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.

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 Rn

Cohort included P. S. Krishnaprassad and Joseph Ja’ Ja”

Sabbatical at Harvard in 1982

Peter Crouch: The reason we are here!

Peter Crouch at the Center: From the Web

Some Lie Theoretic, Least Squares, State Transfer Problems involving Z2 Graded Lie Algebras

The first two have finite Volterra series

Recall

What about regulator versions of these systems?

What it Approximates

Our Quadratic Regulator Problem

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 determinedso as to put x on the right submanifold

This is from the zeroth order term.

This is from the first order term.

Formula for Z

Factoring the Euler-Lagrange Equation

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).

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Here we first define the optimal trajectory using initial conditionsgiving 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

An Example

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

A Further Elaboration

A Further Elaboration

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”.

As for the Cost---

This is not a dead end—Many more possibilities

Peter---

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