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y of time varying systems with unbounded perturbations 10 example 11 see also 12 references 13 further reading toggle the table of contents lyapunov stability 16 languages català esperanto español فارسی français עברית italiano 日本語 қазақша 한국어 polski português русский українська tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from asymptotic stability property of a dynamical system where solutions near an equilibrium point remain so this article is about asymptotic stability of nonlinear systems for stability of linear systems see exponential stability this article s lead section may need to be rewritten please review the lead guide and help improve the lead of this article if you can december 2021 learn how and when to remove this message part of a series on astrodynamics orbital mechanics orbital elements apsis argument of periapsis eccentricity inclination mean anomaly orbital nodes semi major axis true anomaly types of two body orbits by eccentricity circular orbit elliptic orbit transfer orbit hohmann transfer orbit bi elliptic transfer orbit parabolic orbit hyperbolic orbit radial orbit decaying orbit equations dynamical friction escape velocity kepler s equation kepler s laws of planetary motion orbital period orbital velocity surface gravity specific orbital energy vis viva equation celestial mechanics gravitational influences barycenter hill sphere perturbations sphere of influence n body orbits lagrangian points halo orbits lissajous orbits lyapunov orbits engineering and efficiency preflight engineering mass ratio payload fraction propellant mass fraction tsiolkovsky rocket equation efficiency measures gravity assist oberth effect propulsive maneuvers orbital maneuver orbit insertion v t e various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems the most important type is that concerning the stability of solutions near to a point of equilibrium this may be discussed by the theory of aleksandr lyapunov in simple terms if the solutions that start out near an equilibrium point x e displaystyle x_ e stay near x e displaystyle x_ e forever then x e displaystyle x_ e is lyapunov stable more strongly if x e displaystyle x_ e is lyapunov stable and all solutions that start out near x e displaystyle x_ e converge to x e displaystyle x_ e then x e displaystyle x_ e is said to be asymptotically stable see asymptotic analysis the notion of exponential stability guarantees a minimal rate of decay i e an estimate of how quickly the solutions converge the idea of lyapunov stability can be extended to infinite dimensional manifolds where it is known as structural stability which concerns the behavior of different but nearby solutions to differential equations input to state stability iss applies lyapunov notions to systems with inputs history edit lyapunov stability is named after aleksandr mikhailovich lyapunov a russian mathematician who defended the thesis the general problem of stability of motion at kharkov university now vn karazin kharkiv national university in 1892 1 a m lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium his work initially published in russian and then translated to french received little attention for many years the mathematical theory of stability of motion founded by a m lyapunov considerably anticipated the time for its implementation in science and technology moreover lyapunov did not himself make application in this field his own interest being in the stability of rotating fluid masses with astronomical application he did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 2 for several decades the theory of stability sank into complete oblivion the russian soviet mathematician and mechanician nikolay gur yevich chetaev working at the kazan aviation institute in the 1930s was the first who realized the incredible magnitude of the discovery made by a m lyapunov the contribution to the theory made by n g chetaev 3 was so significant that many mathematicians physicists and engineers consider him lyapunov s direct successor and the next in line scientific descendant in the creation and development of the mathematical theory of stability the interest in it suddenly skyrocketed during the cold war period when the so called second method of lyapunov see below was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods a large number of publications appeared then and since in the control and systems literature 4 5 6 7 8 more recently the concept of the lyapunov exponent related to lyapunov s first method of discussing stability has received wide interest in connection with chaos