ASYMPTOTIC ESTIMATES OF ESCAPE TIME FOR LAGRANGIAN SYSTEMS

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ASYMPTOTIC ESTIMATES OF ESCAPE TIME FOR
LAGRANGIAN SYSTEMS

• Agnessa Kovaleva
• Space Research Institute
• Russian Academy of Sciences
• Moscow, Russia

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ABSTRACT

• The paper presents an algorithm of exit time
  estimation for Lagrangian systems with linear
  dissipation and weak noise. The reference domain
  of operation is associated with the domain of
  attraction of a stable equilibrium in the
  noiseless system.
• Small additive noise induces small oscillations
  around the stable point with a very large time
  until escape. Large deviations theory reduces the
  mean exit time problem for rare escape to the
  minimization of the action functional and the
  solution of the associated Hamilton-Jacobi
  equation. Under some non-restrictive assumptions,
  the solution of the Hamilton-Jacobi equation is
  found in the explicit form, as transformations
  of kinetic and potential energy of the system.

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BASIC MODEL

• The Lagrangian equation of motion
• W(t) white noise process in Rr, ? - n?r matrix,
• Lagrangian
• L(q, q?) T(q, q?) - ?(q, q?), T(q, q?) (q?,
  M(q)q?)/2
• T- kinetic energy, ? - potential energy.
• The matrices A ?T ?, B, and M(q) are symmetric
  positive definite

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• The new variable (impulse)
  
• The full energy function
• H(q, p) ? (q?,p) ? L(q, q?) ? T(q, q?) ? ?(q),
• q?(q, p) ? M -1(q)p
• The equations of motion

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ASSUMPTIONS

• (B.1) An admissible domain G is an open bounded
  set in R2n with smooth boundary ?
• (B.2) The function H(q,p) ? C2,2(G?? )
• (B.3) There exists a unique asymptotically stable
  point
• O (q ? 0, p 0) ? G\?e ?e is an internal
  e-neighborhood of ?
• (B.4) All trajectories originating in G?? tend to
  O
• (B.5) The matrices A ??T, B, and M(q) are
  symmetric positive definite.

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1. An orbit from G is attracted to a small
neighborhood of O, tatt 1. 2. Motion evolves
near O, ? e exp(-1/e2) residence time 3.
The system leaves G along a deterministic exit
orbit an extremal of action functional. 4. E? e
is defined through the solution of the
variational problem


G
O



?
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THE PRECISE SOLUTION

• Boundary conditions
• At the initial moment t 0, (q q0, p p0)
  ?intG,
• At the moment of escape t ?e, (q, p) ? ? ,
• Calculation of E?e Ve(q0 p0 )
• The asymptotic solution Ve(q,p) expS(q,p)/e2
  .

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THE MEAN EXIT TIME ESTIMATE (Kushner, 1984, Wu,
2001)

• lim? 2ln (ET?) ?infS(q,p)/(q,p)?? ?
  S0, as ??0

HAMILTON-JACOBI EQUATION(Kovaleva, 2005)
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THE SOLUTION(Kovaleva, 2005)

• The solution is sought as
• Potential energy ? is independent of p,

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THE COMPATIBILITY CONDITIONS

• imply restrictions to the matrix K
• SPECIAL CASES
• 1. K ? kIn, where In is n-th order identity
  matrix, k is a scalar
• S(q, p) ? kH(q, p)
• 2. K diagk1,,kn ,
• 3. The linear system
• S(q,p) (p, KM ?1p) ? (q,KCq)/2

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• EXAMPLES
• The space constraints are in the form q ? Gq ?
  G0.There are no particular restrictions to the
  impulses p.
• Since S(q,p) is positive definite quadratic form
  in p,
• Example 1 the linear system in an l-dimensional
  ellipsoid
• Mq?? Bq? Cq e?W(t)
• Gq (q, Lq) lt 1 , ? (q, Lq) ? 1, L is a
  positive definite lxl matrix

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EXAMPLE 2 THE LINEAR SYSTEM IN THE PLANE
q2

• qi?? bqi? cqi e?W(t), i 1,2
• G q12/?12 q22/?22 lt 1,
• ? q12/?12 q22/?22 1,

?1
?2
q1
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EXAMPLE 3 MOTION IN A POTENTIAL TRAP (THE
HENON-HEILES POTENTIAL)

• The equation of motion
• Potential
• ?(q1,q2) (q12 q22 2q12q2 - 2q23/3)/2

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• The equipotential curves
• ?(q1, q2) const lt 1/6
• the equality ?(q1, q2) 1/6 determines the
  separatrix in the projection on the plane.
• The admissible domain
• Gq ? (q1, q2) lt 1/6, ?q ? (q1, q2) 1/6
  
• The mean escape time asymptotics
• lime?8lne2(ETe) min?k ? (q1, q2) b/ 6s2

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Conclusions

• Theory of large deviations is applied to the
  problem of escape from the reference domain for a
  weakly perturbed Lagrangian system. Formally, the
  system is interpreted as a nonlinear stochastic
  degenerate system.
• The techniques employed involve the reduction of
  the escape problem to a deterministic variational
  problem for the action functional, and the
  solution of an associated Hamilton-Jacobi
  equation. It is shown that, under broad
  conditions, the solution can be found in the
  closed form, and the mean escape time can be
  defined as a function of the system and noise
  parameters.