Past Functional Analysis Seminar

5 May 2015
Stochastic homogenization shows that solutions to an elliptic problem 
with rapidly oscillating, ergodic random coefficients can be effectively 
described by an elliptic problem with homogeneous, deterministic 
coefficients. The definition of the latter is based on the construction 
of a "corrector" and invokes an elliptic operator that acts on the 
probability space of admissible coefficient fields. While qualitative 
homogenization is well understood and classical, quantitative results 
(e.g. estimates on the homogenization error and approximations to the 
homogenized coefficients) have only been obtained recently.  In the talk 
we discuss an optimal estimate on the associated semigroup (usually 
called the "random walk in the random environment") and show that it 
decays with an algebraic rate. The result relies on a link between a 
Spectral Gap of a Glauber dynamics on the space of coefficient fields (a 
notion that we borrow from statistical mechanics) and heat kernel 
estimates. As applications we obtain moment bounds on the corrector and 
an optimal convergence rate for the approximation of the homogenized 
coefficients via periodic representative volume elements.
  • Functional Analysis Seminar
10 March 2015
Amol Sasane
Given an ideal I in the polynomial ring C[x1,...,xn], the variety V(I) of I is the set of common zeros in C^n of all the polynomials belonging to I. In algebraic geometry, one tries to link geometric properties of V(I) with algebraic properties of I. Analogously, given a system of linear, constant coefficient partial differential equations, one can consider its zeros, that is, its solutions in various function and distribution spaces. One could then hope to link analytic properties of the set of solutions with algebraic properties of the polynomials which describe the PDEs. In this talk, we will focus on one such analytic property, called autonomy, and we will provide an algebraic characterization for it.
  • Functional Analysis Seminar
3 March 2015
Lassi Paunonen
We begin by reviewing different stability types for abstract differential equations and strongly continuous semigroups on Hilbert spaces. We concentrate on exponential stability, polynomial stability, and strong stability with a finite number of singularities on the imaginary axis. We illustrate each stability type with examples from partial differential equations and control theory. 

In the second part of the talk we study the preservation of strong and polynomial stabilities of a semigroup under bounded perturbations of its generator. As the main results we present conditions for preservation of these two stability types under finite rank and trace class perturbations. In particular, the conditions require that certain graph norms of the perturbing operators are sufficiently small.

In the final part of the talk we consider robust output tracking for linear systems, and explain how this control problem motivates the study of preservation of polynomial stability of semigroups. In particular, the solution of this problem requires determining which uncertainties in the parameters of the controlled system preserve the stability of the closed-loop system consisting of the system and the dynamic controller. We show that if the reference signal to be tracked is a nonsmooth periodic function, it is impossible to stabilize the closed-loop system exponentially, but polynomial stability is achievable under suitable assumptions. Subsequently, the uncertainties in the parameters of the system can be represented as a bounded perturbation to the system operator of the polynomially stable closed-loop system.
  • Functional Analysis Seminar