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Ch 9.6: Liapunov’s Second Method

Ch 9.6: Liapunov’s Second Method. In Section 9.3 we showed how the stability of a critical point of an almost linear system can usually be determined from a study of the corresponding linear system.

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Ch 9.6: Liapunov’s Second Method

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  1. Ch 9.6: Liapunov’s Second Method • In Section 9.3 we showed how the stability of a critical point of an almost linear system can usually be determined from a study of the corresponding linear system. • However, no conclusion can be drawn when the critical point is a center of the corresponding linear system. • Also, for an asymptotically stable critical point, we may want to investigate the basin of attraction, for which the localized almost linear theory provides no information. • In this section we discuss Liapunov’s second method, or direct method, in which no knowledge of the solution is required. • Instead, conclusions about the stability of a critical point are obtained by constructing a suitable auxiliary function.

  2. Physical Principles • Liapunov’s second method is a generalization of two physical principles for conservative systems. • The first principle is that a rest position is stable if the potential energy is a local minimum, otherwise it is unstable. • The second principle states that the total energy is a constant during any motion. • To illustrate these concepts, we again consider the undamped pendulum, which is a conservative system.

  3. Undamped Pendulum Equation (1 of 5) • The governing equation for the undamped pendulum is • To convert this equation into a system of two first order equations, we let x =  and y = d /dt, obtaining • The potential energy U is the work done in lifting pendulum above its lowest position:

  4. Undamped Pendulum System: Potential Energy (2 of 5) • The critical points of our system are x= n , y = 0, for n = 0, 1, 2,…. • Physically, we expect the points (2n , 0) to be stable, since the pendulum bob is vertical with the weight down, and the points ((2n+1) , 0) to be unstable, since the pendulum bob is vertical with the weight up. • Comparing this with the potential energy U, we see that U is a minimum equal to zero at (2n , 0) and U is a maximum equal to 2mgL at ((2n+1) , 0).

  5. Undamped Pendulum System: Total Energy (3 of 5) • The total energy V is the sum of potential and kinetic energy: • On a solution trajectory x = (t), y = (t), V is a function of t. • The derivative of V((t), (t)) with respect to t is called the rate of change of V following the trajectory. • For x = (t), y = (t), and using the chain rule, we obtain • Since x and y satisfy the differential equations it follows that dV(, )/dt = 0, and hence V is constant.

  6. Undamped Pendulum System: Small Energy Trajectories (4 of 5) • Observe that we computed the rate of change dV(, )/dt of the total energy V without solving the system of equations. • It is this fact that enables us to use Liapunov’s second method for systems whose solution we do not know. • Note that V = 0 at the stable critical points (2n , 0), where we recall • If the initial state (x1, y1) of the pendulum is sufficiently near a stable critical point, then the energy V(x1, y1) is small, and the corresponding trajectory will stay close to the critical point. • It can be shown that if V(x1, y1) is sufficiently small, then the trajectory is closed and contains the critical point.

  7. Undamped Pendulum System: Small Energy Elliptical Trajectories (5 of 5) • Suppose (x1, y1) is near (0,0), and that V(x1, y1) is very small. The energy equation of the corresponding trajectory is • From the Taylor series expansion of cosx about x = 0, we have • Thus the equation of the trajectory is approximately • This is an ellipse enclosing the origin. The smaller V(x1, y1) is, the smaller the axes of the ellipse are. • Physically, this trajectory corresponds to a periodic solution, whose motion is a small oscillation about equilibrium point.

  8. Damped Pendulum System: Total Energy (1 of 2) • If damping is present, we may expect that the amplitude of motion decays in time and that the stable critical point (center) becomes an asymptotically stable critical point (spiral point). • Recall from Section 9.3 that the system of equations is • The total energy is still given by • Recalling it follows that dV/dt = -cLy2  0.

