1 Lyapunov theory of stability

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1 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability properties of an equilibrium point of a dynamical system (or systems of differential equations). The intuitive picture is that of a scalar outputfunction, often thought of as a generalized energy that is bounded below, and decreasing along solutions. If this function has only a single local minimum, and it is strictly decreasing along all nonequilibrium solutions, then one expects that all solutions tend to that equilibrium where the output function has a minimum. This is indeed correct. In the sequel we state and prove theorems, including some that relax the requirement of strictly or globally decreasing, and also discuss converse theorems that guarantee the existence of such functions. Figure 1: Output decreasing along trajectories, V not proper Much of the power of the method comes from its simplicity one does not need to know any solutions: Knowing only the differential (or difference) equation one can easily establish whether such an output function is decreasing along solutions. However, while it is easy to see that for every asymptotically stable system there exists many, even smooth, such Lyapunov functions, in many cases it is almost impossible to get one s hands onto one such Lyapunov function. They are easy to construct for e.g. linear systems, and many strategies are available for special classes in general it is a true art to come up with explicit formulas for good candidate Lyapunov functions.
2 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 2 We begin with one example and some preliminary definitions. For a function V : R n R and a locally Lipschitz continuous vector field f: R n R n define the derivative V (p) of V along the flow Φ of f at the point p R n by the following limit provided it exists 1 V (p) = lim (V (Φ(t, p)) V (p)). t t In the case that V is continuously differentiable, this agrees with the directional derivative of V in the direction of f(p) at p. This is defined for any vector field f: R n R n, and if V is continuously differentiable, one has the formula V = (DV ) f which is convenient for practical applications. In particular, for any times t 1 t 2 and any point p for which the flow is defined, the fundamental theorem of calculus yields that V is decreasing if V is nonpositive along a trajectory. t2 t2 V (Φ(t 2, p)) V (Φ(t 1, p)) = V (Φ(t, p) dt dt =. t 1 t 1 Example 1 For a damped pendulum, write the equation of motion ml φ+mk φ+mg sin φ = as a first order system: ẋ = y ẏ = g sin x ky l Figure 2: Generalized Lyapunov function for pendulum A natural candidate Lyapunov function is obtained from the total energy E = E pot + E kin = mgl(1 cos φ) ml2 φ2. In the (x, y) coordinates this yields, after division by m, the candidate Lyapunov function
3 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 3 V (x, y) = gl(1 cos x) l2 y 2 Near the origin, the graph of this function looks very similar to a paraboloid but globally it has other features which will motivate more detailed studies later. It is easily seen (and discussed in detail below) that the rate of change of V along solutions of the system for positive friction k >, as long as y, (with common abuse of notation) V (x, y) = (DV )(x, y) (ẋ, ẏ) = gly sin x + l 2 y( g l sin x ky) = k l2 y 2 clearly is negative. It makes sense that at the instances when the angular velocity y = φ is zero (i.e. when the pendulum changes directions), there is no loss of energy. But since the pendulum cannot stay in any of these states, it will continue to lose energy, and settle into an equilibrium corresponding to V =. The detailed analysis of situations in which the Lyapunov function is not strictly decreasing along trajectories culminates in Lasalle s Invariance Principle. Definition 1 A function V : R n [, ) is called positive definite if V (x) = if and only if x =. A function V : X Y between topological spaces X and Y is called proper if for every compact set K Y its preimage V 1 (K) X is compact. A function V : R n [, ) is called radially unbounded if for every M R there exists R R such that for all x R n, if x > R then V (x) > M. Proposition 1 A continuous function V : R n [, ) is radially unbounded if and only if it is proper. Suppose V : R n [, ) is continuous and proper, and M [, ). The interval [, M] is compact, and since f is proper, the preimage K = V 1 ([, M]) is compact. In particular, K R n is bounded, i.e., there exists R R such that K B n (, R). If x R n such that x > R, then x K = V 1 ([, M]), and thus V (x) > M. Conversely suppose V is continuous and radially unbounded and let K R be compact. Thus K is closed, and by continuity of V, the preimage V 1 (K) R n is also closed. Since K is bounded, there exists M R such that K [ M, M]. Since V is radially unbounded, there exists R R so that V (R n \ B n (, R)) (M, ). Therefore, V 1 (K) B n (, R) and V 1 (K) is compact.
