Nonautonomous Second-order Systems Research Articles

Approximate controllability of non-autonomous evolution system with infinite delay

This article deals with the existence and approximate controllability results for a class of nonautonomous second-order evolution systems with infinite delay. To establish sufficient conditions for the proposed control problem the theory of evolution operator with Schauder’s fixed point theorem is used. Further, we extend the approximate controllability results to the integro-impulsive differential system. Finally, to emphasize our theoretical concepts, an example is provided.

Finite time stabilization of non-autonomous, nonlinear second-order systems based on minimum time principle

This paper proposes a controller design method to stabilize a class of nonlinear, non-autonomous second-order systems in finite time. This method is developed based on exact-linearization and Pontryagin’s minimum time principle. It is shown that the system can be stabilized in a finite time of which the upper bound can be chosen according to the initial states of the system. Simulation results are given to validate the theoretical analysis

Data-Based H∞Control for the Constrained-Input Nonlinear Systems and its Applications in Chaotic Circuit Systems

In this paper, $H_\infty $ control problem is investigated by off-policy integral reinforcement learning (IRL) method for the nonlinear systems with completely unknown dynamics, disturbances, and constrained-input. Firstly, according to a model-based policy iteration (PI) algorithm, a model-free algorithm is proposed based on the derived iterative equation, and the equivalence of model-based PI algorithm and model-free algorithm is proven. Then, the model-free algorithm is implemented by off-policy IRL technology to solve the Hamilton-Jacobi-Isaacs (HJI) equation with the collected system data by the least-square approach, where three neural networks (NNs) are constructed to approximate the value function, control and the disturbance. Finally, our proposed methods are applied to stabilize an autonomous third-order Chua’s chaotic circuit system and a non-autonomous second-order memristive chaotic circuit system to illustrate the efficiency of the proposed method.

Approximate Controllability of Second-order Non-autonomous System with Finite Delay

In this article, we shall study the approximate controllability of certain non-autonomous second-order nonlinear differential problems with finite delay in the infinite dimensional space. Sufficient conditions are proposed and proved for the controllability of such systems. Further, we briefly discuss the approximate controllability of impulsive as well as integro-differential problem. We establish these results by utilizing Schauder’s fixed-point analysis approach. Finally, the application of the proposed results is presented by giving an example.

INFINITELY MANY SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER SYSTEMS WITH IMPULSIVE EFFECTS

In this paper, we establish the existence of infinitely many solutions for a class of non-autonomous second-order systems with impulsive effects. Our technique is based on the Fountain Theorem of Bartsch and the Symmetric Mountain Pass Lemma due to Kajikiya.

An Analytical and Experimental Study of SC-CNN-Based Simple Nonautonomous Chaotic Circuit

Abstract In this study, we report an explicit analytical solution of state-controlled cellular neural network (SC-CNN) based second-order nonautonomous system. The proposed system is modeled with an aid of a generalized two-state-controlled cellular neural network (CNN) equations and experimentally realized by imposing a suitable connection of simple two-state-controlled generalized CNN cells following the report of Swathi et al. [2014]. The chaotic and quasi-periodic dynamics observed from this system have been investigated through an analytical approach for the first time. The intriguing dynamics observed from the system where further substantiated by phase portraits, Poincaré map, power spectra, and “0−1 test.” We trace the transition of the system from periodic to chaos through analytical solutions, which are in good agreement with hardware experiments. Additionally, we show PSpice circuit simulation results for validating our analytical and experimental studies.

Existence of periodic solutions for a class of second-order p-Laplacian systems

In this paper, the existence of periodic solutions are obtained for a class of non-autonomous second-order p-Laplacian systems by the least action principle.

On Uniqueness and Properties of Periodic Solution of Second-Order Nonautonomous System with Discontinuous Nonlinearity

We consider an automatic control system with discontinuous nonlinearity of non-ideal relay type and continuous external periodic influence. The control object can be either stable or unstable. In both cases, theorems on sufficient conditions for the existence of a unique periodic solution with given properties are obtained.

Precise Estimates of the Walk Speed of Solutions of Second-Order Linear Systems

We consider some classes of nonautonomous second-order systems of differential equations whose coefficients are bounded by a given constant M, in particular, diagonal systems, triangular systems, and systems corresponding to a single equation. It is shown that the walk speeds of solutions of various systems from these classes fill up a certain interval, and precise estimates are obtained for the length of that interval in terms of the value of M.

Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

This paper is concerned with the following nonlinear second-order nonautonomous problem:ü(t)+q(t)u̇(t)-∇K(t,u(t))+∇W(t,u(t))=0, wheret∈R,u∈RN, andK,W∈C1(R×RN,R)are not periodic intandq:R→Ris a continuous function andQ(t)=∫0t‍q(s)dswithlim|t|→+∞⁡Q(t)=+∞. The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.