  9. Damped Pendulum System: Nonincreasing Total Energy (2 of 2) • Thus dV/dt = -cLy2  0, and hence the energy is nonincreasing along any trajectory, and except for the line y = 0, the motion is such that the energy decreases. • Therefore each trajectory must approach a point of minimum energy, or a stable equilibrium point. • If dV/dt < 0 instead of dV/dt  0, we can expect this to hold for all trajectories that start sufficiently close to the origin.

  10. General Autonomous System: Total Energy • To pursue these ideas further, consider the autonomous system and suppose (0,0) is an asymptotically stable critical point. • Then there exists a domain D containing (0,0) such that every trajectory that starts in D must approach (0,0) as t  . • Suppose there is an “energy” function V such that V(x, y)  0 for (x, y) in D with V = 0 only at (0,0). • Since each trajectory in D approaches (0,0) as t  , then following any particular trajectory, V approaches 0 as t  . • The result we want is the converse: If V decreases to zero on every trajectory as t  , then the trajectories approach (0,0) as t  , and hence (0,0) is asymptotically stable.

  11. Definitions: Definiteness • Let V be defined on a domain D containing the origin. Then we make the following definitions. • V is positive definite on D if V(0,0) = 0 and V(x, y) > 0 for all other points (x, y) in D. • V is negative definite on D if V(0,0) = 0 and V(x, y) < 0 for all other points (x, y) in D. • V is positive semi-definite on D if V(0,0) = 0 and V(x, y)  0 for all other points (x, y) in D. • V is negative semi-definite on D if V(0,0) = 0 and V(x, y)  0 for all other points (x, y) in D.

  12. Example 1 • Consider the function • Then V is positive definite on the domain since V(0,0) = 0 and V(x, y) > 0 for all other points (x, y) in D.

  13. Example 2 • Consider the function • Then V is only positive semi-definite on the domain since V(x, y) = 0 on the line y = -x.

  14. Derivative of VWith Respect to System • We also want to consider the function where F and G are the functions given in the system • The function can be identified as the rate of change of V along the trajectory that passes through (x, y), and is often referred to as the derivative of V with respect to the system. • That is, if x = (t), y = (t) is a solution of our system, then

  15. Theorem 9.6.1 • Suppose that the origin is an isolated critical point of the autonomous system • If there is a function V that is continuous and has continuous first partial derivatives, is positive definite, and for which is negative definite on a domain D in the xy-plane containing (0,0), then the origin is an asymptotically stable critical point. • If negative semidefinite, then (0,0) is a stable critical point. • See the text for an outline of the proof for this theorem.

  16. Theorem 9.6.2 • Suppose that the origin is an isolated critical point of the autonomous system • Let V be a function that is continuous and has continuous first partial derivatives. • Suppose V(0,0) = 0 and that in every neighborhood of (0,0) there is at least one point for which V is positive (negative). • If there is a domain D containing the origin such that is positive definite (negative definite) on D, then the origin is an unstable critical point. • See the text for an outline of the proof for this theorem.

  17. Liapunov Function • The function V in Theorems 9.6.1 and 9.6.2 is called a Liapunov function. • The difficulty in using these theorems is that they tell us nothing about how to construct a Liapunov function, assuming that one exists. • In the case where the autonomous system represents a physical problem, it is natural to consider first the actual total energy of the system as a possible Liapunov function. • However, Theorems 9.6.1 and 9.6.2 are applicable in cases where the concept of physical energy is not pertinent. • In these cases, a trial-and-error approach may be necessary.

  18. Example 3: Undamped Pendulum (1 of 3) • For the undamped pendulum system use Theorem 9.6.1 show that (0,0) is a stable critical point, and use Theorem 9.6.2 to show (, 0) is an unstable critical point. • Let V be the total energy function and let • Thus V is positive definite on D, since V > 0 on D, except at the origin, where V(0,0) = 0.

