Second-order Dynamic Equations Research Articles

A novel fast model predictive control for large-scale structures

To protect the engineering structures from natural hazards, especially for large-scale structures, a novel fast model predictive control (NFMPC) method is presented in this paper. Based on the second-order dynamic equation, a novel explicit expression form of Newmark-β method is first derived, from which the future states can be easily predicted without computing matrix exponential. By applying this explicit expression form into the standard model predictive control (MPC) method, the NFMPC method is developed. Based on the explicit expression form, the optimal control input can be computed by two off-line transient analyses and one on-line transient analysis at every sampling instant on the structure. For no computation of matrix exponential, the off-line computation efficiency of NFMPC is several orders of magnitude higher than that of MPC. And the small amount of on-line computation guarantees the on-line computation efficiency. Furthermore, the use of the Newmark-β method also guarantees the computation accuracy. At last, several typical numerical examples are carried out to verify the validity and high efficiency of NFMPC by the comparison with MPC.

Oscillation results for nonlinear second-order damped dynamic equations

Oscillation results for nonlinear second-order damped dynamic equations

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales

Oscillation criteria for certain second-order Emden-Fowler delay functional dynamic equations with damping on time scales

This paper is concerned with oscillations of certain second-order Emden-Fowler variable delay functional dynamic equations with damping and neutral of the form $$\bigl[A(t)\varphi\bigl(y^{\Delta} (t)\bigr)\bigr]^{\Delta} + b(t) \varphi\bigl(y^{\Delta} (t)\bigr) + P(t)F\bigl(\varphi\bigl(x\bigl(\delta(t) \bigr)\bigr)\bigr) - Q(t)f\bigl(\varphi\bigl(x\bigl(\gamma(t)\bigr)\bigr)\bigr) = 0 $$ on an arbitrary time scale T, where $y(t) = x(t) + B(t)x(\tau (t))$ and $\varphi(u) = |u|^{\lambda- 1}u$ ( $\lambda> 0$ ). By using the generalized Riccati transformation and the inequality technique, some new oscillation criteria for the equations are established. Our results extend and improve some known results, but they also unify the oscillation of second-order Emden-Fowler delay differential equations with damping and second-order Emden-Fowler delay difference equations with damping. Examples are given to illustrate the importance of our results.

Oscillation Criteria for Functional Nonlinear Dynamic Equations with $${\gamma}$$ γ -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals

In this paper, we present oscillation criteria for the second-order nonlinear dynamic equation \({[a(t)\phi_{\gamma} (x^{\Delta}(t))]^{\Delta} + p(t)\phi_{\gamma}(x^{\Delta^{\sigma}}(t)) + q_{0}(t) \phi_{\gamma}(x(g_{0}(t)))+\sum_{i=1}^{2}\int_{a_{i}}^{b_{i}}q_{i}(t,s)\phi_{\alpha_{i}(s)}(x(g_{i}(t,s))) \Delta \zeta_{i}(s)=0}\) on a time scale \({\mathbb{T}}\) which is unbounded above. Our results generalize and improve some known results for oscillation of second-order nonlinear dynamic equation. Some examples are given to illustrate the main results.

Oscillatory behavior of second-order half-linear damped dynamic equations

By refining the generalized Riccati transformation technique, we obtain new oscillation criteria for a class of second-order half-linear delay dynamic equations with damping on a time scale. Assumptions in our theorems are less restrictive, these results complement and improve related contributions to the subject. In particular, the results obtained amend related results reported in the literature.

Kamenev-type criteria for nonlinear damped dynamic equations

We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.

Oscillation of second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales

This paper concerns second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales of the form $$\bigl(r(t) \bigl(\bigl(y(t)+p(t)y\bigl(\tau(t)\bigr)\bigr)^{\Delta}\bigr)^{\gamma}\bigr)^{\Delta}+\int_{a}^{b}f \bigl(t,y\bigl(\delta (t,\xi)\bigr)\bigr)\Delta\xi=0, $$ where $\gamma>0$ is a quotient of odd positive integers. By using the generalized Riccati technique and integral averaging techniques, we derive new oscillation criteria for the above equations, which generalize and improve some existing results in the literature.