Infinitely Many Fast Homoclinic Solutions for Some Second-Order Nonautonomous Systems

We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved.

Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects

AbstractIn this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects{uΔ2(t)+A(σ(t))u(σ(t))+∇F(σ(t),u(σ(t)))=0,Δ-a.e. t∈[0,T]Tκ;u(0)−u(T)=uΔ(0)−uΔ(T)=0,(ui)Δ(tj+)−(ui)Δ(tj−)=Iij(ui(tj)),i=1,2,…,N,j=1,2,…,p,wheret0=0<t1<t2<⋯<tp<tp+1=T,tj∈[0,T]T(j=0,1,2,…,p+1),u(t)=(u1(t),u2(t),…,uN(t))∈RN,A(t)=[dlm(t)]is a symmetricN×Nmatrix-valued function defined on[0,T]Twithdlm∈L∞([0,T]T,R)for alll,m=1,2,…,N,Iij:R→R(i=1,2,…,N,j=1,2,…,p) are continuous andF:[0,T]T×RN→R. Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.MSC:34B37, 34N05.

PRACTICAL SYNCHRONIZATION OF NONAUTONOMOUS SYSTEMS WITH UNCERTAIN PARAMETER MISMATCH VIA A SINGLE STATE FEEDBACK CONTROL

Robust practical synchronization of general second-order nonautonomous systems with uncertain parameter mismatch is investigated by using a single state feedback control. Some simple general algebraic criteria are derived based on practical stability theory of nonautonomous dynamical system. A distinctive feature of this work is that the parameter mismatch not only exists in system parameters, but also in the external excitation ones. More reasonably, the values of parameter mismatch can be uncertain. Besides, a single state feedback control including an approximate differentiation filter only needs to know information about one state, which provides an advantage over the use of full-state model-based observers. It is shown that the approaches developed here further extend the ideas and techniques presented in recent literature. As a direct application of the new theoretical results, the obtained results are applied to a typical horizontal platform system and the representative forced Duffing–Van der Pol oscillator. Subsequently, numerical simulations demonstrate the effectiveness of the criteria and the robustness of the control technique.

Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications

This paper is concerned with robust practical synchronization for general second-order nonautonomous systems with parameter mismatch. Some simple yet general algebraic criteria are derived based on practical stability theory of nonautonomous dynamical systems. A distinctive feature of this work is that the parameter mismatch cannot only be existed in system parameters, but also in external excitation ones. Furthermore, the obtained results are applied to a typical horizontal platform system and the representative forced Van der Pol oscillator. Subsequently, numerical simulations demonstrate the effectiveness of the criteria and the robustness of the control technique.

Infinitely many periodic solutions for some second-order differential systems with p(t)-Laplacian

Abstract In this article, we investigate the existence of infinitely many periodic solutions for some nonautonomous second-order differential systems with p(t)-Laplacian. Some multiplicity results are obtained using critical point theory. 2000 Mathematics Subject Classification: 34C37; 58E05; 70H05.

SOME EXISTENCE RESULTS ON PERIODIC SOLUTIONS OF ORDINARY (q, p)-LAPLACIAN SYSTEMS

Some existence theorems are obtained for periodic solutions of nonautonomous second-order differential systems with (q, p)-Laplacian by the minimax methods in critical point theory.

A note on existence of a unique periodic solution of non-autonomous second-order Hamiltonian systems

Abstract In this study, we shall be concerned with a non-autonomous second-order Hamiltonian system. Some criteria for guaranteeing the existence and uniqueness of T -periodic classical solution of this system are presented by employing the least action principle, the Saddle Point Theorem and some analysis techniques.

Some existence results on periodic solutions of nonautonomous second-order differential systems with [formula omitted]-Laplacian

Some existence theorems are obtained for periodic solutions of nonautonomous second-order differential systems with ( q , p ) -Laplacian by using the least action principle and the minimax methods.

Periodic solutions to non-autonomous second-order systems

We establish the existence of periodic solution of a class of non-autonomous second-order systems, x ̈ + μ x + V ( t , x ) = 0 , where V ( t , x ) = ( v 1 ( t , x ) , … , v n ( t , x ) ) , if lim | x | → ∞ v i ( t , x ) | x | = 0 , i = 1 , … , n , uniformly in t and V is bounded below or above for appropriate ranges of μ .

Measurement system for model verification of nonautonomous second-order nonlinear systems

A measurement system for model verification of nonautonomous second-order nonlinear systems is presented. The measurement system is tested on two characteristic examples of nonlinear electrical systems, a boost dc converter and a ferroresonant circuit. In both cases, the models used in simulation are verified.