On Agrachev

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On Agrachevs curvature of optimal control

• Matthias Kawski ?Eric Gehrig ?
• Arizona State University
• Tempe, U.S.A.

? This work was partially supported by NSF grant
DMS 00-72369.
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Outline

• Motivation. WANTED Sufficient conditions
  for optimality
• Review / survey
• Agrachevs definition and main theorem
• Comment connection to recent work on Dubins car
  (Chitour, Sigalotti)
• Best studied case Zermelos navigation problem
  (Ulysse Serres)
• Computational issues,
• Computer Algebra Systems. Live interactive?
• Recent efforts to visualize curvature of
  optimal control
• how to read our pictures
• what one may be able to see in our pictures
• Conclusion / outlook / current work
• Connection with Bang-Bang controls. Relaxation/
  approximation,

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References

• A. Agrachev On the curvature of control systems
  (abstract, SISSA 2000)
• A. Agrachev and Yu. Sachkov Lectures on
  Geometric Control Theory (SISSA 2001)Control
  Theory from the Geometric Viewpoint (Springer
  2004)
• Ulysse Serres, The curvature of 2-dimensional
  optimal control systems' and Zermelos
  navigation problem. (preprint 2002).
• A. Agrachev, N. Chtcherbakova, and I. Zelenko, On
  curvatures and focal points of dynamical
  Lagrangian distributions and their reductions by
  1st integrals (preprint 2004)
• M. Sigalotti and Y. Chitour, Dubins' problem on
  surfaces. II. Nonpositive curvature (preprint
  2004)On the controllability of the Dubins
  problem for surfaces (preprint 2004)

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Purpose/use of curvature in opt. control

• Maximum principle provides comparatively
  straightforward necessary conditions for
  optimality,sufficient conditions are in general
  harder to
• come by, and often comparatively harder to
  apply.Curvature (w/ corresponding comparison
  theorem)suggest an elegant geometric alternative
  to obtain verifiable sufficient conditions for
  optimality
• ? compare classical Riemannian geometry

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Curvature of optimal control

• understand the geometry (very briefly)
• develop intuition in basic examples
• apply to obtain new optimality results

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Curvature and double-Lie-brackets

• Usually, we think of curvature as defined in
  terms of connectionse. g.
• But here it is convenient to think of curvature
  as a measure of the lack of integrability of a
  horizontal distribution of horizontal lifts.In
  the case of a 2-dimensional base manifold, let g
  be the unit vertical field of infinitesimal
  rotation in fibres, and f be the geodesic vector
  field. In this case Gauss curvature is obtained
  as
• Recent beautiful application, analysis and
  controllability results by Chitour Sigalotti
  for control of Dubins car on curved surfaces.

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Curvature of optimal control

• understand the geometry
• develop intuition in basic examples
• apply to obtain new optimality results

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Classical geometry Focusing geodesics
Positive curvature focuses geodesics, negative
curvature spreads them out. Thm. curvature
negative geodesics ? (extremals) are optimal
(minimizers)
The imbedded surfaces view, and the color-coded
intrinsic curvature view
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Definition versus formula
A most simple geometric definition - beautiful
and elegant. but the formula in coordinates is
incomprehensible (compare classical curvature)

(formula from Ulysse Serres, 2001)
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Aside other interests / plans

• What is theoretically /practically feasible to
  compute w/ reasonable resources? (e.g.
  CAS simplify, old controllability is
  NP-hard, MK 1991)
• Interactive visualization in only your browser
• CAS-light inside JAVA (e.g. set up geodesic
  eqns)
• real-time computation of geodesic
  spheres (e.g. drag initial point w/ mouse,
  or continuously vary parameters)
• bait, hook, like Mandelbrot fractals.

Riemannian, circular parabloid
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From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
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From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
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From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
Next Define distinguished parameterization of H x
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The canonical vertical field v
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Jacobi equation in moving frame
Frame
or
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Zermelos navigation problem
Zermelos navigation formula
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formula for curvature ?

• total of 782 (279) terms in num, 23 (7) in denom.
  MAPLE cant factor

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First pictures fields of polar plots

• On the left the drift-vector field (wind)
• On the right field of polar plots of
  k(x1,x2,f)in Zermelos problem u f. (polar
  coord on fibre)polar plots normalized and color
  enhanced unit circle ? zero
  curvature negative curvature ? inside ?
  greenish positive curvature ? outside ?
  pinkish

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Example F(x,y) sech(x),0
k NOT globally scaled. colors for k and k-
scaled independently.
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Example F(x,y) 0, sech(x)
Question What do optimal paths look like?
Conjugate points?
k NOT globally scaled. colors for k and k-
scaled independently.
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Special case linear drift

• linear drift F(x)Ax, i.e., (dx/dt)Axeiu
• Curvature is independent of the base point x,
  study dependence on parameters of the
  drift kA(x1,x2,f) k(f,A)This case was
  studied in detail by U.Serres.Here we only give
  a small taste of the richness of even this very
  special simple class of systems

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Linear drift, preparation I

• (as expected), curvature commutes with
  rotationsquick CAS check

gt k'B'combine(simplify(zerm(Bxy,x,y,theta),tri
g))
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Linear drift, preparation II

• (as expected), curvature scales with
  eigenvalues(homogeneous of deg 2 in space of
  eigenvalues)quick CAS check

gt kdiagzerm(lambdax,muy,x,y,theta)
Note q is even and also depends only on even
harmonics of q
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Linear drift

• if drift linear and ortho-gonally
  diagonalizable ? then no conjugate pts(see U.
  Serres for proof, here suggestive picture only)

gt kdiagzerm(x,lambday,x,y,theta)
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Linear drift

• if linear drift has non-trivial Jordan block ?
  then a little bit ofpositive curvature exists
• Q enough pos curv forexistence of conjugate
  pts?

gt kjordzerm(lambdaxy,lambday,x,y,theta)
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Some linear drifts
Question Which case is good for optimal
control?
diag w/ l10,-1
diag w/ l1i,1-i
jordan w/ l13/12
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Ex A1 1 0 1. very little pos curv
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F(x)0,sech(3x)

• globally scaled.
• colors for k and k- scaled simultaneously.

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Curvature and Bang-Bang extremals

• Current theory of curvature in optimal control
  applies to systems whose set of admissible
  velocities is a topological sphere (circle).
  Current efforts Approximate affine system
  whose set of velocities is a line or plane
  segment by system whose set of velocities is a
  thin ellipsoids,and analyze the limit as the
  ellipsoids degenerate.

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Curvature and Bang-Bang extremals

• Current theory of curvature in optimal control
  applies to systems whose set of admissible
  velocities is a topological sphere (circle).
  What about affine systems whose set of
  velocities is a line or plane segment ?

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Conclusion

• Curvature of control beautiful
  subject promising to yield new sufficiency
  results
• Even most simple classes of systems far from
  understood
• CAS and interactive visualization promise to be
  useful tools to scan entire classes of systems
  for interesting, proof-worthy properties.
• Some CAS open problems (simplify). Numerically
  fast implementation for JAVA not yet.
• Zermelos problem particularly nice because
  everyone has intuitive understanding, wants to
  argue which way is best, then see and compare to
  the true optimal trajectories.
• Current efforts Agrachevs theory applies to
  systems whose set of admissible velocities is a
  topological sphere (circle). Current efforts
  Approximate systems whose set of velocities is a
  line/plane segment by thin ellipsoids and
  analyze the limit as the ellipsoids degenerate.

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