New Oscillation Results of Second-Order Damped Dynamic Equations withp-Laplacian on Time Scales

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation withp-Laplacian on time scales of the form(rtφγ(xΔ(t)))Δ+ptφγ(xΔ(t))+f(t,x(g(t)))=0, where the coefficient functionp(t)may change sign. Two examples are given to demonstrate the obtained results.

Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators

The study of half-linear differential equations has become an important area of research due to the fact that such equations occur in a variety of real-world problems such as in the study of \(p\)-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. On the basis of these background details, we study oscillatory behavior of a class of second-order neutral functional dynamic equations on a time scale. New criteria improve and complement related results reported in the literature. Some examples are included to illustrate the results obtained. In particular, an example regarding the second-order neutral differential equation is also provided to show that these theorems improve those in the continuous case.

Oscillation of Second-Order Neutral Dynamic Equations with Mixed Arguments

We present some new oscillation criteria for a class of half-linear second -order neutral dynamic equations with mixed arguments. An example is given to illustrate the main results.

Critical oscillation constant for Euler-type dynamic equations on time scales

In this paper we study the second-order dynamic equation on the time scale T of the form(r(t)yΔ)Δ+γq(t)tσ(t)yσ=0,wherer,qare positive rd-continuous periodic functions with inf{r(t),t∈T}>0 and γ is an arbitrary real constant. This equation corresponds to Euler-type differential (resp. Euler-type difference) equation for continuous (resp. discrete) case. Our aim is to prove that this equation is conditionally oscillatory, i.e., there exists a constant Γ>0 such that studied equation is oscillatory for γ>Γ and non-oscillatory for γ<Γ.

A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales

The principal goal of this paper is to amend oscillation results obtained in the recent paper by Saker and O’Regan (2011) [9].

Classification and Existence of Nonoscillatory Solutions of the Second-Order Neutral Delay Dynamic Equations on Time Scales

In the present paper, we give a classification of nonoscillatory solutions of the second-order neutral delay dynamic equation on time scales $$ \begin{array}{*{20}{c}} {{{{\left[ {x(t)-c(t)x\left( {t-\tau } \right)} \right]}}^{{\varDelta \varDelta }}}+f\left( {t,x\left( {{g_1}(t)} \right),\ldots,x\left( {{g_m}(t)} \right)} \right)=0,} \hfill & {t\in T.} \hfill \end{array} $$ Some existence results for each kind of nonoscillatory solutions are also established.

Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales

This paper is concerned with oscillations of the second-order delay nonlinear dynamic equation on a time scale , where a and q are real-valued rd-continuous positive functions on , α and β are ratios of odd positive integers, , , , and . We establish some new sufficient conditions for this equation. MSC:34K11, 39A10, 39A99.

Oscillation criteria for second-order dynamic equations on time scales

Erbe’s and Hassan’s contributions regarding oscillation criteria are interesting in the development of oscillation theory of dynamic equations on time scales. The objective of this paper is to amend these results.

On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of𝕋=ℝand𝕋=ℤ.

Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

New Oscillatory Behavior of Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

We establish four new oscillation criteria of Grace-type for the second-order nonlinear dynamic equations with damping. These criteria extend known criteria for corresponding dynamic equations. Our results are new even in the continuous and the discrete cases.

Oscillation Criteria of Second-Order Dynamic Equations with Damping on Time Scales

Using functions in some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order dynamic equations with damping on time scales of the form(r(t)(xΔ(t))γ)Δ+p(t)(xΔ(t)γ)+f(t,x(g(t)))=0. Two examples are included to show the significance of the results.