theory lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems 9 definition for continuous time systems edit consider an autonomous nonlinear dynamical system x f x t x 0 x 0 displaystyle dot x f x t x 0 x_ 0 where x t d r n displaystyle x t in mathcal d subseteq mathbb r n denotes the system state vector d displaystyle mathcal d an open set containing the origin and f d r n displaystyle f mathcal d rightarrow mathbb r n is a continuous vector field on d displaystyle mathcal d suppose f displaystyle f has an equilibrium at x e displaystyle x_ e so that f x e 0 displaystyle f x_ e 0 then this equilibrium is said to be lyapunov stable if for every ϵ 0 displaystyle epsilon 0 there exists a δ 0 displaystyle delta 0 such that if x 0 x e δ displaystyle x 0 x_ e delta then for every t 0 displaystyle t geq 0 we have x t x e ϵ displaystyle x t x_ e epsilon the equilibrium of the above system is said to be asymptotically stable if it is lyapunov stable and there exists δ 0 displaystyle delta 0 such that if x 0 x e δ displaystyle x 0 x_ e delta then lim t x t x e 0 displaystyle lim _ t rightarrow infty x t x_ e 0 the equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and there exist α 0 β 0 δ 0 displaystyle alpha 0 beta 0 delta 0 such that if x 0 x e δ displaystyle x 0 x_ e delta then x t x e α x 0 x e e β t displaystyle x t x_ e leq alpha x 0 x_ e e beta t for all t 0 displaystyle t geq 0 conceptually the meanings of the above terms are the following lyapunov stability of an equilibrium means that solutions starting close enough to the equilibrium within a distance δ displaystyle delta from it remain close enough forever within a distance ϵ displaystyle epsilon from it note that this must be true for any ϵ displaystyle epsilon that one may want to choose asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium exponential stability means that solutions not only converge but in fact converge faster than or at least as fast as a particular known rate α x 0 x e e β t displaystyle alpha x 0 x_ e e beta t the trajectory ϕ t displaystyle phi t is locally attractive if x t ϕ t 0 displaystyle x t phi t rightarrow 0 as t displaystyle t rightarrow infty for all trajectories x t displaystyle x t that start close enough to ϕ t displaystyle phi t and globally attractive if this property holds for all trajectories that is if x belongs to the interior of its stable manifold it is asymptotically stable if it is both attractive and stable there are examples showing that attractivity does not imply asymptotic stability 10 11 12 such examples are easy to create using homoclinic connections if the jacobian of the dynamical system at an equilibrium happens to be a stability matrix i e if the real part of each eigenvalue is strictly negative then the equilibrium is asymptotically stable system of deviations edit instead of considering stability only near an equilibrium point a constant solution x t x e displaystyle x t x_ e one can formulate similar definitions of stability near an arbitrary solution x t ϕ t displaystyle x t phi t however one can reduce the more general case to that of an equilibrium by a change of variables called a system of deviations define y x ϕ t displaystyle y x phi t obeying the differential equation y f t y ϕ t ϕ t g t y displaystyle dot y f t y phi t dot phi t g t y this is no longer an autonomous system but it has a guaranteed equilibrium point at y 0 displaystyle y 0 whose stability is equivalent to the stability of the original solution x t ϕ t displaystyle x t phi t lyapunov s second method for stability edit lyapunov in his original 1892 work proposed two methods for demonstrating stability 1 the first method developed the solution in a series which was then proved convergent within limits the second method which is now referred to as the lyapunov stability criterion or the direct method makes use of a lyapunov function v x which has an analogy to the potential function of classical dynamics it is introduced as follows for a system x f x displaystyle dot x f x having a point of equilibrium at x 0 displaystyle x 0 consider a function v r n r displaystyle v mathbb r n rightarrow mathbb r such that v x 0 displaystyle v x 0 if and only if x 0 displaystyle x 0 v x 0 displaystyle v x 0 if and only if x 0 displaystyle x neq 0 v x d d t v x i 1 n v x i f i x v f x 0 displaystyle dot v x frac d dt v x sum _ i 1 n frac partial v partial x_ i f_ i x nabla v cdot f x leq 0 for all values of x 0 displaystyle x neq 0 note for asymptotic stability v x 0 displaystyle dot v x 0 for x 0 displaystyle x neq 0 is required then v x is called a lyapunov function and the system is