4 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 4 Note: Even if a continuous function V goes to infinity along along every ray r (r cos θ, r sin θ) as r goes to infinity, it need not be radially unbounded as the following simple example shows. V ( r, ) = r2 + r 2 and V (r cos θ, r sin θ) = r2 + r 2 1 (1 ) if θ π. 1+r 2 1+r 2 1+(r tan( θ 2 ))2 There are many versions of Lyapunov stability theorems that make different regularity assumptions, that are local or global in nature, that concern continuous or discrete dynamical systems (differential or difference equations), deterministic or stochastic, finite or infinite (PDE) dimensional. They all have in common the picture sketched above, but the technical details may offer many challenges. Theorem 1 Suppose f: R n R n is locally Lipschitz continuous, f() =, and V : R n R is continuously differentiable, proper, positive definite such that V is negative definite. Then the origin is a globally asymptotically stable equilibrium of x = f(x). A function V satisfying the hypotheses of this theorem is called a Lyapunov function for the (flow of the) system x = f(x). Since f is locally Lipschitz continuous, solutions to initial value problems of x = f(x) exist locally and are unique. The only way that they can fail to exist for all positive times is if they escape to infinity in finite time. But since for every q R n, and every t, V (Φ(t, q)) V (q), and the sublevelset V 1 ([, V (q)]) is compact, every solution curve stays bounded in forward time. To show Lyapunov stability suppose ε > is given. Choose ε 1 (, ε). Since V is continuous and the closed ball B n (, ε 1 ) B n (, ε) is compact, there exists M R such that V is bounded above by M on B n (, ε 1 ). Since V is continuous, there exists δ > such that for all x B n (, δ), V (x) < M. Hence for every p B n (, δ) and every t, since V is nonpositive, V (Φ(t, p)) = V (p) + t V (Φ(s, p)) ds V (p) M. Hence for every p B n (, δ) and all t, Φ(t, p) B n (, ε 1 ) B n (, ε). To show that the origin is attractive, fix any p R n and define c = lim inf t V (Φ(t, p)) = lim t V (Φ(t, p)) If c = then we are done since V is positive definite. Otherwise, choose δ > such that < c δ < c + δ < V (p), and consider the annular region A = V 1 ([c δ < c + δ]) R n. Since V is proper, A is compact.
5 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 5 Figure 3: No positive limit for V (Φ(t, p)) Since V is continuously differentiable and f is continuous, V is continuous. Since V is negative definite, there exists µ > such that for all x A, V (x) µ. By choice of c there exists T R such that V (Φ(T, p)) < c + δ, i.e., Φ(T, p) A. Calculate: 2δ V (Φ(T + 2δ, p)) = V (Φ(T, p)) + µ µ which contradicts the choice of c. V (Φ(T + s, p)) ds (c + δ) + Thus c = and the origin is globally asymptotically stable. 2δ µ µ ds (c δ) Theorem 2 Suppose Ω R n is open, Ω, f: Ω R n is locally Lipschitz continuous, f() =, and V : Ω R is continuously differentiable, positive definite (on Ω) and such that V is negative definite. Then the origin is a locally asymptotically stable equilibrium of x = f(x), i.e. there exists an open neighborhood U R n of the origin such that every solution starting at any p U is defined for all positive times, will converge to the origin as t goes to infinity, and the equilibrium is Lyapunov stable. For timevarying systems x = f(t, x) there exists a parallel rich theory which utilizes time varying Lyapunov functions V (t, x). The main additional technical feature that one needs
6 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 6 to be attentive of is that definiteness properties be uniform in time, e.g. V (t, x, y) = x 2 + e t y 2 are not sufficient. functions like Theorem 3 (Lasalle s Invariance Principle) Suppose Ω R n is open, f: Ω R n is locally Lipschitz continuous, and V : Ω R is continuously differentiable. Suppose K Ω is compact, invariant, and for all x K, V (x). Let E = V 1 () and let M E be the largest invariant set contained in E. Then every solution starting in K approaches M as t. Comments: In many common applications of Lasalle s Invariance Principle V is a positive definite function which fails to guarantee that V is negative definite only because there are some points, like in the example of the damped pendulum, where V vanishes. One usually proceeds by demonstrating that f is nontangent to the set E = V 1 () except at the equilibrium point to conclude asymptotic stability However, Lasalle s theorem has much broader applications e.g., in the statement above there is no requirement that V be positive definite. Thus it also applies to the example of the energy function for the damped pendulum which has infinite numbers of local minima and saddle points, compare figure 1 there is no need to first restrict one s attention to a small neighborhood of one equilibrium point on which V is positive definite. Suppose K, f, V, E and M are as in the statement of the theorem. Suppose p K and consider the curve t Φ(t, p). Since V is continuous it is bounded from below on the compact set K. Since K is invariant by hypothesis, for all t, Φ(t, p) K. Moreover, using that K is compact, and hence bounded, the ωlimit set ω(p) is nonempty. Since K is closed, ω(p) K. Using lemma 4, for all q ω(p), V (q) = and therefore ω(p) {y : V (y) = } = V 1 () = E M. Since ωlimit sets are invariant, ω(p) M. Since ωlimit sets are attracting, for every ɛ > there exists T R such that for every t > T there exists q ω(p) such that Φ(t, p) q < ε. Since ω(p) M, it follows that q M. This proves that the solution curve starting at an arbitrary p K approaches M as t. Lemma 4 Suppose Ω R n is open, f: Ω R n is locally Lipschitz continuous, and V : Ω R is continuously differentiable and bounded from below. Suppose that p Ω and for all t, Φ(t, p) is defined and is contained in Ω. Then for all q ω(p), V (q) =.
