Sufficient Conditions For Global Stability Research Articles

A global stabilization theorem for planar nonlinear systems

AbstractWe provide sufficient conditions for global stabilization of planar nonlinear systems by means of static and dynamic feedback laws. The results of the paper extend previous work on the same problem. Our approach uses some ideas from a recent paper of the author based on the well‐known Artstein theorem on stabilization as well as a recent result of Coron and Praly.

Necessary and sufficient conditions for global stability and uniqueness in finite element simulations of adaptive bone remodeling

Conditions which can guarantee the global stability and uniqueness of the solution to a bone remodeling simulation are derived using a specific rate equation based on strain energy density. We modeled bone tissue as isotropic with a constant Poisson ratio and the elastic modulus proportional to volumetric density of calcified tissue raised to the power n. Our remodeling rate equation took the rate of change of volumetric hard tissue density as proportional to the difference between a stimulus (strain energy density divided by volumetric density taken to the power m) and a set point. In previous studies we defined state variables which are conjugate to the remodeling stimulus, and the function which acts as a variational indicator for the remodeling stimulus. In this study, we use the properties of this variational indicator to establish the stability and the uniqueness of the solution to the remodeling rate equations for all possible density distributions. We show that the solution is the global minimum of a weighted sum of the total strain energy and the integral of density to the power m over the remodeling elements. These results are proven for n < m, and we show that taking n > m will eliminate the possibility that a unique solution exists.

Global stability of the competitive economy involving complementary relations among commodities

In this paper, we provide some sufficient conditions for global stability of the competitive economy involving complementary relations among commodities. We show that our result gives a generalization of the well-known result regarding weak gross substitutability. Our result also gives a global stability condition different from that due to Morishima.

Global stabilization of a class of quadratic systems

This paper gives a necessary and sufficient condition for global stabilization of quadratic systems in R n as well as the stabilizing feedbacks which are written under invariant form.

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.

Nonlinear stability of a flow with bound eddies

Kovasznay [Proc. Cambridge Philos. Soc. 44, 58 (1948)] obtained an exact solution of the Navier–Stokes equations that describes a flow with periodic bound eddies. The stability of this nonparallel flow with respect to three-dimensional finite amplitude disturbances is analyzed by use of the energy method. A sufficient condition for global stability is obtained. It is shown that the additional nonhomogeneity in shear stress distribution in a nonparallel flow is a destabilizing factor.

Global stability of a biological model with time delay

This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay.

The equilibrium and stability of rotating plasmas

In a rotating equilibrium state, the velocity and magnetic fields are shown to share the same flux surfaces. A simplified derivation is given of a second-order (not necessarily elliptic) partial differential equation which determines axisymmetric equilibrium states. For general configurations, equations on flux surfaces which determine the Alfvén and cusp continuous spectrum are derived and the stability investigated. These equations are written without the use of any particular coordinate system. Similar equations yield a sufficient condition for global stability of axisymmetric equilibria if the flow is parallel to the magnetic field up to a rigid rotation of the plasma. This condition is also necessary for stability in a mirror configuration with no toroidal field and a pure rigid rotation.

Global Stability in Ecological Systems with Continuous Time Delay

A population of interacting species is considered which is governed by a system of first order integro-differential equations. In the case that the integral kernels satisfy linear differential equations with constant coefficients, a sufficient condition for global stability of an equilibrium state is derived. The result is applied to the growth of one species whose growth rate depends on the population density in the past and to a predator-prey system where the numerical response of the predators contains a continuous time delay.

A global stability criterion for simple control loops

A method for studying biological control loops has been developed, which suffices to prove global stability for the Goodwin equations when the Hill coefficient θ is equal to 1. This holds for arbitrary reaction constants, even if time delays are included in the system. For a generalized class of repfessible systems, including the Goodwin Equations for θ>1, the method gives a sufficient condition for global stability, in terms of solutions of an algebraic equation in a single variable. When the criterion is not satisfied, the same equation gives bounds on any possible limit cycles. The method also shows that inducible systems with a unique equilibrium are globally stable. The system of equations studied allows each reaction rate equation to be non-linear, and to include a time delay.

Global stability in two species interactions.

It is proved that sufficient conditions for global stability in a Lotka-Volterra model of a two species interactions are (i) the equilibrium is feasible, (ii) the equilibrium is locally stable and (iii) at the equilibrium each species sustains density dependent mortalities due to intraspecific interactions.