No abstract

The output feedback pole assignment problem is a classical problem in linear systems theory. In this paper we calculate the number of complex dynamic compensators of order $q$ assigning a given set of poles for a $q$-nondegenerate $m$-input, $p$-output system of McMillan degree $n = q(m + p -1) + mp$. As a corollary it follows that when this number is odd, the generic system can be arbitrarily pole assigned by output feedback with a real dynamic compensator of order at most $q$ if and only if $q(m + p -1) + mp \geq n$. ©1996 Society for Industrial and Applied Mathematics SIAM J. CONTROL AND OPTIMIZATION Vol. 34, No. 3, pp. 813-832, May 1996 Abstract. The output feedback pole assignment problem is a classical problem in linear systems theory. In this paper we calculate the number of complex dynamic compensators of order q assigning a given set of poles for a q-nondegenerate m-input, p-output system of McMillan degree n q (m + p 1) + mp. As a corollary it follows that when this number is odd, the generic system can be arbitrarily pole assigned by output feedback with a real dynamic compensator of order at most q if and only if q(m + p 1) +mp > n.

No abstract

No abstract

We compute the degree of the generalized Plücker embedding κ of a Quot scheme X over IP 1 . The space X can also be considered as a compactification of the space of algebraic maps of a fixed degree from IP 1 to the Grassmanian Grass(m, n). Then the degree of the embedded variety κ(X) can be interpreted as an intersection product of pullbacks of cohomology classes from Grass(m, n) through the map ψ that evaluates a map from IP 1 at a point x ∈ IP 1 . We show that our formula for the degree verifies the formula for these intersection products predicted by physicists through Quantum cohomol-. We arrive at the degree by proving a version of the classical Pieri's formula on the variety X, using a cell decomposition of a space that lies in between X and κ(X).Let X ′ be the space of all algebraic maps, of a fixed degree q, from the projective line IP 1 to Grass(m, n), the Grassmannian of all m-dimensional subspaces of a fixed n-dimensional vector space V . In various papers, physicists have been discussing so called correlation functions on this space ([Int91] [Vaf92] [Wit93]) which specify the intersection products on X ′ of pullbacks of cohomology classes from the Grassmannian (a precise formulation can be found in Section 4). Further, based on certain physical arguments, there have been some conjectured formulas for these correlation functions (actually these conjectures deal with the more general case of maps from any Riemann surface to Grass(m, n), but we will restrict our attention to maps from IP 1 ). In [BDW93] the authors give a mathematically rigorous proof of the conjectured formula in the case of maps from a Riemann surface of genus one to Grass(2, n). In