Second-order Linear Delay Differential Equations Research Articles

Properties of second order differential equations with advanced and delay argument

In this paper, we study the second-order linear delay differential equation of the form (E)y′′(t)+p(t)y(τ(t))=0.We establish new oscillation criteria for (1.1), which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of (1.1) than those presented in known works. Two main approaches for the investigation of (1.1) will be used, namely a comparison principle with first-order delay differential inequalities and generalization of very effective Koplatadze’s technique. We illustrate the improvement over the known results by applying and comparing our method with the other known results for (1.1).

An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

Asymptotic integration of a certain second-order linear delay differential equation

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrodinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.

Asymptotic Integration of a Certain Second-Order Linear Delay Differential Equation

We construct some asymptotic formulas for solutions of a certain linear second-order delay differential equation when the independent variable tends to infinity. Two features concerning the considered equation should be emphasized. First, the coefficient of this equation has an oscillatory decreasing form. Second, when the delay equals zero, this equation turns into the so-called one-dimensional Schr¨odinger equation at energy zero with Wigner–von Neumann type potential. Dynamics of the latter is well-known. The question of interest is how the behavior of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. This equation also attracts interest from the standpoint of the theory of oscillations of solutions of functional differential equations. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients. The essence of the method is to construct a so-called critical manifold in the phase space of the considered dynamical system. This manifold is attractive and positively invariant, and, therefore, the dynamics of all solutions of the initial equation is determined by the dynamics of the solutions lying on the critical manifold. The system that describes the dynamics of the solutions lying on the critical manifold is a linear system of two ordinary differential equations. To construct the asymptotics for solutions of this system, we use the averaging changes of variables and transformations that diagonalize variable matrices. We reduce the system on the critical manifold to what is called the L-diagonal form. The asymptotics of the fundamental matrix of L-diagonal system may be constructed by the use of the classical Levinson’s theorem.

Myshkis type oscillation criteria for second-order linear delay differential equations

New Myshkis type oscillation criteria for second-order linear delay differential equations are obtained which generalise and, under some natural additional assumptions, improve previously known results.

Spacecraft Attitude Stabilization Using Nonlinear Delayed Multiactuator Control and Inverse Dynamics

The dynamics of a rigid spacecraft with nonlinear delayed multiactuator feedback control are studied in this paper. It is assumed that the time delay occurs in one of the actuators, whereas the other actuator has a negligible time delay. Therefore, a nonlinear feedback controller using both delayed and nondelayed states is sought for the controlled system to have the desired linear delayed closed-loop dynamics using an inverse dynamics approach. The closed-loop stability is shown to be approximated by a second-order linear delay differential equation for the modified Rodriguez parameter attitude coordinates for which the Hsu–Bhatt–Vyshnegradskii stability chart can be used to choose the control gains that result in a stable closed-loop response. An analytical derivation of the boundaries of this chart for the case of no derivative feedback control is shown, whereas a numerical method is used to obtain the stability chart for the general case. Then, to achieve a specified performance, the criteria for a critically damped closed-loop response are studied. Further, an integral feedback control is also implemented, which is capable of eliminating the steady-state attitude error caused by any unmodeled external torque.

On oscillations of solutions to second-order linear delay differential equations

New oscillation criteria are proved for second-order two-term linear delay differential equations with locally integrable coefficients and measurable delays. In particular, a suitable a priori lower bound of non-oscillatory solutions is found, which plays a crucial role in the proofs of the results obtained.

Some oscillation criteria for the second-order linear delay differential equation

Some Wintner and Nehari type oscillation criteria are established for the second-order linear delay differential equation.

Oscillation of second-order linear delay differential equations

The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality and the Sturm comparison theorem.

Exact region of stability for a linear orthogonal cutting model

In this work, stability of chatter in orthogonal cutting is investigated in order to identify the critical spindle speeds. The process is modeled by a second-order linear delay-differential equation and the characteristic equation is analyzed in formulating the continuous stability boundaries, explicitly for any given range of the spindle speeds. Moreover, a simple algorithm is introduced for checking whether the system is in the stable zone without determining the whole stability diagram. Numerical simulations and comparison with some other methods are presented to justify the theoretical results.

Stability criteria for certain second-order delay differential equations with mixed coefficients

In this paper, we study asymptotic stability of the zero solution of the second-order linear delay differential equation: y″(t)=p 1y′(t)+p 2y′(t−τ)+q 1y(t)+q 2y(t−τ), where p 1, p 2, q 1, and q 2 are constants with p 1 p 2⩾0 and q 1 q 2<0. Here τ>0 is a constant. In proving our results we make use of Pontryagin's theory of quasi-polynomials.

Impulsive stabilization of two kinds of second-order linear delay differential equations

In this paper, we consider the impulsive stabilization problems for two kinds of 2nd-order linear delay differential equations. By the method of Lyapunov functionals, criteria on the exponential stabilization by impulses and exponential stabilization by periodical impulses are gained. The control effects of impulses are especially stressed in the definitions and results. Some examples are also given to show the applications of our results.

On the stability of the second-order delay differential equations with variable coefficients

The sufficient conditions for the stability of the solutions of the second-order linear delay differential equations with variable coefficients is constructed.

Oscillation of a class of second-order delay differential equation with impulses

In this paper, oscillation of a class of second-order linear delay differential equation (DDE) with impulses is investigated, sufficient conditions of oscillation for all solutions of DDE with impulses are obtained by using differential inequality.