matthias kawski. “on agrachev’s curvature of optimal control” ams 1006. lubbock tx. april,...

http://math.asu.edu/~kawski [email protected]

Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005

On Agrachev’s 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.



Outline• Motivation. WANTED: Sufficient conditions for optimality

• Review / survey:– Agrachev’s definition and main theorem

– Comment: connection to recent work on Dubins’ car (Chitour, Sigalotti)

– Best studied case: Zermelo’s 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,



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 Zermelo’s 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)



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



Curvature of optimal control

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



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”.



Curvature of optimal control

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



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



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)



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



From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001



Next: Define distinguished parameterization of H x

From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001



The canonical vertical field v



Jacobi equation in moving frameFrame

or:



Zermelo’s navigation problem

“Zermelo’s navigation formula”



formula for curvature ?

total of 782 (279) terms in num, 23 (7) in denom. MAPLE can’t factor…



First pictures: fields of polar plots

• On the left: the drift-vector field (“wind”)

• On the right: field of polar plots of (x1,x2,)

in Zermelo’s problem u* = . (polar coord on fibre)

polar plots normalized and color enhanced: unit circle zero curvature

negative curvature inside greenish positive curvature outside pinkish



Example: F(x,y) = [sech(x),0]

NOT globally scaled. colors for + and - scaled independently.



Example: F(x,y) = [0, sech(x)]

NOT globally scaled. colors for + and - scaled independently.

Question: What do optimal paths look like? Conjugate points?



Special case: linear drift

• linear drift F(x)=Ax, i.e., (dx/dt)=Ax+eiu

• Curvature is independent of the base point x, study dependence on parameters of the drift

(x1,x2,) = ()

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



Linear drift, preparation I

• (as expected), curvature commutes with rotations

quick CAS check:, ,B

( )cos ( )sin

( )sin ( )cos

a b

c d

( )cos ( )sin

( )sin ( )cos

kB19 a d

16

3

8d

2( )cos 2 2 3

4b

2( )cos 2 2 3

4c2

( )cos 2 2 3

8a

2( )cos 2 2 3

32c

2( )cos 4 4 :=

3

32b

2( )cos 4 4 3

8c a ( )sin 2 2 21 c b

16

9

8b a ( )sin 2 2 9

8c d ( )sin 2 2 23 a

2

32

21 b2

32

23 d2

32

3

32d

2( )cos 4 4 3

32a

2( )cos 4 4 3

16c a ( )sin 4 4 3

16c d ( )sin 4 4 3

16b a ( )sin 4 4

3

16a d ( )cos 4 4 3

16c b ( )cos 4 4 3

16d b ( )sin 4 4 3

8d b ( )sin 2 2 21 c

2

32

> k['B']:=combine(simplify(zerm(Bxy,x,y,theta),trig));



Linear drift, preparation II

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

quick CAS check:

:= kdiag 23 ( ) 2

32

4

3

8( )cos 2 ( )2 2 3

32( )cos 4 ( ) ( )

> kdiag:=zerm(lambda*x,mu*y,x,y,theta);

Note: is even and also depends only on even harmonics of



Linear drift

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

> kdiag:=zerm(x,lambda*y,x,y,theta);

:= kdiag 23 ( )1 2

32

4

3

8( )cos 2 ( ) 1 2 3

32( )cos 4 ( ) 1 ( )1



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?

> kjord:=zerm(lambda*x+y,lambda*y,x,y,theta);

:= kjord 21

32

2

4

3

4( )cos 2 3

4( )sin 2 3

32( )cos 4



Some linear drifts

jordan w/ =13/12

diag w/ =10,-1

diag w/ =1+i,1-i

Question: Which case is good for optimal control?



Ex: A=[1 1; 0 1]. very little pos curv



F(x)=[0,sech(3x)]

globally scaled. colors for + and - scaled simultaneously.



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.



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 ?



Conclusion

• Curvature of control: beautiful subjectpromising 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.

• Zermelo’s 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: Agrachev’s 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.

matthias kawski. “on agrachev’s curvature of optimal control” ams 1006. lubbock tx. april,...

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