stable in the sense of lyapunov note that v 0 0 displaystyle v 0 0 is required otherwise for example v x 1 1 x displaystyle v x 1 1 x would prove that x t x displaystyle dot x t x is locally stable an additional condition called properness or radial unboundedness is required in order to conclude global stability global asymptotic stability gas follows similarly it is easier to visualize this method of analysis by thinking of a physical system e g vibrating spring and mass and considering the energy of such a system if the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state this final state is called the attractor however finding a function that gives the precise energy of a physical system can be difficult and for abstract mathematical systems economic systems or biological systems the concept of energy may not be applicable lyapunov s realization was that stability can be proven without requiring knowledge of the true physical energy provided a lyapunov function can be found to satisfy the above constraints definition for discrete time systems edit the definition for discrete time systems is almost identical to that for continuous time systems the definition below provides this using an alternate language commonly used in more mathematical texts let x d be a metric space and f x x a continuous function a point x in x is said to be lyapunov stable if ϵ 0 δ 0 y x d x y δ n n d f n x f n y ϵ displaystyle forall epsilon 0 exists delta 0 forall y in x left d x y delta rightarrow forall n in mathbf n d left f n x f n y right epsilon right we say that x is asymptotically stable if it belongs to the interior of its stable set i e if δ 0 d x y δ lim n d f n x f n y 0 displaystyle exists delta 0 left d x y delta rightarrow lim _ n to infty d left f n x f n y right 0 right stability for linear state space models edit a linear state space model x a x displaystyle dot textbf x a textbf x where a displaystyle a is a finite matrix is asymptotically stable in fact exponentially stable if all real parts of the eigenvalues of a displaystyle a are negative this condition is equivalent to the following one 13 a t m m a displaystyle a textsf t m ma is negative definite for some positive definite matrix m m t displaystyle m m textsf t the relevant lyapunov function is v x x t m x displaystyle v x x textsf t mx correspondingly a time discrete linear state space model x t 1 a x t displaystyle textbf x _ t 1 a textbf x _ t is asymptotically stable in fact exponentially stable if all the eigenvalues of a displaystyle a have a modulus smaller than one this latter condition has been generalized to switched systems a linear switched discrete time system ruled by a set of matrices a 1 a m displaystyle a_ 1 dots a_ m x t 1 a i t x t a i t a 1 a m displaystyle textbf x _ t 1 a_ i_ t textbf x _ t quad a_ i_ t in a_ 1 dots a_ m is asymptotically stable in fact exponentially stable if the joint spectral radius of the set a 1 a m displaystyle a_ 1 dots a_ m is smaller than one stability for systems with inputs edit a system with inputs or controls has the form x f x u displaystyle dot textbf x textbf f textbf x textbf u where the generally time dependent input u t may be viewed as a control external input stimulus disturbance or forcing function it has been shown 14 that near to a point of equilibrium which is lyapunov stable the system remains stable under small disturbances for larger input disturbances the study of such systems is the subject of control theory and applied in control engineering for systems with inputs one must quantify the effect of inputs on the stability of the system the main two approaches to this analysis are bibo stability for linear systems and input to state stability iss for nonlinear systems example edit this example shows a system where a lyapunov function can be used to prove lyapunov stability but cannot show asymptotic stability consider the following equation based on the van der pol oscillator equation with the friction term changed y y ε y 3 3 y 0 displaystyle ddot y y varepsilon left frac dot y 3 3 dot y right 0 let x 1 y x 2 y displaystyle x_ 1 y x_ 2 dot y so that the corresponding system is x 1 x 2 x 2 x 1 ε x 2 3 3 x 2 displaystyle begin aligned dot x _ 1 x_ 2 dot x _ 2 x_ 1 varepsilon left frac x_ 2 3 3 x_ 2 right end aligned the origin x 1 0 x 2 0 displaystyle x_ 1 0 x_ 2 0 is the only equilibrium point let us choose as a lyapunov function v 1 2 x 1 2 x 2 2 displaystyle v frac 1 2 left x_ 1 2 x_ 2 2 right which is clearly positive definite its derivative is v x 1 x 1 x 2 x 2 x 1 x 2 x 1 x 2 ε x 2 4 3 ε x 2 2 ε x 2 4 3 ε x 2 2 displaystyle dot v x_ 1 ...