Usual Definiteness Research Articles

On Boundedness of Solutions of State Periodic Systems: A Multivariable Cell Structure Approach

Many dynamical systems are periodic with respect to several state variables. This periodicity typically leads to the coexistence of multiple invariant solutions (equilibria or limit cycles). As a consequence, while there are many classical techniques for analysis of boundedness and stability of such systems, most of these only permit to establish local properties. Motivated by this, a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented. The proposed method is inspired by the cell structure approach developed by Leonov and Noldus and characterized by two main advances. First, the conventional cell structure framework is extended to the case of dynamics, which are periodic with respect to multiple states. Second, by introducing the notion of a Leonov function, the usual definiteness requirements of standard Lyapunov functions are relaxed to sign-indefinite functions. Furthermore, it is shown that under (mild) additional conditions the existence of a Leonov function also ensures input-to-state stability, i.e., robustness with respect to exogenous perturbations. The performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system.

Conditionally positive definite kernels in Hilbert $$C^*$$ C ∗ -modules

We investigate the notion of conditionally positive definite in the context of Hilbert \(C^*\)-modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert \(C^*\)-modules. As a consequence, we show that a \(C^*\)-metric space (S, d) is \(C^*\)-isometric to a subset of a Hilbert \(C^*\)-module if and only if \(K(s,t)=-d(s,t)^2\) is a conditionally positive definite kernel. We also present a characterization of the order \(K'\le K\) between conditionally positive definite kernels.

Strictly positive definite kernels on a product of spheres

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.

Conic positive definiteness and sharp minima of fractional orders in vector optimization problems

Motivated by the fact that the usual positive definiteness does not work in an infinite space, we introduce the concept of S-positive definiteness with respect to an ordering cone in a general Banach space and show that the S-positive definiteness plays the same role as the usual positive definiteness in the finite dimensional case. As applications, we study sharp and weak sharp minima of fractional orders in vector optimization.

Quadratic Eigenvalue Problems for Second Order Systems

We consider the spectral structure of a quadratic second order system boundary-value problem. In particular we show that all but a finite number of the eigenvalues are real and semi-simple. We develop the eigencurve theory for these problems and show that the order of contact between an eigencurve and the parabola gives the Jordan chain associated with the eigenvector corresponding to that eigencurve. Following this we use our knowledge of the eigencurves to obtain eigenvalue asymptotics. Finally the completeness of the eigenfunctions is studied using operator matrix techniques. It should be noted here that the usual left definiteness assumptions have been overcome in this study.

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS WITH PERIODIC BOUNDARY CONDITIONS

Abstract The oscillation theory for two simultaneous systems of second order linear differential equations in two parameters with periodic boundary conditions is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theory under another important definiteness condition.

AN OSCILLATION THEOREM FOR A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS

Abstract The oscillation theorem for two simultaneous Sturm-Liouville systems in two parameters is well known when the coefficients of the differential equations are subjected to the usual definiteness condition. However, in practical applications the usual definiteness condition may fail to hold, and hence in this paper we consider the oscillation theorem under another important definiteness condition.

An isothermal theory of anisotropic rods

To provide an isothermal, deterministic theory of anisotropic rods is the primary objective of this paper. Our starting point is the 3-D linear theory of micropolar elastodynamics. First, the governing equations of the theory are established by the use of a suitable averaging procedure together with a separation of variables solution for kinematic variables. Next, without making the usual definiteness assumption for the strain energy density, a dynamic uniqueness theorem is constructed for the solutions of the governing equations. Logarithmic convexity arguments are then used to enumerate a set of conditions sufficient for uniqueness. The theory includes the effects of warping and shearing deformations, and in fact, it incorporates as many higher order effects as deemed necessary in any special case. Also, the application of the theory is illustrated in a sample example.

Quantitative analysis of simple and interconnected systems: Stability, boundedness, and trajectory behavior

In practice one is not only interested in the qualitative type of information obtainable from the stability (in the Lyapunov sense) of a dynamic system, but also in quantitative data, such as specific trajectory bounds and specific transient behavior. A system could, for example, be stable and still be completely useless because it may exhibit undesirable transient characteristics (e.g., it may exceed certain limits imposed on the trajectory bounds). In order to develop a meaningful quantitative theory for the analysis of dynamic systems, stability is defined here in terms of subsets of the state space that are prespecified in a given problem and, in general, may be time varying. The properties of these subsets yield not only information about the stability of a system, but they also yield estimates of trajectory bounds and of trajectory behavior. The theory developed here is general enough to include autonomous and nonautonomous systems, linear and nonlinear systems, simple systems and interconnected systems. The composite, or interconnected systems considered are analyzed and treated in terms of their subsystems. After stating various definitions of stability and instability, theorems that yield sufficient conditions for stability and instability are stated and proved. These theorems involve the existence of Lyapunov-like functions which in general do not possess the usual definiteness requirements on V and V. In order to demonstrate the developed theory, several examples are considered.

On the bounds of the trajectories of differential systems†

The behaviour of differential systems is investigated by considering the stability and instability of such systems with respect to certain sub-sets of the state space These Sets may in general be time-varying, and their properties do not only yield information about the stability of a system but also estimates of the bounds of the system trajectories. In all cases the results which are established yield sufficient conditions for stability and instability, and in general involve the existence of Lyapunov-like functions which do not appear to possess the usual definiteness requirements on V and [Vdot]. The developed theory is applied to two special cases: in one case, only time-invariant sub-sets are considered: in the second case, the time interval [t0, ∞) over which the systems me defined is truncated to [t 0, t a + T), T < ∞, So the that case of stability over a finite time interval may be considered.