1. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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.

2. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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,

3. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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)

4. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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

5. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Curvature of optimal control understand the geometry (very briefly) develop intuition in basic examples apply to obtain new optimality results

6. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Curvature and double-Lie-brackets Usually, we think of curvature as defined in terms of connections e. 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”.

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

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

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

10. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu

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

12. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

13. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

14. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Next: Define distinguished parameterization of H x From: Agrachev / Sachkov: “Lectures on Geometric Control Theory”, 2001

15. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu The canonical vertical field v

16. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Jacobi equation in moving frame Frame or:

17. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Zermelo’s navigation problem “Zermelo’s navigation formula”

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

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

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

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

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

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

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

28. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu Ex: A=[1 1; 0 1]. very little pos curv

29. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu F(x)=[0,sech(3x)]  globally scaled.  colors for  + and  - scaled simultaneously.

30. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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.

31. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu 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 ?

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

33. Matthias Kawski. “On Agrachev’s curvature of optimal control” AMS 1006. Lubbock TX. April, 2005 http://math.asu.edu/~kawski kawski@asu.edu