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Liapunov - Introduction 82 Liapunov Functions Besides the...

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82 Introduction Liapunov Functions Besides the Liapunov spectral theorem, there is another basic method of proving stability that is a generalization of the energy method we have seen in the introductory mechanical examples. Definition (Liapunov Function) . . Let X be a C r vector field on R n , r 1 , and let x e be an equilibrium point for X , that is, X ( x e ) = 0 . A Liapunov function for X at x e is a continuous function L : U R defined on a neighborhood U of x e , di ff erentiable on U \{ x e } , and satisfying the following conditions: (i) L ( x e ) = 0 and L ( x ) > 0 if x = x e ; (ii) The directional derivative of L along X , denoted X [ L ] 0 on U \{ x e } ; this means that d dt L 0 along solution curves of X . The Liapunov function L is said to be strict , if (ii) is replaced by the condition (ii) : X [ L ] < 0 in U \{ x e } . Condition (i) is sometimes called the potential well hypothesis . 7 By the Chain Rule for the time derivative of V along integral curves, condition (ii) is equivalent to the statement: L is nonincreasing along integral curves of X . Theorem (Liapunov Stability Theorem.) . Let X be a C r vector field on R n , r 1 , and let x e be an equilibrium point for X , that is, X ( x e ) = 0 . If there exists a Liapunov function for X at x e , then x e is stable. Proof. Since the statement is local, we can assume that x e = 0. By the local existence theory, there is a neighborhood U of 0 such that all solutions starting in U exist for time t [ - δ , δ ], with δ depending only on X and U , but not on the solution. Now fix ε > 0 and an open ball D ε (0) that is included in U . Let ρ ( ε ) > 0 be the minimum value of L on the sphere of radius ε , and define the open set U = { x D ε (0) | L ( x ) < ρ ( ε ) } . By (i), U = , 0 U , and by (ii), no solution starting in U can meet the sphere of radius ε (since L is decreasing on integral curves of X ). Thus all solutions starting in U never leave D ε (0) U and therefore by uniformity of time of existence, these solutions can be extended indefinitely in time. This shows that 0 is stable. There is a more global concept that is related to this circle of ideas that we discuss somewhat informally. Namely, a region R R n with a (smooth) 7 In infinite dimensions, one needs to augment (i) by the additional condition that there is an ε > 0 such that for all 0 < ε ε , inf { L ( ϕ - 1 ( x )) | x - x e = ε } > 0 . In finite dimensions, this condition is automatic by the compactness of spheres.
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1.7 Liapunov Functions 83 boundary with a well defined inside and outside, is called a positively trapping region if for initial conditions that start in R , the solution stays in R for t 0. If the given vector field is pointing inwards (or is tangent) to the boundary of R , this is a su ffi cient condition for R to be a trapping region. If R is bounded, necessarily the dynamical system is positively complete on R . Often sublevel sets R = { x R n | L ( x ) C } of Liapunov functions provide such trapping regions because the vector field is inwards corresponds to the condition that L is decreasing along solutions at the boundary.
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