Lyapunov stability

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 formula_1 stay near formula_1 forever, then formula_1 is Lyapunov stable. More strongly, if formula_1 is Lyapunov stable and all solutions that start out near formula_1 converge to formula_1, then formula_1 is asymptotically stable. 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.
In the restricted three-body problem, Lyapunov orbits are curved paths around a Lagrangian point that lie entirely in the plane of the two primary bodies, in contrast with halo orbits and Lissajous orbits, which move above and below the plane also.
History.
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 in 1892. A. M. Lyapunov was a pioneer in successful endeavoring to develop the 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 the Russian revolution of 1917 . 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. Actually, his figure as a great scientist is comparable to the one of A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev 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.
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.
Definition for continuous-time systems.
Consider an autonomous nonlinear dynamical system
where formula_9 denotes the system state vector, formula_10 an open set containing the origin, and formula_11 is a continuous vector field on formula_10. Suppose formula_13 has an equilibrium at formula_1 so that formula_15 then
Conceptually, the meanings of the above terms are the following:
The trajectory "formula_32" is (locally) "attractive" if
for all trajectories formula_35 that start close enough to formula_36, 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. 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.
Instead of considering stability only near an equilibrium point (a constant solution formula_37), one can formulate similar definitions of stability near an arbitrary solution formula_32. However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations". Define formula_39, obeying the differential equation:
This is no longer an autonomous system, but it has a guaranteed equilibrium point at formula_41 whose stability is equivalent to the stability of the original solution formula_32. 
Lyapunov's second method for stability.
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 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 formula_43 having a point of equilibrium at formula_44. Consider a function formula_45 such that
Then "V(x)" is called a Lyapunov function and the system is stable in the sense of Lyapunov (Note that formula_54 is required; otherwise for example formula_55 would "prove" that formula_56 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.
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,
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.
A linear state space model
where formula_60 is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of formula_60 are negative. This condition is equivalent to the following one:
is negative definite for some positive definite matrix formula_63. (The relevant Lyapunov function is formula_64.)
Correspondingly, a time-discrete linear state space model
is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of formula_60 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 formula_67)
is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set formula_67 is smaller than one.
Stability for systems with inputs.
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". It has been shown 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.
Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:
Here is a good example of an unsuccessful try to find a Lyapunov function that proves stability.
Let
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 formula_77 is positive, stability is asymptotic for formula_78 But this is wrong, since formula_79 does not depend on formula_80, and will be 0 everywhere on the formula_80 axis. The equilibrium is Lyapunov stable.
Barbalat's lemma and stability of time-varying systems.
Assume that f is a function of time only.
Barbalat's Lemma says:
An alternative version is as follows:
In the following form the Lemma is true also in the vector valued case:
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 formula_114 is a function of time. Assume that the input formula_115 is bounded.
Taking formula_116 gives formula_117
This says that formula_118 by first two conditions and hence formula_119 and formula_120 are bounded. But it does not say anything about the convergence of formula_119 to zero. Moreover, the invariant set theorem cannot be applied, because the dynamics is non-autonomous.
Using Barbalat's lemma:
This is bounded because formula_119, formula_120 and formula_114 are bounded. This implies formula_126 as formula_84 and hence formula_128. This proves that the error converges.