  19. Example 3: Critical Point at (0,0) (2 of 3) • Thus V is positive definite on D, • Further, as we have seen, for all x and y. Thus is negative semidefinite on D. • Thus by Theorem 9.6.1, it follows that the origin is a stable critical point for the undamped pendulum. • To examine the critical point (, 0) using Theorem 9.6.2, we cannot use the same Liapunov function since is not positive or negative definite.

  20. Example 3: Critical Point at (, 0) (3 of 3) • Consider the change of variable x =  + u, and y = v. Then our system of differential equations becomes with critical point (0, 0) in the uv-plane. Let V be defined by and let D be the domain • Then V(u, v) > 0 in the first and third quadrants, and is positive definite on D. • Thus (0, 0) in the uv-plane, or (, 0) in xy-plane, is unstable.

  21. Theorem 9.6.3 • Suppose that the origin is an isolated critical point of the autonomous system • Let V be a function that is continuous and has continuous first partial derivatives. • If there exists a bounded domain DK containing the origin on which V(x, y) < K, with V is positive definite and negative definite, then every solution of the system above that starts at a point in DK approaches the origin as t  . • Thus DK gives a region of asymptotic stability, but may not be the entire basin of attraction.

  22. Liapunov Function Discussion • Theorems 9.6.1 and 9.6.2 give sufficient conditions for stability and instability, respectively. • However, these conditions are not necessary, nor does our failure to determine a suitable Liapunov function mean that there is not one. • Unfortunately, there are no general methods for the construction of Liapunov functions. • However, there has been extensive work on the construction of Liapunov functions for special classes of equations. • An algebraic result that is often useful in constructing positive or negative definite functions is stated in the next theorem.

  23. Theorem 9.6.4 • Let V be the function defined by • Then V is positive definite if and only if and is negative definite if and only if

  24. Example 4 • Consider the system • We try to construct a Liapunov function of the form • Then • If we choose b = 0, and a, c to be any positive numbers, then is negative definite, and V positive definite by Theorem 9.6.4. • Hence (0,0) is asymptotically stable, by Theorem 9.6.1.

  25. Example 5: Competing Species System (1 of 7) • Consider the system • In Example 1 of Section 9.4 we found that this system models a certain pair of competing species, and that the point (0.5,0.5) is asymptotically stable. We confirm this conclusion by finding a suitable Liapunov function. • We transform (0.5,0.5) to the origin by letting x = 0.5 + u, and y = 0.5 + v. Our system then becomes

  26. Example 5: Liapunov Function (2 of 7) • We have • Consider a possible Liapunov function of the form • Then V is positive definite, so we only need to determine whether there is region of the origin in the uv-plane where is negative definite.

  27. Example 5: Derivative With Respect to System (3 of 7) • To show that is negative definite, it suffices to show that is positive definite, at least for u and v sufficiently small. • Observe that the quadratic terms of H can be written as and hence are positive definite. • The cubic terms may be of either sign. We show that in some neighborhood of the origin, the following inequality holds:

  28. Example 5: Negative Definite (4 of 7) • To estimate the left side of the desired inequality we introduce polar coordinates u = rcos and v = rsin. • Then since |cos |, |sin |  1. It is then sufficient to require

  29. Example 5: Asymptotically Stable Critical Point (5 of 7) • Thus for the domain D defined by the hypotheses of Theorem 9.6.1 are satisfied, so the origin is an asymptotically stable critical point of the system • It follows that the point (0.5,0.5) is an asymptotically stable critical point of the original system of equations

  30. Example 5: Region of Asymptotic Stability (6 of 7) • Recall that the Liapunov function for this example is • If we refer to Theorem 9.6.3, then the preceding argument also shows that the disk is a region of asymptotic stability for the system of equations

  31. Example 5: Estimating Basin of Attraction (7 of 7) • The disk is a severe underestimate of the full basin of attraction. • To obtain a better estimate of the actual basin of attraction from Theorem 9.6.3, we would need to estimate the terms in more accurately, use a better (and possibly more complicated) Liapunov function, or both.

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