Geometry of adaptive control

Two incompatible topologies appear in the study of adaptive systems: the graph topology in control design, and the coefficient topology in system identification. Their incompatibility is manifest in the stabilization problem of adaptive control. We argue that this problem can be approached by changing the geometry of the sets of control systems under consideration: estimating n parameters in an n-dimensional manifold whose points all correspond to stabilizable systems. One way to accomplish this is using the properties of the algebraic Riccati equation. To illustrate the ideas we pose a simple parameter estimation problem as a constrained optimization problem, and show that it admits a unique minimum. Search algorithms in a hypersurface lead to adaptive controllers that combine ideas classified as direct and indirect adaptive control in the literature.