Mathematics
Stability
In mathematics, stability refers to the property of a system or solution that remains unchanged or converges to a steady state over time, despite perturbations or disturbances. A stable system or solution is resistant to small changes and maintains its equilibrium. Stability is a fundamental concept in various mathematical fields, including differential equations, control theory, and dynamical systems.
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10 Key excerpts on "Stability"
- Urabe(Author)
- 1967(Publication Date)
- Academic Press(Publisher)
6. Stability The notion of Stability comes from the study of physical motions. A ball at the bottom of a vessel is stable, while a ball at the top of a roof is unstable. Stability is related ‘to the behavior of motion after a long time. In Theorem 2.3, we have seen that the solution of a differential equation is in general continuously dependent on the initial condition in the finite inter- val; but this does not give any information about the Stability of the solution, since Stability is connected with the behavior of the solution in the infinite interval [to, m). In the present chapter, first, mathematical definitions of Stability will be given, and some fundamental theorems concerning Stability will be proved. Then Stability for autonomous systems will be considered, and the Stability of critical points and periodic solutions of autonomous systems will be dis- cussed. References are made to Coddington and Levinson [l] and Niemytski and Stepanov [ 11 concerning fundamental theorems and to Urabe [4], con- cerning Stability for autonomous systems. 6.1. DEFUVITION OF Stability Consider a system of differential equations dx/dt = X(t, x), (6.1) where x and X(t, x) are vectors of the same dimension. Suppose the system (6.1) has a solution xi = qo(t) on the interval [to, m) and the function X(t, x) is continuous in the domain SZ of the tx-space containing the trajectory x = Let F be a family of solutions of (6.1) containing the solution x = qo(t). Then the solution x = qo(t) is said to be stable with respect to the family 9 if, for any positive number E, there are two numbers 6 and T 2 to such that for any solution x = q(t) E F, PO(t). I q(t) - qo(t) I c E on the interval [T, a), (6.2) 54 6.1. Definition of Stability 55 whenever I Cp(t0) - VO(t0) 1 < 6. (6.3) The solution x = qo(t) is said to be asymptotically stable with respect to 9 if it is stable and, in addition, lim I - 4oo(t) I = 0 r+m for any solution x = q(t) E 9 satisfying (6.3).
- Anatoliy A. Martynyuk, Andrzej Szadkowski, Bogusław Radziszewski(Authors)
- 2020(Publication Date)
- Sciendo(Publisher)
1. Stability { What Is It, and What Is It For? Over the years, the perception of Stability and the Stability concept referred not only to mathematics and mechanics but also to numerous and amaz- ingly diverse elds of human activity. Obviously, representatives of diverse sectors of life, dissimilar industries, and dierent sciences are absorbed in separate problems and often, they use dierent terms and even unlike jar- gons when describing the same or similar things. The popular understanding of the Stability concept by dierent factions brings a lot of colloquial, im- precise denitions { no wonder, the science has been trying to sort out, and maybe even unify, them. What has been common in those various and not similar to each other approaches to express a Stability concept? Well, it can be seen that using the notion of Stability to dierent spheres of life and science, one generally refers to \something" what is an object, system, solution, process, or phenomenon, and this \thing" behaves in a way which is pronounced by a motion, state, equilibrium, status, or reaction. In various areas of life, these situations are innumerable. For simplicity, instead of distinguishing between dierent pos- sible circumstances and describing each time a particular situation, it seems natural to generalise the approach. According to an informal agreement, the Stability of motion or Stability of solutions of equations are understood, in general terms as the Stability of some \state" or \system". No matter what the \thing" under consideration, for accuracy and precision of the delibera- tion is, some model of the occurrence is created. By eliminating secondary attributes, the situation is simplied and its abstract physical model suit- able for a mathematical representation is generated. Not surprisingly in our civilization of prevailing pictorial culture, the model having clear graphical interpretation is mostly appreciated.
