Visualizing Agrachvs curvature of optimal control

1
Visualizing Agrachëvs curvature of optimal
control

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

? This work was partially supported by NSF grant
DMS 00-72369.
2
Outline

• Motivation of this work
• Brief review of some of Agrachëvs theory, and
  of last years work by Ulysse Serres
• Some comments on ComputerAlgebraSystems ideally
  suited ? practically impossible
• Current efforts to see curvature of optimal
  cntrl.
• how to read our pictures
• what one may be able to see in our pictures
• Conclusion A useful approach? Promising 4 what?

3
Purpose/use of curvature in opt.cntrl

• 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

4
Curvature of optimal control

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

5
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
6
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)
7
(No Transcript)
8
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
9
References

• Andrei Agrachev On the curvature of control
  systems (abstract, SISSA 2000)
• Andrei Agrachev and Yu. Sachkov Lectures on
  Geometric Control Theory, 2001, SISSA.
• Ulysse Serres On the curvature of
  two-dimensional control problems and Zermelos
  navigation problem. (Ph.D. thesis at SISSA)
  ONGOING WORK ???

10
From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
11
From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
12
From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
Next Define distinguished parameterization of H x
13
The canonical vertical field v
14
From Agrachev / Sachkov Lectures on Geometric
Control Theory, 2001
15
Jacobi equation in moving frame
Frame
or
16
Zermelos navigation problem
Zermelos navigation formula
17
formula for curvature ?

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

18
Use U. Serres form of formula
polynomial in f and first 2 derivatives, trig
polynomial in q, interplay of 4 harmonics
so far have still been unable to coax MAPLE into
obtaining this without doing all
simplification steps manually
19
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

20
Example F(x,y) sech(x),0
k NOT globally scaled. colors for k and k-
scaled independently.
21
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.
22
Example F(x,y) - tanh(x), 0
k NOT globally scaled. colors for k and k-
scaled independently.
23
From now on color code only (i.e., omit
radial plots)
24
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 is being
  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

25
Linear drift, preparation I

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

gt k'B'combine(simplify(zerm(Bxy,x,y,theta),tri
g))
26
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
27
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)
28
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)
29
Some linear drifts
Question Which case is good for optimal
control?
diag w/ l10,-1
diag w/ l1i,1-i
jordan w/ l13/12
30
Ex A1 1 0 1. very little pos curv
31
Scalings / - , local / global
same scale for pos. neg. parts
global color-scale, same for every fibre
here F(x) ( 0, sech(3x1))
local color-scales, each fibre independ.
pos. neg. parts color-scaled independently
32
ExampleF(x)0,sech(3x)
scaled locally / globally
33
F(x)0,sech(3x)

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

34
F(x)0,sech(3x)

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

35
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????
• 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.

36
(No Transcript)