7 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 7 Suppose Ω, f, V and p are as in the statement of the lemma. If ω(p) = then there is nothing to prove. Suppose q 1 = q 3 ω(p), q 2 ω(p) and fix any ε >. Since V is continuous at q 1 and at q 2, there exists δ > such that for all x Ω, for i = 1, 2, if x q i < δ then V (x) V (q i ) < ε 4. Since q 1, q 2 ω(p), there exist t 1 t 2 t 3 such that for i = 1, 2, 3, Φ(t i, p) q i < δ, and hence for i = 1, 2, 3, V (Φ(t i, p)) V (q i ) < ε 4. This implies that V (Φ(t 3, p)) V (Φ(t 1, p)) V (Φ(t 3, p)) V (q 1 ) + V (q 1 ) V (Φ(t 1, p)) < ε 2. Since V does not change sign this implies Therefore V (Φ(t 2, p)) V (Φ(t 1, p)) = t2 t 1 t3 t 1 V (Φ(s, p) ds V (Φ(s, p) ds V (Φ(t 3, p)) V (Φ(t 1, p)) < ε 2. V (q 2 ) V (q 1 ) V (q 2 ) V (Φ(t 2, p)) + V (Φ(t 2, p)) V (Φ(t 1, p)) + V (Φ(t 1, p)) V (q 1 ) < ε 4 + ε 2 + ε 4 = ε. Since ε > was arbitrary, this establishes that V is constant on ω(p). Now use that ω(p) is invariant, i.e., for every q ω(p) and all t R, Φ(t, q) ω(p). By the foregoing, for every q ω(p) and all t R, V (Φ(t, q)) = V (q) and therefore V (q) = d 1 dt t= (V (Φ(t, q)) V (q)) =, t proving that V vanishes identically on ω(p). There are many theorems that guarantee for every asymptotically stable system the existence of Lyapunov functions, even with various additional properties such as being C, or being homogeneous. But these theorems are mainly useful for theoretical purposes, since, even though their proofs are generally very constructive, they rely on knowing the solutions of the system! In practical applications, where one tries to find a candidate Lyapunov function to establish asymptotic stability, it generally is considered an art to come up with good candidates. Even if one has found a Lyapunov function, it may be very difficult to rigorously prove that its derivative along solution curves is negative definite. A few general suggestions: For asymptotically stable linear systems one always can find a quadratic Lyapunov function. There is a rich literature in engineering about effectively computing suitable Lyapunov functions. One of their main uses is to establish robustness properties, i.e. showing that suitable perturbations of an asymptotically stable system are still asymptotically stable.
8 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 8 In systems arising from modeling physical models, the total energy of the system, and various similarly inspired output functions often serve as good starting points. Taking Taylor approximations of smooth systems yields systems with polynomial right hand sides. These suggest that one try polynomial functions as candidate Lyapunov functions. These are finitely parameterized, and one obtains inequalities for definiteness of V and V that have a finite number of parameters, and thus may be amenable to plain calculus methods for establishing that some minimum is nonnegative for suitable choices of parameters. In general it may be easier to start with a desired negative definite V, use this to compute the corresponding V, and then show that V is positive definite! This is demonstrated in the following simple theorem for linear systems. Theorem 5 Suppose A R n n. The origin is an asymptotically stable equilibrium point of the system x = Ax if and only for every positive definite Q = Q T R n n there exists a positive definite P = P T R n n such that P A + A T P = Q. Moreover, such P is unique. Sufficiency is immediate. Conversely suppose that all eigenvalues of A have strictly negative real parts and Q = Q T is positive definite. Define P = e tat Qe ta dt. This integral converges absolutely because the norm of the integrand is bounded by a function of the form Ct N e αt for some positive constants C, N, α. It is clear that P = P R is symmetric since the integrand is symmetric at every t To show that P is positive definite, suppose x R n is such that x T P x =. Then ( = e ta x ) T ( Q e ta x ) dt Since Q is positive definite this implies that for all t e ta x =. Since for every t, e ta is invertible, this implies x = proving that P is positive definite. To show that the function V (x) = x T P x has the desired derivative along solutions, first note that formally V (x) = ẋ T P x + x T P dotx = x T (P A + A T P )x. Now calculate ) P A + A T P = (e tat Qe ta A + A T e tat Qe ta dt = ) d (e tat Qe ta dt dt = e tat Qe ta = Q.