eBook - PDFDifferential Equations
An Introduction to Basic Concepts, Results and Applications
- Ioan I Vrabie(Author)
- 2004(Publication Date)
- WSPC(Publisher)
Chapter 5 Elements of Stability This chapter is entirely dedicated to the study of the Stability of solutions to certain systems of differential equations. In the first section we introduce and illustrate the main concepts referring to Stability. The second one is concerned with several necessary and sufficient conditions for various types of Stability in the particular case of first-order systems of linear differential equations. In the third section we present some sufficient conditions under which the asymptotic Stability of the null solution of a first-order differential system is inherited by the null solution of a certain perturbed system, provided the perturbation is small enough. In the fourth section we prove several sufficient conditions for Stability expressed by means of some functions decreasing along the trajectories, while in the fifth section we include several results regarding the Stability of solutions of dissipative systems. In the sixth section we analyze the Stability problem referring to automatic control systems, while the seventh section is dedicated to some considerations concerning inStability and chaos. As each chapter of this book, this one also ends with an Exercises and Problems section. 5.1 Types of Stability In its usual meaning, Stability is that property of a particular state of a given system of preserving the features of its evolution, as long as the perturbations of the initial data are sufficiently small. This meaning comes from Mechanics, where it describes that property of the equilibrium state of a conservative system of being insensitive ‘‘h la Zongue” to any kind of perturbations of “small intensity”. Mathematically speaking, this notion 159 160 Elements of Stability has many other senses, all coming from the preceding one, and describing various kinds of continuity of a given global solution of a system as function of the initial data, senses which are more or less different from one another.
eBook - PDFDifferential Equations
An Introduction to Basic Concepts, Results and Applications
- Ioan I Vrabie(Author)
- 2011(Publication Date)
- WSPC(Publisher)
Chapter 5 Elements of Stability This chapter is entirely dedicated to the study of the Stability of solutions to certain systems of differential equations. In the first section we introduce and illustrate the main concepts referring to Stability. The second one is concerned with several necessary and sufficient conditions for various types of Stability in the particular case of first-order systems of linear differential equations. In the third section we present some sufficient conditions under which the asymptotic Stability of the null solution of a first-order differential system is inherited by the null solution of a certain perturbed system, provided the perturbation is small enough. In the fourth section we prove several sufficient conditions for Stability expressed by means of some functions decreasing along the trajectories, while in the fifth section we include several results regarding the Stability of solutions of dissipative systems. In the sixth section we analyze the Stability problem referring to automatic control systems, while the seventh section is dedicated to some considerations concerning inStability and chaos. As each chapter of this book, this one also ends with an Exercises and Problems section. 5.1 Types of Stability In its usual meaning, Stability is that property of a particular state of a given system of preserving the features of its evolution, as long as the perturbations of the initial data are sufficiently small. This meaning comes from Mechanics, where it describes that property of the equilibrium state of a conservative system of being insensitive “ `a la longue ” to any kind of perturbations of “small intensity”. Mathematically speaking, this notion 155 156 Elements of Stability has many other senses, all coming from the preceding one, and describing various kinds of continuity of a given global solution of a system as function of the initial data, senses which are more or less different from one another.
eBook - PDFOrdinary Differential Equations
Introduction and Qualitative Theory, Third Edition
- Jane Cronin(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
Chapter 4 Stability Introduction The material in the previous chapters is basic to all further study of differential equa-tions. The topic of this chapter, Stability, is certainly fundamental, but our emphasis and comparatively lengthy treatment are partly motivated by applications to problems in the physical world, especially biological problems. To some extent, by placing a strong emphasis on Stability, we are choosing now a particular path in our study of differential equations. The subject of Stability can be approached from two viewpoints. The less important and less interesting viewpoint is that of pure mathematics. That is, a reasonable mathematical problem is to generalize or extend some of the results we obtained concerning the orbits of two dimensional linear homogeneous systems to orbits of nonlinear systems of dimension n > 2. It is easy to see that if we attempt as fine an analysis in the more general situation, the results become extremely complicated. The example given by equation (3.19) in Chapter 3 shows how complicated the results can become in the nonlinear two dimensional case, and if we consider even the linear problem in the n -dimensional case, where n > 2, the results become quite complicated (see Exercise 1). We are forced to ask for a more modest result than a detailed description of the orbits. Indeed, one of the few questions we can ask which has a reasonable and uncomplicated answer is: Under what conditions do solutions approach the equilibrium point or stay close to the equilibrium point for all sufficiently large t ? Actually, the question can be made somewhat more general. We can ask under what conditions solutions approach or stay close to a given solution? Thus our first step is to say precisely what we mean by “approach” or “stay close to.” These are Stability properties. Then we seek sufficient conditions that these Stability properties hold.
