Monodromy-Prescribed Polystable Bundles on Punctured Riemann Surfaces and the Geometry of Singular Control Strategies

This paper establishes a functorial algebraic isomorphism between the moduli space BCps(Σ,G) of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface Σ of genus g≥2, for a complex reductive Lie group G, and the character variety MCK(Σ*,G) of representations of its fundamental group with relatively compact image. The dimension formula dimBCps(Σ,G)=2(g−1)dimC(G)+∑i=1kdimR(Ci), where C1,…,Ck are conjugacy classes in a maximal compact subgroup K⊂G, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups.