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1. A Convex Reformulation of the Controller Synthesis Problem for MIMO Single-Delay Systems with Implementation in SOS

   Peet, M. (June 2018, Proceedings of the American Control Conference)

   In this paper, we propose a new dual class of stability condition for MIMO single-delay systems which is based on the implicit existence of a Lyapunov-Krasovskii functional but does not explicitly construct such a functional. This new type of stability condition allows the controller synthesis problem to be formulated as a convex optimization problem with little or no conservatism using a variable transformation. Furthermore, we show how to invert this variable transformation in order to obtain the stabilizing controller. The stability and controller synthesis conditions are then enforced using the SOS framework exploiting recent advances in this field. Numerical testing verifies there is little to no conservatism in either the “dual” stability test or the controller synthesis condition. 

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2. All real projective measurements can be self-tested

   https://doi.org/10.1038/s41567-024-02584-z  

   Chen, Ranyiliu; Mančinska, Laura; Volčič, Jurij (October 2024, Nature Physics)

   Abstract Entangled quantum systems feature non-local correlations that are stronger than could be realized classically. This property makes it possible to perform self-testing, the strongest form of quantum functionality verification, which allows a classical user to deduce the quantum state and measurements used to produce a given set of measurement statistics. While self-testing of quantum states is well understood, self-testing of measurements, especially in high dimensions, remains relatively unexplored. Here we prove that every real projective measurement can be self-tested. Our approach employs the idea that existing self-tests can be extended to verify additional untrusted measurements, known as post-hoc self-testing. We formalize the method of post-hoc self-testing and establish the condition under which it can be applied. Using this condition, we construct self-tests for all real projective measurements. We build on this result to develop an iterative self-testing technique that provides a clear methodology for constructing new self-tests from pre-existing ones. 

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3. Feedback control of transitional flows: A framework for controller verification using quadratic constraints

   https://doi.org/10.2514/6.2021-2825  

   Mushtaq, Talha; Seiler, Peter; Hemati, Maziar (July 2021, AIAA AVIATION 2021 FORUM)

   The dynamics of incompressible fluid flows are governed by a non-normal linear dynamical system in feedback with a static energy-conserving nonlinearity. These dynamics can be altered using feedback control but verifying performance of a given control law can be challenging. The conventional approach is to perform a campaign of high-fidelity direct numerical simulations to assess performance over a wide range of parameters and disturbance scenarios. In this paper, we propose an alternative simulation-free approach for controller verification. The incompressible Navier-Stokes equations are modeled as a linear system in feedback with a static and quadratic nonlinearity. The energy conserving property of this nonlinearity can be expressed as a set of quadratic constraints on the system, which allows us to perform a nonlinear stability analysis of the fluid dynamics with minimal complexity. In addition, the Reynolds number variations only influence the linear dynamics in the Navier-Stokes equations. Therefore, the fluid flow can be modeled as a parameter-varying linear system (with Reynolds number as the parameter) in feedback with a quadratic nonlinearity. The quadratic constraint framework is used to determine the range of Reynolds numbers over which a given flow will be stable, without resorting to numerical simulations. We demonstrate the framework on a reduced-order model of plane Couette flow. We show that our proposed method allows us to determine the critical Reynolds number, largest initial disturbance, and a range of parameter variations over which a given controller will stabilize the nonlinear dynamics. 

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4. Unified Necessary and Sufficient Conditions for the Robust Stability of Interconnected Sector-Bounded Systems

   https://doi.org/10.1109/CDC40024.2019.9029605  

   Cyrus, Saman; Lessard, Laurent (December 2019, IEEE Conference on Decision and Control)

   Classical conditions for ensuring the robust stability of a linear system in feedback with a sector-bounded nonlinearity include small gain, circle, passivity, and conicity theorems. In this work, we present a similar stability condition, but expressed in terms of relations defined on a general semi-inner product space. This increased generality leads to a clean result that can be specialized in a variety of ways. First, we show how to recover both sufficient and necessary-and-sufficient versions of the afore-mentioned classical results. Second, we show that suitably choosing the semi-inner product space leads to a new necessary and sufficient condition for weighted stability, which is in turn sufficient for exponential stability. 

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5. The non-symmetric strong multiplicity property for sign patterns

   https://doi.org/10.13001/ela.2025.8827  

   Curtis, Bryan; Garnett, Colin; Shader, Bryan L; Vander_Meulen, Kevin N (January 2025, The Electronic Journal of Linear Algebra)

   We develop a non-symmetric strong multiplicity property for matrices that may or may not be symmetric. We say a sign pattern allows the non-symmetric strong multiplicity property if there is a matrix with the non-symmetric strong multiplicity property that has the given sign pattern. We show that this property of a matrix pattern preserves multiplicities of eigenvalues for superpatterns of the pattern. We also provide a bifurcation lemma, showing that a matrix pattern with the property also allows refinements of the multiplicity list of eigenvalues. We conclude by demonstrating how this property can help with the inverse eigenvalue problem of determining the number of distinct eigenvalues allowed by a sign pattern. 

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