1. The work of Peter Crouch the control theorist* Conference on Decision and Control December 11, 2011 Canonical Geometrical Control Problems: New and Old Roger Brockett Engineering and Applied Sciences Harvard University *Not to be confused with the bad-boy English footballer of Tottenham Hotspur, Stoke City, Abigail Clancy, etc., etc.

2. Some of my early interactions: The London NATO meeting– September, 1973 Student at Harvard, 1974-1977: Thesis: “Dynamical Realizations of Finite Volterra Series” It showed that the natural state space for a finite Volterra series is diffeomorphic to R n Cohort included P. S. Krishnaprassad and Joseph Ja’ Ja” Sabbatical at Harvard in 1982 Peter Crouch: The reason we are here!

3. Peter Crouch at the Center: From the Web

4. Some Lie Theoretic, Least Squares, State Transfer Problems involving Z 2 Graded Lie Algebras

5. The first two have finite Volterra series

6. Recall

7. What about regulator versions of these systems?

8. What it Approximates

9. Our Quadratic Regulator Problem

10. The Euler-Lagrange Equations We need to factor the linear operator into a stable and unstable factors. The value of x(0) is given. Its derivative is to be determined so as to put x on the right submanifold

11. This is from the zeroth order term. This is from the first order term. Formula for Z Factoring the Euler-Lagrange Equation

12. Relating Properties of x and Z through Q It is important that we are now dealing with initial values Theta and Q are functions of x(0) and Z(0). - -

14. Here we first define the optimal trajectory using initial conditions giving an open loop control. Actually it is true at all times and states! If considered as a “gain” From the perspective of achieving the correct homogeneity, this is quite remarkable, even miraculous. is homogeneous of degree zero

15. An Example These solutions are stable for all $a$ and generate a Z displacement.

16. A Further Elaboration

17. As x(0) approaches 0 the cost is upper bounded by the cost of the u-only optimal trajectory. However, this cost is not differentiable on the “Z axis”.

18. As for the Cost---

19. This is not a dead end—Many more possibilities

20. Peter--- Congratulations on a distinguished career based on talent, hard work, discipline, service to the community.