1. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Visualizing Agrachëv’s 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Outline Motivation of this work Brief review of some of Agrachëv’s theory, and of last year’s 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Curvature of optimal control understand the geometry develop intuition in basic examples apply to obtain new optimality results

5. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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, andthe color-coded intrinsic curvature view

6. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu

8. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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 Zermelo’s navigation problem”. (Ph.D. thesis at SISSA) ONGOING WORK ???

10. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

11. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

12. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Next: Define distinguished parameterization of H x From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

13. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu The canonical vertical field v

14. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

15. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Jacobi equation in moving frame Frame or:

16. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Zermelo’s navigation problem “Zermelo’s navigation formula”

17. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu formula for curvature ? total of 782 (279) terms in num, 23 (7) in denom. MAPLE can’t factor…

18. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Use U. Serre’s form of formula so far have still been unable to coax MAPLE into obtaining this without doing all “simplification” steps manually polynomial in f and first 2 derivatives, trig polynomial in , interplay of 4 harmonics

19. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu First pictures: fields of polar plots On the left: the drift-vector field (“wind”) On the right: field of polar plots of  (x 1,x 2,  ) 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

20. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Example: F(x,y) = [sech(x),0]  NOT globally scaled. colors for  + and  - scaled independently.

21. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Example: F(x,y) = [0, sech(x)]  NOT globally scaled. colors for  + and  - scaled independently. Question: What do optimal paths look like? Conjugate points?

22. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Example: F(x,y) = [ - tanh(x), 0]  NOT globally scaled. colors for  + and  - scaled independently.

23. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu From now on: color code only (i.e., omit radial plots)

24. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Special case: linear drift linear drift F(x)=Ax, i.e., (dx/dt)=Ax+e iu Curvature is independent of the base point x, study dependence on parameters of the drift   (x 1,x 2,  ) =  (  ) 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. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Linear drift, preparation I (as expected), curvature commutes with rotations quick CAS check: > k['B']:=combine(simplify(zerm(Bxy,x,y,theta),trig));

26. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Linear drift, preparation II (as expected), curvature scales with eigenvalues (homogeneous of deg 2 in space of eigenvalues) quick CAS check: > kdiag:=zerm(lambda*x,mu*y,x,y,theta); Note:  is even and also depends only on even harmonics of 

27. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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);

28. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Linear drift if linear drift has non- trivial Jordan block  then a little bit of positive curvature exists Q: enough pos curv for existence of conjugate pts? > kjord:=zerm(lambda*x+y,lambda*y,x,y,theta);

29. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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?

30. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Ex: A=[1 1; 0 1]. very little pos curv

31. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Scalings: + / -, local / global same scale for pos.& neg. parts pos.& neg. parts color-scaled independently local color-scales, each fibre independ. global color-scale, same for every fibre here: F(x) = ( 0, sech(3*x 1 ))

32. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu Example: F(x)=[0,sech(3x)] scaled locally / globally

33. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu F(x)=[0,sech(3x)]  globally scaled.  colors for  + and  - scaled simultaneously.

34. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu F(x)=[0,sech(3x)]  globally scaled.  colors for  + and  - scaled simultaneously.

35. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu 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???? 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.

36. Matthias Kawski. “ Visualizing Agrachëv’s curvature” Banach Institute, Bedlevo June, 2003 http://math.asu.edu/~kawski kawski@asu.edu