Unsolved Problems in Mathematical Systems and Control Theory

Unsolved Problems in Mathematical Systems and Control Theory

Vincent D. Blondel
Alexandre Megretski
Copyright Date: 2004
Pages: 360
  • Cite this Item
  • Book Info
    Unsolved Problems in Mathematical Systems and Control Theory
    Book Description:

    This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science.

    The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.

    eISBN: 978-1-4008-2615-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. iii-vi)
  2. Table of Contents
    (pp. vii-xii)
  3. Preface
    (pp. xiii-xiv)
    The editors
  4. Associate Editors
    (pp. xv-xvi)
  5. Website
    (pp. xvii-xviii)
    • Problem 1.1 Stability and composition of transfer functions
      (pp. 3-7)
      Guillermo Fernández-Anaya and Juan Carlos Martínez-García

      As far as the frequency-described continuous linear time-invariant systems are concerned, the study of control-oriented properties (like stability) resulting from the substitution of the complex Laplace variable s by rational transfer functions have been little studied by the Automatic Control community. However, some interesting results have recently been published:

      Concerning the study of the so-called uniform systems, i.e., LTI systems consisting of identical components and amplifiers, it was established in [8] a general criterion for robust stability for rational functions of the form$D(f(s))$, where$D(s)$is a polynomial and$f(s)$is a rational transfer function. By applying such...

    • Problem 1.2 The realization problem for Herglotz-Nevanlinna functions
      (pp. 8-13)
      Seppo Hassi, Henk de Snoo and Eduard Tsekanovskiῐ

      Roughly speaking, realization theory concerns itself with identifying a given holomorphic function as the transfer function of a system or as its linear fractional transformation. Linear, conservative, time-invariant systems whose main operator isboundedhave been investigated thoroughly. However, many realizations in different areas of mathematics including system theory, electrical engineering, and scattering theory involveunboundedmain operators, and a complete theory is still lacking. The aim of the present proposal is to outline the necessary steps needed to obtain a general realization theory along the lines of M. S. Brodskiῐ and M. S. Livšic [8], [9], [16], who have...

    • Problem 1.3 Does any analytic contractive operator function on the polydisk have a dissipative scattering nD realization?
      (pp. 14-17)
      Dmitry S. Kalyuzhniy-Verbovetzky

      Let$(x,u,y)$be finite-dimensional or infinite-dimensional separable Hilbert spaces. Consider nD linear systems of the form

      $\alpha :\;\left\{ {\begin{array}{*{20}{c}} {x(t)\;\; = \;\;\sum\limits_{k = 1}^n {({A_k}x(t - {e_k})\; + \;{B_k}u(t - {e_k})),} } \\ {y(t)\;\; = \;\;\sum\limits_{k = 1}^n {({C_k}x(t - {e_k})\; + \;{D_k}u(t - {e_k})),} } \\ \end{array} } \right.\;\;(t\; \in \;{\mathbb{Z}^n}:\;\sum\limits_{k = 1}^n {{t_k}} > \;0)$(1)

      where${e_k}: = (0,\; \ldots ,\;0,\;1,\;0,\; \ldots ,\;0)\; \in \;{\mathbb{Z}^n}$(here unit is on the$k$-th place), for all$\;t\; \in \;{\mathbb{Z}^n}$such that$\sum\nolimits_{k = 1}^n {{t_k} \geqslant 0} $one has$x(t)\; \in \;x$(the state space),$u(t)\; \in \;u$(the input space),$y(t)\; \in \;y$(the output space),${A_k},{B_k},{C_k},{D_k}$are bounded linear operators, i.e.,${A_k}\; \in \;L(x),\;{B_k} \in L(u,\;x),\;{C_k} \in L(x,\;y),\;{D_k}\; \in \;L(u,\;y)$for all$k\; \in \;\{ 1,\; \ldots ,\;n\} $. We use the notation$\alpha \; = \;(n;\;{\mathbf{A}},\;{\mathbf{B}},\;{\mathbf{C}},\;{\mathbf{D}};\;x,\;u,\;y)$for such a system (here${\mathbf{A}}: = ({A_1},\; \ldots ,\;{A_n})$, etc.)....

