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. Defnition (Liapunov Function) . . Let X be a C r vector feld 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 defned on a neighborhood U oF x e , di±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 feld on R n , r 1 , and let x e be an equilibrium point For X , that is, X ( x e . 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 ±x 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 de±ne 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 inde±nitely 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 infnite dimensions, one needs to augment (i) by the additional condition that there is an 0 such that For all 0 < ε ± ε , inF { L ( ϕ - 1 ( x )) | ± x - x e ± = ε ± } > 0 . In fnite dimensions, this condition is automatic by the compactness oF spheres.
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1.7 Liapunov Functions 83 boundary with a well defned 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 feld is pointing inwards (or is tangent) to the boundary oF R , this is a su±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 feld is inwards corresponds to the condition that L is decreasing along solutions at the boundary.
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Liapunov - Introduction 82 Liapunov Functions Besides the...

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