eBook - PDF- Dean Karnopp(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
41 3 Stability of Motion: Concepts and Analysis Before dealing with the specific topic of vehicle Stability, it is useful first to discuss some general concepts of Stability. Mathematicians sometimes joke that there is no Stability to the definition of Stability. This is because, for non-linear differential equations, numbers of definitions of Stability and instabil-ity have been developed. Each definition has a precise use and each differs from the others enough that one cannot say they are all essentially equiva-lent. For the purposes of this book, however, the situation is not as hopeless as it might at first seem. The concern here is mainly the Stability of motion of vehicles traveling in a straight line or in a steady turn. The idea will be to try to predict whether a small perturbation from a basic motion will tend to die out or will tend to grow. The analysis will be based on linearized differential equations and the analysis of the Stability properties of linear differential equations is fairly straightfor-ward. The appropriate analytical techniques are discussed in this chapter. In addition, engineers realize that no mathematical model is ever exactly representative of a real device. They therefore do not expect that the results of analysis or even of simulation of a mathematical model will exactly predict how a real device will behave. An engineer’s use of analytical and compu-tational techniques is an attempt to reduce the necessity of experimentation and to foresee trends rather than to obtain precise results in the physical world. A good engineer is never completely surprised to find that some new phenomenon arises when a prototype is tested that was not predicted by analytical or even complicated computational results. The use of mainly linear techniques to study vehicle Stability has two main benefits.
eBook - PDF- Robert M May(Author)
- 2019(Publication Date)
- Princeton University Press(Publisher)
C H A P T E R T W O Mathematical Models and Stability THE MEANINGS OF Stability A variety of ecologically interesting interpretations can be , and have been , attached to the term “ Stability . ” The most common meaning corresponds to neighborhood Stability , that is , Stability in the vicinity of an equilibrium point in a deterministic system . This circumstance is not only the most tractable mathematically , but also ( as we shall see ) it often relates to more general stochastic situations , or to large amplitude disturbances . For population models in deterministic environments , with the environmental parameters all well -defined con -stants , one is interested in the community equilibria where all the species ’ populations have time -independent values , that is where all net growth rates are zero . Such an equilib -rium may be called stable if , when the populations are per -turbed , they in time return to their equilibrium values ; the return may be achieved either as damped oscillations or monotonically . Conversely , if such a disturbance tends to amplify itself , the system may be called unstable ; again such inStability may appear as oscillatory or as monotonic growth in the disturbance . The general cases of Stability and inStability are divided by the razor ’ s edge of neutral Stability , where the perturbed system either remains sta -tionary or oscillates with a constant amplitude set by the magnitude of the initial disturbance . The pathological 1 3 M A T H E M A T I C A L M O D E L S A N D S T A B I L I T Y ( a ) ( b ) ( c ) t t FIGURE 2.1 . Schematic illustration of a deterministic , mechanical system which when disturbed from equilibrium is ( a ) unstable , ( b ) stable , ( c ) neutrally stable . “ frictionless pendulum ” exhibits neutral Stability . These remarks are illustrated by Figure 2.1 . The graphical visualization of these ideas is familiar .
