity of cascades of type a nonlinear part the jet two-port network by criterion of maintenance of the set frequency characteristics is developed. Nonlinear parts are presented in the form of a nonlinear element and parallel either consecutive on a current or pressure of a feedback. According to this criterion systems of the algebraic equations are generated and solved. Models of optimum two-port networks of one of cascades in the form of mathematical expressions for definition of interrelations between elements of their classical matrix of transfer and for search of dependences of resistance of two-poles from frequency are as a result received. It is spent mathematical and circuit simulation of the two-cascade amplifier. It is shown, that the increase in quantity of cascades with the optimised parametres leads to substantial growth of a working strip of frequencies. The comparative analysis of the theoretical results received by mathematical modelling in system MathCad, and the experimental results received by circuit simulation in systems OrCad and MicroCap, shows their satisfactory coincidence.

Parametrical synthesis of radio devices with the set quantity of identical cascades for variants of inclusion of jet two-port networks between a source of a signal and a nonlinear part

The algorithm of parametrical synthesis of various radio devices with any quantity of cascades of type the jet two-port network a nonlinear part by criterion of maintenance of the set frequency characteristics is developed. Nonlinear parts are presented in the form of a nonlinear element and parallel either consecutive on a current or pressure of a feedback. According to this criterion systems of the algebraic equations are generated and solved. Models of optimum two-port networks in the form of mathematical expressions for definition of interrelations between elements of their classical matrix of transfer and for search of dependences of resistance of two-poles from frequency are as a result received. It is shown, that frequency characteristics of investigated radio devices from identical cascades are identical or similar to frequency characteristics of radio devices from one cascade, but with resistance of a source of a signal and the loading, changed definitely. Such schemes are named by equivalent. The comparative analysis of the theoretical results received by mathematical modelling in system MathCad, and the experimental results received by circuit simulation in systems OrCad and MicroCap, shows their satisfactory coincidence.

Realization of combinatorial optimization of small-scale S-box

This method describes a new technology for combinatorial logic optimization in detail, which can combine multiple criteria to reduce the hardware implementation area of the S-box. This technique can be achieved in two steps. The first is to optimize the non-linear part of the S-box. According to the optimization criterion of multiplication complexity, the quality of the S-box nonlinear part optimization can be judged, and the realization of the S-box with the smallest multiplication complexity can be obtained. The second step is to optimize the linear part of the S-box, and optimize on the basis of the results of the first step, focusing on reducing the number of XOR gates, and the optimization is performed through a heuristic-based algorithm. The above combinatorial logic optimization technology can be applied to any small S-box (5 × 5 and below). Finally, the S-box of PRESENT algorithm and CTC2 algorithm are used as examples to illustrate the optimization effect, and the optimal realization under the minimum AND gate condition is obtained.

Conductivity Mechanism of Ion–DNA Interaction in Dilute Solution

In this paper, we propose an impurity scattering model of quasi-one-dimensional disordered system for ion–DNA interaction in dilute solution based on the density of state in non-periodic DNA. This disordered system is composed of cations and DNA, the hydrogen ions adsorbed on the surface of DNA with negative charges are considered as impurities. It is hydrogen ions in hydration layer that cause the variations of the density of state near the Fermi level. The classical theory describes the linear dependence of conductivity on concentration. By developing the Green function approach of ion–DNA interaction in the dilute solution, the quantum theory not only gives the linear part but also demonstrates the nonlinear part of the conductivity.

Shear Flow in Cylindrical Open Channel Under Precession

The study of forced oscillations in open cylindrical channel under precession is extended to include the shear effect, that is induced by inertial waves in such systems. The linear part of the problem led to two equations for stability one for the viscous part similar to Orr-Sommerfeld equation and one for the inviscid part similar to Rayleigh equation, the second was solved and discussed depending on the stream function observation. The linear part also led to relationship that connects between the stream velocity and the disturbance one is derived in a form similar to Burns conditions for open flows under normal conditions. Experimentally measuring the horizontal velocity distribution with depth showed that this distribution is sinusoidal one. Burns condition was solved based on this assumption. The nonlinear part of the problem led to a new version of Koteweg De-Vries (KdV) equation that is solved numerically by applying the leapfrog method for time discretization, Fourier transformation for the space one, and the trapezoidal rule for solving the integrals with depth, the results showed that the shear has no specific impact on the wave form which is similar to the classical results obtained by the theories under normal conditions.

Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

BVPs with the nonlinearity depending on the fractional derivative

In this paper, we study the existence and the numerical estimates of the solutions for a set of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for the exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at origin and infinity. The novelty of this paper is to use variational methods to obtain the multiplicity of solutions of boundary value problems with the nonlinearity depending on the fractional derivative.