    • Problem 1.4 Partial disturbance decoupling with stability
      (pp. 18-21)
      Juan Carlos Martínez-García, Michel Malabre and Vladimir Kučera

      Consider a linear time-invariant system$(A,B,C,E)$described by:

      $\left\{ {\begin{array}{*{20}{c}} {\sigma x(t)\; = \;Ax\,(t)\; + \;Bu\,(t)\; + \;Ed\,(t),} \\ {z(t)\; = \;Cx(t),} \\ \end{array} } \right.$(1)

      where σ denotes either the derivation or the shift operator, depending on the continuous-time or discrete-time context;$x(t)\; \in \;x\; \simeq \;{\mathbb{R}^n}$denotes the state;$u(t)\; \in \;u\; \simeq \;{\mathbb{R}^m}$denotes the control input;$z(t)\; \in \;z\; \simeq \;{\mathbb{R}^m}$denotes the output, and$d(t)\; \in \;D \simeq \;{\mathbb{R}^p}$denotes the disturbance.$A:\;x\; \to \;x,\;B:\;u\; \to \;x,$$C:\;x\; \to \;z$, and$E:\;D\; \to \;x$denote linear maps represented by real constant matrices.

      Let a system$(A,B,C,E)$and an integer$k \le 1$be given. Find necessary and sufficient conditions for the existence of a static state feedback control law$u(t) = Fx(t)\; + \;Gd(t)$, where$F:\;x\; \to \;u$and$G:\;D\; \to \;u$are linear maps such as zeroing the first$k$...

    • Problem 1.5 Is Monopoli’s model reference adaptive controller correct?
      (pp. 22-28)
      A. S. Morse

      In 1974 R. V. Monopoli published a paper [1] in which he posed the now classical model reference adaptive control problem, proposed a solution and presented arguments intended to establish the solution’s correctness. Subsequent research [2] revealed a flaw in his proof, which placed in doubt the correctness of the solution he proposed. Although provably correct solutions to the model reference adaptive control problem now exist (see [3] and the references therein), the problem of deciding whether or not Monopoli’s original proposed solution is in fact correct remains unsolved. The aim of this note is to review the formulation of...

    • Problem 1.6 Model reduction of delay systems
      (pp. 29-32)
      Jonathan R. Partington

      Our concern here is with stable single input single output delay systems, and we shall restrict to the case when the system has a transfer function of the form$G(s)\; = \;{e^{ - sT}}R(s)$, with$T\; > \;0$and$R$rational, stable, and strictly proper, thus bounded and analytic on the right half plane${\mathbb{C}_ + }$. It is a fundamental problem in robust control design to approximate such systems by finite-dimensional systems. Thus, for a fixed natural number$n$, we wish to find a rational approximant${G_n}(s)$of degree at most$n$in order to make small the approximation error...

    • Problem 1.7 Schur extremal problems
      (pp. 33-35)
      Lev Sakhnovich

      In this paper we consider the well-known Schur problem the solution of which satisfy in addition the extremal condition

      $w{\kern 1pt} *(z)w(z)\; \leqslant \;\rho _{min}^2,\;\;|\,z\,|\; < \;1$, (1)

      where$w(z)$and${\rho _{min}}$are$m\; \times \;m$matrices and${\rho _{min}} > \;0$. Here the matrix${\rho _{min}}$is defined by a certain minimal-rank condition (see Definition 1). We remark that the extremal Schur problem is a particular case. The general case is considered in book [1] and paper [2]. Our approach to the extremal problems does not coincide with the superoptimal approach [3],[4]. In paper [2] we compare our approach to the extremal problems with the superoptimal approach. Interpolation has found great applications in...

    • Problem 1.8 The elusive iff test for time-controllability of behaviors
      (pp. 36-39)
      Amol J. Sasane

      Problem: Let$R\; \in \;\mathbb{C}{[{\eta _1},\; \ldots ,\;{\eta _{\text{m}}},\;\xi ]^{{\text{g}}\; \times \;{\text{w}}}}$and let$\mathfrak{B}$be the behavior given by the kernel representation corresponding to$R$. Find an algebraic test on$R$characterizing the time-controllability of$\mathfrak{B}$.

      In the above, we assume$\mathfrak{B}$to comprise of only smooth trajectories, that is,

      $\mathfrak{B} = \{ w\; \in \;{C^\infty }\;({\mathbb{R}^{{\text{m + 1}}}},\;{\mathbb{C}^{\text{w}}})\;|\;{D_R}w\; = \;0\} $,

      where${D_R}:\;{C^\infty }\;({\mathbb{R}^{{\text{m}} + 1}},\;{\mathbb{C}^{\text{w}}})\; \to \;{{\text{C}}^\infty }\;({\mathbb{R}^{{\text{m}} + 1}},\;{\mathbb{C}^{\text{g}}})$is the differential map that acts as follows: if$R\; = \;{[{r_{{\text{ij}}}}]_{{\text{g}}\; \times \;{\text{w}}}}$, then

      ${D_R}\;\left[ {\begin{array}{*{20}{c}} {{w_1}} \\ \vdots \\ {{w_{\text{w}}}} \\ \end{array} } \right] = \;\left[ {\begin{array}{*{20}{c}} {\sum\nolimits_{{\text{k}} = 1}^{\text{w}} {{r_{1{\text{k}}}}\;\left( {\frac{\partial } {{\partial {x_1}}},\; \ldots ,\;\frac{\partial } {{\partial {x_{\text{m}}}}},\;\frac{\partial } {{\partial t}}} \right)\;{w_{\text{k}}}} } \\ \vdots \\ {\sum\nolimits_{{\text{k}} = 1}^{\text{w}} {{r_{{\text{gk}}}}\;\left( {\frac{\partial } {{\partial {x_1}}},\; \ldots ,\;\frac{\partial } {{\partial {x_{\text{m}}}}},\;\frac{\partial } {{\partial t}}} \right)\;{w_{\text{k}}}} } \\ \end{array} } \right]\;$.

      Time-controllability is a property of the behavior, defined as follows. The behavior$\mathfrak{B}$is said to betime-controllableif for any${w_1}$and${w_2}$in$\mathfrak{B}$, there exits a$w\; \in \;\mathfrak{B}$and a$\tau \geqslant 0$such that

      $w( \bullet ,\;t)\; = \;\left\{ {\begin{array}{*{20}{c}} {{w_1}( \bullet ,\;t)} & {{\text{for}}\;{\text{all}}\;t\; \leqslant \;0} \\ {{w_2}( \bullet ,\;t - \tau )} & {{\text{for}}\;{\text{all}}\;t\; \geqslant \;\tau } \\ \end{array} } \right.$.

      The behavioral theory for systems described...

    • Problem 1.9 A Farkas lemma for behavioral inequalities
      (pp. 40-43)
      A. A. (Tonny) ten Dam and J. W. (Hans) Nieuwenhuis

      Within the systems and control community there has always been an interest in minimality issues. In this chapter we conjecture a Farkas Lemma for behavioral inequalities that, when true, will allow to study minimality and elimation issues for behavioral systems described by inequalities.

      Let${\mathbb{R}^{n\, \times \,m}}[s,\;{s^{ - 1}}]$denote the$(n\; \times \;m)$polynomial matrices with real coefficients and positive and negative powers in the indeterminate$s$. Let$\mathbb{R}_ + ^{n\, \times \,m}[s,\;{s^{ - 1}}]$denote the set of matrices in${\mathbb{R}^{n\, \times \,m}}[s,\;{s^{ - 1}}]$with non-negative coefficients only. In this chapter we consider discrete-time systems with time-axis$\mathbb{Z}$. Let$\sigma $denote the (backward) shift operator, and let$R(\sigma ,\;{\sigma ^{ - 1}})$denote polynomial operators in...

