arXiv subjects

Andrew Wynn

Publications and source records attributed to Andrew Wynn.

25 records · Page 2Linked to original sources

arXiv2017

Exact energy stability of Bénard-Marangoni convection at infinite Prandtl number

Using the energy method we investigate the stability of pure conduction in Pearson's model for Bénard-Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh-Bénard convection setup, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations.

physics.flu-dyn↗

arXiv2017

Optimization with affine homogeneous quadratic integral inequality constraints

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs.

math.OC↗

arXiv2015

$β$-admissibility of observation and control operators for hypercontractive semigroups

We prove a Weiss conjecture on $β$-admissibility of control and observation operators for discrete and continuous $γ$-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case $γ=2$, which corresponds to the unweighted Bergman shift.

math.AP↗

arXiv2012

Observer design for systems with an energy-preserving non-linearity

Observer design is considered for a class of non-linear systems whose non-linear part is energy preserving. A strategy to construct convergent observers for this class of non-linear system is presented. The approach has the advantage that it is possible, via convex programming, to prove whether the constructed observer converges, in contrast to several existing approaches to observer design for non-linear systems. Finally, the developed methods are applied to the Lorenz attractor and to a low order model for shear fluid flow.

math.OC↗

arXiv2009

Composition operators on weighted Bergman spaces of a half plane

We use induction and interpolation techniques to prove that a composition operator induced by a map $ϕ$ is bounded on the weighted Bergman space $\A^2_α(\mathbb{H})$ of the right half-plane if and only if $ϕ$ fixes $\infty$ non-tangentially, and has a finite angular derivative $λ$ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by $λ^{(2+α)/2}$.

math.FA↗

arXiv2009

$α$-admissibility of the right-shift semigroup on $L^2(\mathbb{R}_+)$

It is shown that the right shift semigroup on $L^2(\mathbb{R}_+)$ does not satisfy the weighted Weiss conjecture for $α\in (0,1)$. In other words, $α$-admissibility of scalar valued observation operators cannot always be characterised by a simple resolvent growth condition. This result is in contrast to the unweighted case, where 0-admissibility can be characterised by a simple growth bound. The result is proved by providing a link between discrete and continuous $α$-admissibility and then translating a counterexample for the unilateral shift on $H^2(\mathbb{D})$ to continuous time systems.

math.FA↗

arXiv2009

Counterexamples to the discrete and continuous weighted Weiss conjectures

Counterexamples are presented to weighted forms of the Weiss conjecture in discrete and continuous time. In particular, for certain ranges of $α$, operators are constructed that satisfy a given resolvent estimate, but fail to be $α$-admissible. For $α\in (-1,0)$ the operators constructed are normal, while for $α\in (0,1)$ the operator is the unilateral shift on the Hardy space $H^2(\mathbb{D})$.

math.FA↗