Every smooth function on a contractible complete Riemannian manifold satisfying the semiglobal Polyak-Lojasiewicz inequality can be written as f* + ||phi||^2 for a submersion phi, extending the Boumal-Criscitiello-Rebjock theorem.
On Topological Properties of the Set of Stabilizing Feedback Gains
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abstract
This work presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input-single-output (SISO) continuous and discrete time linear-time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states has at most $\lceil{\frac{n}{2}}\rceil$ connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LIT systems; the proposed algorithm also computes the number of unstable roots in each unstable interval. Along the way, we also make a number of observations on the set of stabilizing state feedback gains for MIMO systems.
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Smooth globally PLI functions are nonlinear least-squares, and so are their gradient-dominated cousins
Every smooth function on a contractible complete Riemannian manifold satisfying the semiglobal Polyak-Lojasiewicz inequality can be written as f* + ||phi||^2 for a submersion phi, extending the Boumal-Criscitiello-Rebjock theorem.