    • Problem 1.10 Regular feedback implementability of linear differential behaviors
      (pp. 44-48)
      H. L. Trentelman

      In this short paper, we want to discuss an open problem that appears in the context of interconnection of systems in a behavioral framework. Given a system behavior, playing the role of plant to be controlled, the problem is to characterize all system behaviors that can be achieved by interconnecting the plant behavior with a controller behavior, where the interconnection should be a regular feedback interconnection.

      More specifically, we will deal with linear time-invariant differential systems, i.e., dynamical systems$\Sigma $given as a triple$\{ \mathbb{R},\;{\mathbb{R}^{\text{w}}},\;B)$, where$\mathbb{R}$is the timeaxis, and where$B$, called thebehaviorof the...

    • Problem 1.11 Riccati stability
      (pp. 49-53)
      Erik I. Verriest

      Given two$n\; \times \;n$real matrices,$A$and$B$, consider the matrix Riccati equation

      $A'P\; + \;PA\; + \;Q\; + \;PB{Q^{ - 1}}B'P\; + \;R\; = \;0$. (1)

      Can onecharacterizethe pairs$(A,B)$for which the above equation has a solution for positive definite symmetric matrices$P,Q,$and$R$?

      In [8] a pair$(A,B)$was defined to beRiccati stableif a triple of positive definite matrices$P,Q,R$exists such that (1) holds.

      The problem may be stated equivalently as an LMI:

      Can one characterize all pairs$(A,B)$without invoking additional matrices, for which there exist positive definite matrices$P$and$Q$such that

      $\left[ {\begin{array}{*{20}{c}} {A'P\; + \;PA\; + \;Q} & {PB} \\ {B'P} & { - Q} \\ \end{array} } \right]\; < \;0$. (2)

      Equation (1)...

    • Problem 1.12 State and first order representations
      (pp. 54-57)
      Jan C. Willems

      We conjecture that the solution set of a system of linear constant coefficient PDEs is Markovian if and only if it is the solution set of a system of first order PDEs. An analogous conjecture regarding state systems is also made.

      First, we introduce our notation for the solution sets of linear PDEs in the n real independent variables$x = ({x_1},\; \ldots ,\;{x_{\text{n}}})$. Let${{\mathfrak{D}'}_{\text{n}}}$denote, as usual, the set of real distributions on${\mathbb{R}^{\text{n}}}$, and$\mathfrak{L}_{\text{n}}^{\text{w}}$the linear subspaces of${({{\mathfrak{D}'}_{\text{n}}})^w}$consisting of the solutions of a system of linear constant coefficient PDEs in...

    • Problem 1.13 Projection of state space realizations
      (pp. 58-64)
      Antoine Vandendorpe and Paul Van Dooren

      We consider two$m\; \times \;p$strictly proper transfer functions

      $T(s)\; = \;C{(s{I_n} - \;A)^{ - 1}}B,\;\;\;\hat T(s)\; = \;\hat C{(s{I_k} - \hat A)^{ - 1}}\hat B$, (1)

      of respective Mc Millan degrees$n$and$k < n$. We want to characterize the set of projecting matrices$Z,\;V\; \in \;{\mathbb{C}^{n\, \times \,k}}$such that

      $\hat C\; = \;CV,\;\;\;\hat A = {Z^T}AV,\;\;\;\hat B = {Z^T}B,\;\;\;\;{Z^T}V = {I_k}$. (2)

      Given only$T(s)$, we are interested in characterizing the set of all transfer functions$\hat T(s)$that can be obtained via the projection equations (1,2). Here is our first conjecture.

      Conjecture 1. Any minimal state space realization of$\hat T(s)$can be obtained by a projection from any minimal state space realization of$T(s)$if

      $\frac{{m\; + \;p}} {2} \leqslant n - k$. (3)

      In the case that condition (3) is not satisfied, we give a...

    • Problem 2.1 On error of estimation and minimum of cost for wide band noise driven systems
      (pp. 67-70)
      Agamirza E. Bashirov

      The suggested open problem concerns the error of estimation and the minimum of the cost in the filtering and optimal control problems for a partially observable linear system corrupted by wide band noise processes.

