Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance
Bernard Bonnard Monique Chyba Alain Jacquemard John Marriott
The analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on $B_1 \times B_2$, where $B_1$ and $B_2$ are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points.
keywords: contrast imaging Mayer problem geometric optimal control invariant theory.
On periodic orbits of polynomial relay systems
Alain Jacquemard Weber Flávio Pereira
We present an algorithm which determines global conditions for a class of discontinuous vector fields in 4D (called polynomial relay systems) to have periodic orbits. We present explicit results relying on constructive proofs, which involve classical Effective Algebraic Geometry algorithms.
keywords: semi-algebraic sets. discontinuous differential equations periodic orbits

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