19,688 research outputs found
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
Maximum Likelihood for Matrices with Rank Constraints
Maximum likelihood estimation is a fundamental optimization problem in
statistics. We study this problem on manifolds of matrices with bounded rank.
These represent mixtures of distributions of two independent discrete random
variables. We determine the maximum likelihood degree for a range of
determinantal varieties, and we apply numerical algebraic geometry to compute
all critical points of their likelihood functions. This led to the discovery of
maximum likelihood duality between matrices of complementary ranks, a result
proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur
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