dynamical system ଗୁଡ଼ିକୁ ବୁଝାଉଥିବା ବିଭେଦକ ସମୀକରଣ (differential equation) ଗୁଡ଼ିକର ସମାଧାନ ପାଇଁ ବିଭିନ୍ନ ପ୍ରକାରର ସ୍ଥାୟୀତ୍ତ୍ୱ/ସ୍ଥିରତା (stability) କଥା ବିଚାରକୁ ନିଆଯାଇପାରେ । ତା'ଭିତରୁ equilibrium ବିନ୍ଦୁର ନିକଟସ୍ଥ ସମାଧାନ ଗୁଡ଼ିକର ସ୍ଥାୟୀତ୍ତ୍ୱକୁ ଗ୍ରହଣ କରିବା ଅଧିକ ଗୁରୁତ୍ତ୍ୱପୂର୍ଣ୍ଣ । ଲ୍ୟାପୁନୋଭ୍ଙ୍କ ତତ୍ତ୍ୱଦ୍ୱାରା ଏହା ଆଲୋଚନା ହୋଇପାରିବ । ସରଳ ଭାଷାରେ, ଏକ equilibrium ବିନ୍ଦୁ ନିକଟରୁ ଉତ୍ପନ୍ନ ହେଉଥିବା ସମାଧାନଗୁଡ଼ିକ ଯଦି ସର୍ବଦା ନିକଟବର୍ତ୍ତୀ ହୋଇ ରହନ୍ତି, ତା'ହେଲେ ଲ୍ୟାପୁନୋଭ୍ ସ୍ଥାୟୀ (Lyapunov stable) ଅଟେ । ଭଲକରି କହିଲେ, ଯଦି ଲ୍ୟାପୁନୋଭ୍ ସ୍ଥାୟୀ ଅଟେ ଏବଂ ନିକଟରୁ ଉତ୍ପନ୍ନ ହୋଇଥିବା ସମସ୍ତ ସମାଧାନଗୁଡ଼ିକ ସହ converge ହୁଅନ୍ତି, ତା'ହେଲେ asymptotically ସ୍ଥାୟୀ ଅଟେ । exponential ସ୍ଥାୟୀତ୍ତ୍ୱର ଧାରଣା କ୍ଷୟର ଏକ ସର୍ବନିମ୍ନ ହାରକୁ ସୁଚାଏ, ମାନେ, କେତେ ଶୀଘ୍ର ସମାଧାନ ଗୁଡ଼ିକ converge ହେବ ଏହା ତା'ହାର ଏକ ଆକଳନ । Lyapunov ସ୍ଥିରତାର ଏହି ଧାରଣାକୁ infinite-dimensional manifoldsକୁ ସଂପ୍ରସାରଣ କରାଯାଇପାରିବ, ଯେଉଁଠାରେ ଏହା structural stability ଭାବେ ଜଣା, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
Lyapunov was the first to consider the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. Interest in it started suddenly 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.
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.
ଲ୍ୟାପୁନୋଭ୍ ସ୍ଥାୟୀତ୍ତ୍ୱ, ଆଲେକ୍ଜାଣ୍ଡର୍ ଲ୍ୟାପୁନୋଭ୍ଙ୍କ (Aleksandr Lyapunov)ଙ୍କ ନାମ ଅନୁସାରେ ନାମିତ । ସେ ଜଣେ ରୁଷୀୟ ଗଣିତଜ୍ଞ ଥିଲେ ଓ ୧୮୯୨ରେ ସେ ତାଙ୍କର ପୁସ୍ତକ The General Problem of Stability of Motionକୁ ପ୍ରକାଶ କରିଥିଲେ । Lyapunov ନିକଟବର୍ତ୍ତୀ linearizing ଉପରେ ଆଧାରିତ ସ୍ଥିରତାର ରୈଖିକ ତତ୍ତ୍ୱକୁ nonlinear ତନ୍ତ୍ରରେ ପରିବର୍ତ୍ତନ ଆବଶ୍ଯକ ବିଚାର ପ୍ରଥମ ଥିଲା 1892.ରେ ଗତିର ସ୍ଥାୟୀତ୍ତ୍ୱର ସାଧାରଣ ସମସ୍ୟା ପ୍ରକାଶିତ କିଏ ଏକ ରୁଷି ଗଣିତଜ୍ଞ ପରେ ନାମିତ equilibriumର ଏକ ବିନ୍ଦୁ. ତାଙ୍କର କାମ, ପ୍ରାରମ୍ଭିକ ରୁଷିରେ ପ୍ରକାଶିତ ଏବଂ ତାପରେ ଫ୍ରେଞ୍ଚ୍କୁ ଅନୁବାଦ, ଅନେକ ବର୍ଷ ପାଇଁ ଅଳ୍ପ ଧ୍ୟାନ ଗ୍ରହଣ. ତେଣୁ-କୁହାଯାଏ "Lyapunov ଦ୍ୱିତୀୟ ପଦ୍ଧତି" (ନିମ୍ନରେ ଦେଖନ୍ତୁ) ସାଧାରଣତଃ ଅନ୍ୟ ପଦ୍ଧତିଦ୍ୱାରା treatable ନାହିଁ ଶକ୍ତିଶାଳୀ nonlinearities ଧାରଣ ଯାହା Aerospace ପରାମର୍ଶ ତନ୍ତ୍ରର ସ୍ଥିରତାକୁ ପ୍ରଜୁଯ୍ୟ ବୋଲି ମିଳିଲା ଯେତେବେଳେ ଏହାରେ ଆଗ୍ରହ ଶୀତଳ ଯୁଦ୍ଧରେ ସମୟରେ ହଠାତ୍ ଆରମ୍ଭ. ପ୍ରକାଶନର ଏକ ସଂଖ୍ୟକ ନିୟନ୍ତ୍ରଣ ଏବଂ ତନ୍ତ୍ର ସାହିତ୍ୟରେ ତାପରେ ଏବଂ ଯେହେତୁ ଦର୍ଶନ.      Lyapunov exponentର ଅଧିକ ନିକଟରେ ଅବଧାରଣା (ସ୍ଥିରତା ଆଲୋଚନାର Lyapunov's ପ୍ରଥମ ପଦ୍ଧତି ସମ୍ପର୍କୀୟ) କରିବା ବିଶୃଙ୍ଖଳା ସିଦ୍ଧାନ୍ତ ସହିତ ସଂଯୋଗରେ ଗ୍ରହଣ ବିସ୍ତୃତ ସୁଧ. Lyapunov ସ୍ଥିରତା ପଦ୍ଧତି ମଧ୍ୟ ଟ୍ରାଫିକ୍ କାର୍ଯ୍ୟଭାର ସମସ୍ୟାରେ equilibrium ସମାଧାନ ଖୋଜି ପ୍ରୟୋଗ କରାଯାଇଛି. 
