International Workshop on Operator Theory (IWOTA) 2012
Thematic Session on Systems and Control Theory
July 16-18, 2012.

Session Organizer: Hendra Nurdin (The University of New South Wales)

Monday, July 16, 2012
  1. 16:30-17.00: Bill Helton, University of California San Diego
    Linear Matrix Inequalities in Matrix Variables.
    Abstract. Bill's talks at IWOTA will describe the recent development of free analogs of two different classical subjects, real algebraic geometry and convex optimization. This one focuses on recent developments surrounding LMIs which have unknowns which are matrices. It will be co-ordinated with Igor Klep's plenary talk. One of the main developments in optimization in the last 15 years is the rise of Linear Matrix Inequalities (LMIs). This tool now extends into most areas of science and engineering. Any problem treatable by LMIs is convex, but what about the converse? Convexity of a {\it free semi-algebraic set} $D:= \{X:p(X)\succ 0\}$ is of serious interest. Convexity turns out to be a very strong hypothesis. For instance, if $D$ is convex, bounded and contains $0$, then $D$ is the set of solutions to some (monic) LMI. Thus for free situations convexity and LMI techniques have the same scope. Less well understood are the possibilities for mappings between convex and not necessarily convex semi-algebraic sets as well as construction of free convex hulls. A bit will be said about motivation in optimization and engineering. System engineering problems described by a signal flow diagram lead to a feasible set described by free inequalities. In this setting, convexity is highly desirable, because it guarantees reliability in a numerical optimization. In another direction, free considerations provide a systematic framework for "relaxing" basic problems regarding LMIs. For example, these encompass Nemirovski's relaxation of the matrix cube problem arising in spectral synthesis. While relating to a variety of areas, the ideas and techniques are functional analytic in nature and have a decided operator systems - operator spaces and matrix convexity flavor. The talk describes recent results obtained jointly with Igor Klep, Scott McCullough, Harry Dym, Damon Hay, Chris Nelson, Nick Slingeland and Victor Vinnikov. For references see the arXiv.
  2. 17:10-17.40: Matthew James, The Australian National University
    Towards a Quantum-Classical Systems Theory
    Abstract. Classical systems and control theory is well developed with strong intellectual foundations and wide ranging applicability in engineering and elsewhere. Quantum technology is rapidly developing in laboratories around the world and quantum control theory has been progressing correspondingly. However, the approaches to quantum mechanics commonly used in physics differ significantly from classical systems. Furthermore, quantum technologies inevitably involve some combination of classical and quantum devices. It is therefore desirable to have an expanded framework for systems and control theory that seamlessly includes both quantum and classical systems. This talk will discuss these ideas using the framework of quantum probability.
  3. 17:50-18.20: Vladimir Gaitsgory, University of South Australia
    Linear Programming and Averaging Approaches to Singularly Perturbed Optimal Control Problems
    Abstract. Abstract. It has been some time since it was understood that a reduction technique based on equating of the singular perturbations parameter to zero may not lead to "near optimality" in nonlinear singularly perturbed (SP) optimal control problems (OCPs) and that, if this is the case, then an appropriate way of dealing with such problems is an averaging approach. In this presentation, we will revisit some of the existing results and discuss new results on averaging of SP OCPs based on linear programming relaxations of the latter. Theoretical results will be illustrated with numerical examples.
Tuesday, July 17, 2012

  1. 16:30-17.00: Ian Petersen, University of New South Wales at ADFA
    Robust Stability of Quantum Systems with a Nonlinear Coupling Operator
    Abstract. This talk considers the problem of robust stability for a class of uncertain quantum systems subject to unknown perturbations in the system coupling operator. A general stability result is given for a class of perturbations to the system coupling operator. Then, the special case of a nominal linear quantum system is considered with non-linear perturbations to the system coupling operator. In this case, a robust stability condition is given in terms of a scaled strict bounded real condition.
  2. 17:10-17.40: Victor Solo, The University of New South Wales
    Averaging Analysis Off SO(3); Stability of an Adaptive Attitude Estimation Algorithm
    Abstract. The attitude or pose of a rigid body is its orientation with respect to a fixed reference frame. In applications such as computer vision, robotics and satellite tracking the attitude has to be estimated in real time from body centered position measurements as well as known reference measurements such as star sight measurements. Other observations may also be available such as angular velocity. The fundamental object of interest is a time varying rotation matrix describing the rigid body motion. Here we sketch an averaging stability analysis of a new algorithm whichis unusual in not estimating the rotation matrix (or its quaternion equivalent) directly.
  3. 17:50-18.20: Rolf Gohm, Aberystwyth University
    Asymptotic Completeness for Weak Markov Processes
    Abstract. A weak Markov process in quantum probability for a discrete time parameter is essentially the same thing as a dilation of a row contraction by a row isometry. But the quantum probabilistic point of view suggests interesting additional questions. In this presentation we define subprocesses and quotient processes and the notion of a $\gamma$-cascade which allows a classification of the resulting structures. Considering the control theoretic notion of observability is particularly interesting for such a cascade. We show that if we start from a (noncommutative) Markov chain with an invariant state then we automatically get an associated subprocess and the problem of observability in this case is closely related to a theory of asymptotic completeness for Markov chains. This motivates a general definition of asymptotic completeness in the category of weak processes. Reference: R.G., Weak Markov Processes as Linear Systems, preprint
Wednesday, July 18, 2012
  1. 15:05-15.35: Bruce Watson, University of the Witwatersrand
    Eigencurves of non-definite Sturm-Liouville problems for the p-Laplacian
                  (λ,μ) eigencurves are studied for an indefinite weight
                  quasi-linear Sturm-Liouville-type problem of the form
                  −∆py = (p − 1)(λr − q − μs)sgny|y|p−1, on (0, 1) with
                  Sturmian-type boundary conditions (∆p being the
                  p-Laplacian). Joint work with Paul A. Binding and
                  Patrick J. Browne.

    (Joint work with Paul A. Binding and Patrick J. Browne)

  2. 16:25-16.55: John Gough, Aberystwyth University
    Fractional Linear Transformation for Quantum Feedback Networks
    Abstract. We present the theory of quantum feedback networks, and focus on the algebraic features emerging from model reduction through adiabatic elimination of slow degrees of freedom, feedback reduction, etc.
  3. 17:05-17:35: Hendra Nurdin, The University of New South Wales
    On Structure Preserving Transformations of the Ito Generator Matrix for Model Reduction of Quantum Feedback Networks
    Abstract. Two standard operations of model reduction for quantum feedback networks, elimination of internal connections under the instantaneous feedback limit, and adiabatic elimination of fast degrees of freedom, are cast as structure preserving transformations of It\={o} generator matrices. It is shown, under certain technical conditions, that the order in which they are applied is inconsequential. That is, the two model reduction operations can be commuted. Joint work with John Gough.
  4. 17:45-18.15: Sonja Currie, University of the Witwatersrand
    Quadratic Eigenvalue Problems for Second Order Systems
    Abstract. We consider the spectral structure of a quadratic second order system boundary-value problem. In particular we show that all but a finite number of the eigenvalues are real and semi-simple. We develop the eigencurve theory for these problems and show that the order of contact between an eigencurve and the parabola gives the Jordan chain associated with the eigenvector corresponding to that eigencurve. Following this we use our knowledge of the eigencurves to obtain eigenvalue asymptotics. Finally the completeness of the eigenfunctions is studied using operator matrix techniques. It should be noted here that the usual left definiteness assumptions have been overcome in this study.