Published in Tall, I. A. (2010). Time-invariant quadratic Hamiltonians via generalized transformations. American Control Conference (ACC), 2010, 1428-1433. ©2010 AACC / IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the publisher. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.


In this paper we give necessary and sufficient conditions for achieving a quadratic positive definite time-invariant Hamiltonian for time-varying generalized Hamiltonian control systems using canonical transformations. Those necessary and sufficient conditions form a system of partial differential equations that reduces to the matching conditions obtained earlier in the literature for time-invariant systems. Their theoretical solvability is proved via the Cauchy-Kowalevskaya theorem and their practical solvability discussed in some particular cases. Systems with time-invariant positive definite Hamiltonian are known to yield a passive input-output map and can be stabilized by unity feedback, which underlines the importance of achieving the positive definiteness and time-invariancy for the Hamiltonian. We illustrate the results with few examples including the rolling coin.