continuous-time ସିଷ୍ଟମ ଗୁଡ଼ିକ ପାଇଁ ସଂଜ୍ଞା[ସମ୍ପାଦନା]
ଗୋଟିଏ autonomous nonlinear dynamical ସିଷ୍ଟମ କଥା ବିଚାରକୁ ନିଅନ୍ତୁ
ଯେଉଁଠାରେ system state vectorକୁ ସୁଚାଏ, ହେଉଛି originକୁ ଧାରଣ କରିଥିବା ଏକ open ସେଟ୍, ଏବଂ continuous on . ମନେକର ଠାରେ ର ଏକ equilibrium ଅଛି ଯେପରିକି ତା'ହେଲେ
- ଏହି equilibriumକୁ ଲ୍ୟାପୁନୋଭ୍ ସ୍ଥାୟୀ କୁହାଯିବ, ଯଦି, for every , there exists a such that, if , then for every we have .
- The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and there exists such that if , then .
- The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and there exist such that if , then , for .
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 from it) remain "close enough" forever (within a distance from it). Note that this must be true for any 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 .
The trajectory x is (locally) attractive if
(where y(t) denotes the system output) for for all trajectories that start close enough, 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 counterexamples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using homoclinic connections.)
Lyapunov's second method for stability[ସମ୍ପାଦନା]
Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. The first method developed the solution in a series which was then proved convergent within limits. The second method, which is almost universally used nowadays, 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 having a point of equilibrium at x=0. Consider a function such that
- with equality if and only if (positive definite)
- with equality not constrained to only (negative semidefinite). Note: for asymptotic stability, is required to be negative definite.
Then V(x) is called a Lyapunov function candidate and the system is stable in the sense of Lyapunov. (Note that is required; otherwise for example would "prove" that is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability.) Furthermore, the system is asymptotically stable, in the sense of Lyapunov, if with equality if and only if . 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[ସମ୍ପାଦନା]
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.