      Recent results allow to construct a wide band noise process in a certain integral form on the basis of its autocovariance function and design the optimal filter and the optimal control for a partially observable linear system corrupted by such wide band noise processes. Moreover, explicit formulae for the error of estimation and for the minimum of the cost have been obtained.

      But, the information about...

    • Problem 2.2 On the stability of random matrices
      (pp. 71-75)
      Giuseppe C. Calafiore and Fabrizio Dabbene

      In the theory of linear systems, the problem of assessing whether the omogeneous system$\dot x = Ax$,$A\; \in \;{\mathbb{R}^{n,\,n}}$, is asymptotically stable is a well understood (and fundamental) one. Of course, the system (and we shall say also the matrix$A$) is stable if and only if Re${\lambda _i} < \;0,\;i\; = \;1,\; \ldots ,\;n,$, being${\lambda _i}$the eigenvalues of$A$.

      Evolving from this basic notion, much research effort has been devoted in recent years to the study ofrobuststability of a system. Without entering in the details of more than thirty years of fruitful research, we could...

    • Problem 2.3 Aspects of Fisher geometry for stochastic linear systems
      (pp. 76-81)
      Bernard Hanzon and Ralf Peeters

      Consider the space$S$of stable minimum phase systems in discrete-time, of order (McMillan degree)$n$, having$m$inputs and$m$outputs, driven by a stationary Gaussian white noise (innovations) process of zero mean and covariance$\Omega $. This space is often considered, for instance in system identifi-cation, to characterize stochastic processes by means of linear time-invariant dynamical systems (see [8, 18]). The space$S$is well known to exhibit a differentiable manifold structure (cf. [5]), which can be endowed with a notion of distance between systems, for instance by means of a Riemannian metric, in various meaningful ways....

    • Problem 2.4 On the convergence of normal forms for analytic control systems
      (pp. 82-86)
      Wei Kang and Arthur J. Krener

      A fruitful technique for the local analysis of a dynamical system consists of using a local change of coordinates to transform the system to a simpler form, which is called a normal form. The qualitative behavior of the original system is equivalent to that of its normal form which may be easier to analyze. A bifurcation of a parameterized dynamics occurs when a change in the parameter leads to a change in its qualitative properties. Therefore, normal forms are useful in analyzing when and how a bifurcation will occur. In his dissertation, Poincaré studied the problem of linearizing...

    • Problem 3.1 Minimum time control of the Kepler equation
      (pp. 89-92)
      Jean-Baptiste Caillau, Joseph Gergaud and Joseph Noailles

      We consider the controlled Kepler equation in three dimensions

      $\ddot r = - k\frac{r} {{|\,r\,{|^3}}} + \gamma $(1)

      where$r = ({r_1},{r_2},{r_3})$is the position vector–the double dot denoting the second order time derivative–,$k$a strictly positive constant,$\left| . \right|$the Euclidean norm in R³, and where$\gamma ({\gamma _1},\;{\gamma _2},\;{\gamma _3})$is the control. The minimum time problem is then stated as follows: find a positive time$T$and a measurable function$\gamma $defined on$[0,T]$such that (1) holds almost everywhere on$[0,T]$and:

      $T \to \min $

      $r(0)\; = \;{r^0},\;\;\dot r(0)\; = \;{{\dot r}^0}$(2)

      $h(r(T),\;\dot r(T)) = 0$(3)

      $\,\gamma \,\; \leqslant \;\Gamma $. (4)

      In (2),${{\text{r}}^0}$and${{\dot r}^0}$are the known initial...

    • Problem 3.2 Linearization of linearly controllable systems
      (pp. 93-96)
      R. Devanathan

      We consider a class of systems of the form

      $\dot \xi = f(\xi )\; + \;g(\xi )\zeta $(1)

      where$\xi $is an$n$-tuple vector and$f(\xi )$and$g(\xi )$are vector fields, i.e.,$n$-tuple vectors whose elements are, in general, functions of$\xi $. For simplicity, we assume a scalar input$\zeta $. We require that the system (1) be linearly controllable [1], i.e., the pair$(F,G)$is controllable where$F = \frac{{\partial f}} {{\partial \xi }}(0)$and$G = g(0)$at the assumed equilibrium point at the origin.

