Stabilization for a class of second-order switched systems

This paper considers the asymptotic stabilization problem of second-order linear time-invariant (LTI) autonomous switched systems consisting of two subsystems with unstable focus equilibrium. More precisely, we find a necessary and sufficient condition for the origin to be asymptotically stable under the predesigned switching law. The result is obtained without looking for a common Lyapunov function or multiple Lyapunov function, but studying the locus in which the two subsystem's vector fields are parallel. Then the “most stabilizing” switching laws are designed which have translated the switched system into a piecewise linear system. Two numerical examples are presented to show the potential of the proposed techniques.     

Stabilization criteria of a class of switched systems

In this paper, stabilizability property for a switched system under arbitrary switching is considered from an algebraic point of view by means of the existence of a set of block‐diagonal Lyapunov solutions with common Schur complement of certain order—or, equivalently, with common block (1,1)—for the matrix bank. It is shown that the existence of that set is equivalent to the existence of solutions for some Riccati inequalities done in terms of the blocks of matrices of the bank. In addition, we conclude that a particular class of systems with matrix bank constituted by Metzler matrices—Positive Switched Systems—are stabilizable by partial state reset.     

Asymptotic linearization and stabilization for a class of nonlinear systems

This paper presents a nonlinear output feedback which asymptotically linearizes the class of nonlinear, continuous-time, affine in the control systems having relative degree equal to the state space dimension. Moreover, we show that any set of eigenvalues can be assigned for the asymptotic closed-loop linear system. The controller is based on a nonlinear observer, presented in a previous paper, and on the linearizing state feedback proposed by Isidori and computed in the estimated state. The main result obtained is equivalent to the separation theorem in the linear case.This work was supported by the Italian Ministry for University and Scientific and Technological Research.     

Homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems , where t∈R, u∈Rn and W₁,W₂∈C¹(R×Rn,R) and f∈C(R,Rn) are not necessary periodic in t. This result generalizes and improves some existing results in the literature.     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

一类二阶Hamiltonian系统的无穷多周期解

研究一类超线性二阶Hamiltonian系统,且非线性项是奇的,不需要假设Ambros-etti-Rabinowitz的超二次条件,利用对称型山路引理得到无穷多周期解存在性结果.     

Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results.     

Modal stabilization of a certain class of uniformly controllable systems

The system $\dot x = A( \cdot )x + b( \cdot )u,$ where A(·) ∈ ? ^n×n and b(·) ∈ ? ^n×1, is considered. The elements of the matrix A(·) and the column b(·) are bounded by nonanticipating functionals of an arbitrary nature that satisfy the condition $\mathop {\inf }\limits_{( \cdot )} A^{n - 1} ( \cdot )b( \cdot ),...,A( \cdot )b( \cdot ),b( \cdot )| > 0$ . From a given constant spectrum contained in the left half-plane, a feedback u = (s(·), x) is constructed, the coefficients of which are expressed in terms of A(·) and b(·). Conditions for the closed system to be globally exponentially stable are found. A similar result is obtained for the system $x(k + 1) = A(k)x(k) + b(k)u(k)$ .     

Existence of homoclinic solutions for a class of second-order Hamiltonian systems

A new result for existence of homoclinic orbits is obtained for the second-order Hamiltonian systems under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary-value problems which are obtained by the minimax methods.     

Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems

In this paper we consider the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system:

(HS)     

Infinitely many periodic solutions for a class of second-order Hamiltonian systems

In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems

$$\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right.$$

, where F(t, u) is even in u, and ?F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.

     

Stability analysis of a class of nonlinear positive switched systems with delays

This paper addresses the stability problem of delayed nonlinear positive switched systems whose subsystems are all positive. Both discrete-time systems and continuous-time systems are studied. In our analysis, the delays in systems can be unbounded. Two conditions are established to test the local asymptotic stability of the considered systems. The method to compute the domains of attraction is also proposed provided that the system is locally asymptotically stable. When reduced to general nonlinear positive systems, that is, the considered switched system consists of only one mode, an interesting conclusion follows that the proposed nonlinear positive system is locally asymptotically stable for all admissible delays and nonnegative nonlinearities which satisfy an extra condition at the origin, if and only if the system represented by the linear part is asymptotically stable for all admissible delays. Finally, a numerical example is presented to illustrate the obtained results.     

Estimate of the attraction domain for a class of nonlinear switched systems

We consider a hybrid dynamical system composed of a family of subsystems of nonlinear differential equations and a switching law which determines the order of their operation. It is assumed that subsystems are homogeneous with homogeneity degrees less than one, and zero solutions of all subsystems are asymptotically stable. Using the Lyapunov direct method and the method of differential inequalities, we determine classes of switching laws providing prescribed estimates of domains of attraction for zero solutions of the corresponding hybrid systems. The developed approaches are used for the stabilization of a double integrator.     

Stability analysis and control synthesis for a class of switched neutral systems

In this paper, the problems of asymptotical stability and stabilization of a class of switched neutral control systems are investigated. A delay-dependent stability criterion is formulated in term of linear matrix inequalities (LMIs) by using quadratic Lyapunov functions and inequality analysis technique. The corresponding switching rule is obtained through dividing the state space properly. Also, the synthesis of stabilizing state-feedback controllers are done such that the close-loop system is asymptotically stable. Two numerical examples are given to show the proposed method.     

Existence of homoclinic solutions for a class of second-order non-autonomous Hamiltonian systems

By using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of second-order Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.     

Existence of positive solutions for a class of second-order ordinary differential systems

In this paper, we present some new concepts of superlinearity and sublinearity for nonlinear terms of second-order ordinary differential systems, and then consider the existence of positive solutions for systems with such superlinear and sublinear nonlinear terms, especially proving the existence of positive solutions for systems in which one nonlinear term is superlinear and the other is sublinear.     

Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems

Some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second-order Hamiltonian systems by the minimax methods in critical point theory.     

Existence of periodic solutions for a class of second-order discrete Hamiltonian systems

By using the variant fountain theorem, we study the existence of periodic solutions for a class of superquadratic non-autonomous second-order discrete Hamiltonian systems.