9 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 9 To show uniqueness, suppose P, P R n n are both positive definite, symmetric and satisfy P A + A T P = P A + A T P = Q. Then (P P )A + A T (P P ( ) =, and hence for every t e tat (P P )A + A T (P P ) ) e ta = But the expression on the left hand side is the time derivative d dt etat (P P ))e ta. Since the derivative vanishes identically, the product e tat (P P )e ta must be a constant function of t, and in particular, must at every time be equal to its value at t =. Therefore for all t, e tat (P P ))e ta = e AT (P P )e A = P P. On the other hand, since the origin was assumed to be asymptotically stable, as t goes to infinity, e ta goes to zero, and therefore (P P ) = lim t etat (P P ))e ta = (P P )) =. Proofs of converse theorem for nonlinear systems use a similar constructions. One technical difficulty is that the solution might not converge exponentially fast to the equilibrium, and hence integrals like V (x) = Φ(t, x) 2 dt need not converge. Moreover, one needs to establish that even if this defines a function value V (x) at every point, that the sodefined function is indeed continuously differentiable. It turns out that one can always achieve that V is even C, compare classic theorems by Massera and Kurzweil. As a handson example to illustrate the construction, consider the system x = Ax where ( ) ( ) 1 1 A =, and define Q = I = Make the Ansatz V (x) = x T P x = p 11 x p 12 x 1 x 2 + p 22 x 2 2 with p ij R. Calculate the derivative along solutions of x = Ax. V (x) = (Ax) T P x + x T P (Ax) = 2p 11 x 1 x 2 2p 12 x p 12 x 2 1 2p 12 x 1 x 2 + p 22 x 1 x 2 p 22 x 2 2 = 2p 12 x ( p 11 p 12 + p 22 )x 1 x 2 + 2( p 12 p 22 )x 2 2! = x 2 1 x 2 2. This defines a linear system for the entries of P 2 p p p 22 which has the unique solution ( 3 1 P = = When trying to adapt Lyapunov methods for stability to an instability theorem, a first thought might be to keep V positive definite and argue with indefinite V. But unlike in stability theorems, one does need to consider all trajectories. Hence it suffices to identify a suitable region in which both V and V have the same sign. The following statement and arguments use the delicate intersection of a closed set F with the open set V 1 (, ). ). 1 1
10 M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 Theorem 6 (adapted from Chetaev 1934). Suppose E R n is open, E, f: E R n and V : E R are continuously differentiable, f() =, V () =. Suppose U an open neighborhood of with compact closure F = U E such that the restriction of V = (DV )f to F V 1 (, ) is strictly positive. If for every open neighborhood W R n of the set W V 1 (, ) is nonempty, then the origin is an unstable equilibrium of ẋ = f(x). Assume above hypotheses. Let W be an open neighborhood of R n. There exists z W V 1 (, ). Define a = V (z) >. The set K = F V 1 [a, ) is compact, and it is nonempty since z K. Using the continuity of V and V there exist m = min x K V (x) and M = max x K V (x). Since K = F V 1 [a, ) F V 1 (, ) it follows that m >. Let I [, ) be the maximal interval of existence for φ: I E such that φ() = z, and φ = f φ. Define the set T = {t I: for all s [, t], φ(s) K}. For all t T, Hence for all t T, d (V φ)(t) = ( V φ)(t) m. dt V (φ(t)) = V (φ() + t V (φ(s)) ds a + mt. Since V is bounded above by M, T is also bounded above. Let t the the least upper bound of T. Since K E, I [, t ] and φ(t ) is welldefined. Moreover t lies in the interior of I. Since for all t [, t ), φ(t) K, and for every δ > there exist t I, t < t < t + δ such that φ(t) K it follows that φ(t ) lies on the boundary of the set K. The boundary of K is contained in the union K V 1 (a) (F \ U). Since V (φ(t )) a + mt > a it follows that φ(t ) V 1 (a). Therefore φ(t ) (F \ U), and, in particular, φ(t ) U. Since W was arbitrary, this shows that the origin is an unstable equilibrium of f. This is a very first draft, likely with many many typeohs. Comments are appreciated.
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