We say that x is asymptotically stable if it belongs to the interior of its stable set, i.e. if,
Stability for linear state space models[ସମ୍ପାଦନା]
A linear state space model
is negative definite for some positive definite matrix . (The relevant Lyapunov function is .)
Correspondingly, a time-discrete linear state space model
is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of 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 )
is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set is smaller than one.
Stability for systems with inputs[ସମ୍ପାଦନା]
A system with inputs (or controls) has the form
where the (generally time-dependent) input u(t) may be viewed as a control, external input, stimulus, disturbance, or forcing function. 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 (ISS) stability (for nonlinear systems)
Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:
The equilibrium is at :
Here is a good example of an unsuccessful try to find a Lyapunov function that proves stability:
so that the corresponding system is
Let us choose as a Lyapunov function
which is clearly positive definite. Its derivative is
It seems that if the parameter is positive, stability is asymptotic for But this is wrong, since does not depend on , and will be 0 everywhere on the axis. The system is Lyapunov stable.
Barbalat's lemma and stability of time-varying systems[ସମ୍ପାଦନା]
Assume that f is function of time only.
- Having does not imply that has a limit at . For example, .
- Having approaching a limit as does not imply that . For example, .
- Having lower bounded and decreasing () implies it converges to a limit. But it does not say whether or not as .
Barbalat's Lemma says:
- If has a finite limit as and if is uniformly continuous (or is bounded), then as .
Usually, it is difficult to analyze the asymptotic stability of time-varying systems because it is very difficult to find Lyapunov functions with a negative definite derivative.
We know that in case of autonomous (time-invariant) systems, if is negative semi-definite (NSD), then also, it is possible to know the asymptotic behaviour by invoking invariant-set theorems. However, this flexibility is not available for time-varying systems. This is where "Barbalat's lemma" comes into picture. It says:
- IF satisfies following conditions:
- is lower bounded
- is negative semi-definite (NSD)
- is uniformly continuous in time (satisfied if is finite)
- is lower bounded
- then as .
The following example is taken from page 125 of Slotine and Li's book Applied Nonlinear Control.
Consider a non-autonomous system
This is non-autonomous because the input is a function of time. Assume that the input is bounded.
This says that by first two conditions and hence and are bounded. But it does not say anything about the convergence of to zero. Moreover, the invariant set theorem cannot be applied, because the dynamics is non-autonomous.
Using Barbalat's lemma:
This is bounded because , and are bounded. This implies as and hence . This proves that the error converges.
- Letov, A. M. (1955). Устойчивость нелинейных регулируемых систем (in Russian). Moscow: Gostekhizdat. Unknown parameter
|trans_title=ignored (help)CS1 maint: unrecognized language (link) English tr. Princeton 1961
- Kalman, R. E.; Bertram, J. F (1960). "Control System Analysis and Design via the Second Method of Lyapunov". J. Basic Engrg. 88: 371, 394.
- LaSalle, J. P.; Lefschetz, S. (1961). Stability by Lyapunov's Second Method with Applications. New York: Academic.
- Parks, P. C. (1962). "Liapunov's method in automatic control theory". Control. I Nov 1962 II Dec 1962.
- Kalman, R. E. (1963). "Lyapunov functions for the problem of Lur'e in automatic control". Proc Nat Acad.Sci USA. 49 (2): 201–205. PMC 299777.
- Smith, M. J.; Wisten, M. B. (1995). "A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium". Annals of Operations Research. 60 (1): 59–79. doi:10.1007/BF02031940.
- Lyapunov A. M. The General Problem of the Stability of Motion (In Russian), Doctoral dissertation, Univ. Kharkov 1892 English translations: (1) Stability of Motion, Academic Press, New-York & London, 1966 (2) The General Problem of the Stability of Motion, (A. T. Fuller trans.) Taylor & Francis, London 1992. Included is a biography by Smirnov and an extensive bibliography of Lyapunov's work.
- Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 407–428. ISBN 3-540-60988-1.
- Parks, P. C. (1992). "A. M. Lyapunov's stability theory—100 years on". IMA Journal of Mathematical Control & Information. 9 (4): 275–303. doi:10.1093/imamci/9.4.275.
- Slotine, Jean-Jacques E.; Weiping Li (1991). Applied Nonlinear Control. NJ: Prentice Hall.
- Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
- Wiggins, S. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos (2 ed.). New York: Springer Verlag. ISBN 0-387-00177-8.
- http://www.mne.ksu.edu/research/laboratories/non-linear-controls-lab (login required)