- Bellman(Author)
- 1967(Publication Date)
- Academic Press(Publisher)
For example, we may be interested in the existence of periodic solutions, in the asymptotic behavior of the solution as t becomes infinite, or in various analytic approximations involving power series or the solution of differential equations of lower degree. This type of investigation has been fruitfully extended by considering more general types of functional equations arising in the study of other physical systems, partial differential equations such as and differential-difference equations such as u” = g(u, u’; u(t - z), u’(t - z)). (3.1.5) Furthermore, by considering stochastic processes of different kinds, many more intriguing categories of equations are engendered. In this volume, we will restrict our attention to finite-dimensional processes of deterministic type, and even here we will consider only particularly simple kinds of equations. We try as carefully as we can to avoid a confusion of purely analytic difficulties due to the nature of the equation with conceptual complexities inherent in a theory of control. It is for this reason that we consider the one-dimensional control process first for orientation purposes. 3.2. Stability One of the most interesting and important qualitative properties of a solution is that of Stability. By this term, we mean intuitively the ability of the solution of the equation to preserve certain structural features un ler various types of changes in the values of the initial conditions, or the form of the equation, or both. To illustrate one aspect of this fundamental concept of both mathe- matics and science, consider the equation w” + aw’ + sin w = g(t), (3.2.1) 3.2. Stability 31 which we interpret as describing the pendulum of Section 3.1, subject to some external forces. If g(t) = 0, we know that w(t) = 0 is a solution. This corresponds to a position of equilibrium.
eBook - PDFDifferential Equations
Stability, Oscillations, Time Lags
- Halanay(Author)
- 1966(Publication Date)
- Academic Press(Publisher)
Let us suppose that perturbations of short duration have occurred during the development of the phenomenon which cannot be exactly known and hence cannot be taken into account in the mathematical description of the phenomenon. After the stopping of the action of these perturbations, the phenomenon will be described by the same system of differential equations as before their occurrence. But what has happened? The phenomenon was modified under the influence of the perturbations; hence the value corresponding to the moment when the perturbations have stopped their action will be different from the one given by the solution initially considered. It follows that after the stopping of the perturbations’ action, the phenomenon will be described by a solution 13 14 1. Stability THEORY other than the initial one. In other words, the effect of some perturbations of short duration consists in the passing from a solution with certain initial conditions to a solution with other initial conditions, the initial moment being considered the moment when the perturbations stop their action. Since these kinds of perturbations are neglected in every mathe- matical model of natural phenomena, it is necessary, to correctly describe the phenomenon and confer a physical sense upon the mathematical solution, that slight modifications of the initial conditions should not have too serious effects on the solution. This condition is always assured on a given interval (a, b) by the theorem of continuity with respect to initial conditions. The same reasoning as before lets us consider as necessary the independence of 6 with respect to the size of the interval and leads us therefore to the notion of Stability of the solution. 1.1. Theorems on Stability and Uniform Stability We shall now be concerned with the precise definition of Stability. Let us consider the system and let Z(t) be a solution of the system defined for t 3 to .
- Wolfgang Hahn(Author)
- 2019(Publication Date)
- Dover Publications(Publisher)
The function v entering the Stability Definition 34.1 is chosen as a quadratic form of all variables. As usual, a condition for absolute Stability (Definition 14.1) is obtained by discussing the sign of its total derivative for the equations of motion: A certain quadratic form of the quantities p j (t) and some auxiliary parameters, must be negative definite. However, the practical evaluation of this condition is rather difficult. Štelik [1] and Czan [1, 3] established a relation between the concept of Stability in a finite interval and the determination of optimal control (Sec. 6, Remark 6; cf. also Lebedev [4], S. K. Persidskii [2], Kudakova [1]). 35. DIFFERENTIAL EQUATIONS WITH BOUNDED SOLUTIONS Applying the considerations of Sec. 3 to the case where the function v is positive in the exterior of a certain sphere and where v is negative in that domain, it can be concluded that the phase trajectories penetrate those hyper-surfaces v = const which are sufficiently far away from the origin from the outside to the inside. Consequently, all solutions with bounded initial points are bounded themselves. If t is greater than a sufficiently large t -value, the absolute values of these solutions are even smaller than a fixed upper bound. Criteria for boundedness can be derived for special differential equations, cf. e.g., Reuter [1, 2]. Yoshizawa developed this method systematically in several publications ([2 to 8], in particular [9]). Apparently, his investigations are originally independent of the considerations of Liapunov. The analogy to the direct method is obvious and is emphasized by the fact that the boundedness of the solutions can be interpreted in the sense of a Stability property of the trivial solution. LaSalle [2] introduced the terminology: “Stability in the sense of Lagrange.” To the different types of Stability (cf. Secs. 4 and 17) there correspond different types of boundedness
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