      The power series expansion of (1) about the origin can be written,with an appropriate change of variable and input, as

      $\dot x = Fx\; + \;G\phi \; + \;{O_1}{(x)^{(2)}} + {\gamma _1}{(x,\;\phi )^{(1)}}$(2)

      where, without...

    • Problem 3.3 Bases for Lie algebras and a continuous CBH formula
      (pp. 97-102)
      Matthias Kawski

      Many time-varying linear systems$\dot x\; = \;F(t,\;x)$naturally split into timeinvariant geometric components and time-dependent parameters. A special case are nonlinear control systems that are affine in the control$u$, and specified by analytic vector fields on a manifold${M^n}$

      $\dot x\; = \;{f_0}(x)\; + \;\sum\limits_{k = 1}^m {{u_k}} {f_k}(x)$. (1)

      It is natural to search for solution formulas for$x(t)\; = \;x(t,\;u)$that separate the time-dependent contributions of the controls$u$from the invariant, geometric role of the vector fields${f_k}$. Ideally, one may be able to a priori obtain closed-form expressions for the flows of certain vector fields. The quadratures of the control might be done in real-time, or the...

    • Problem 3.4 An extended gradient conjecture
      (pp. 103-106)
      Luiz Carlos Martins Jr. and Geraldo Nunes Silva

      Let$f:\;{\mathbb{R}^n} \to \mathbb{R}$be a locally Lipschitz function, i.e., for all$x\; \in \;\mathbb{R}$there is$\varepsilon \; > \;0$and a constant$K$depending on$\varepsilon$such that

      $\,f({x_1})\; - \;f({x_2})\,|\; \leqslant \;K\parallel \,{x_1}\; - \;{x_2}\,\parallel ,\;\forall {x_1},\;{x_2}\; \in \;x\; + \;\varepsilon B$.

      Here$B$denotes the open unit ball of${\mathbb{R}^n}$.

      Let$v\; \in \;{\mathbb{R}^n}$. Thegeneralized directional derivativeof$f$at$x$, in the direction$v$, denoted by${f^0}(x;v)$, is defined as follows:

      ${f^0}(x;\:v)\: = \mathop {\mathop {{\text{lim}}}\limits_{y \to x} }\limits_{s \to {0^ + }} {\text{sup}}\frac{{f(y\; + \;sv)\; - \;f(y)}} {s}$.

      Here$y\; \in \;{\mathbb{R}^n}$,$s\; \in \;(0,\; + \infty )$. Thegeneralized gradientof$f$at$x$, denoted by$\partial f(x)$is the subset of${\mathbb{R}^n}$given by

      $\{ \xi \; \in \;{\mathbb{R}^n}:\;{f^0}(x;\;v)\; \geqslant \;\langle \xi ,\;v\rangle ,\;\;\;\;\forall v\; \in \;\mathbb{R}\} $.

      For the properties and basic calculus of the generalized gradient,standard references are [1] and[2].

      The problem we propose here is...

    • Problem 3.5 Optimal transaction costs from a Stackelberg perspective
      (pp. 107-110)
      Geert Jan Olsder

      The problem to be considered is

      $\dot x = f(x,\;u),\;x(0)\; = \;{x_0}$, (1)

      $\mathop {{\text{max}}}\limits_u \;{J_{\text{F}}} = \mathop {{\text{max}}}\limits_u {\text{\{ }}q(x(T))\; + \;\int_0^T {g(x,\;u)} {\text{d}}t - \int_0^T {\gamma (u(t)){\text{d}}t\} } $, (2)

      $\mathop {{\text{max}}}\limits_{\gamma ( \cdot )} \;{J_{\text{L}}} = \mathop {{\text{max}}}\limits_{\gamma ( \cdot )} \;\int_0^T {\gamma (u(t)){\text{d}}t} $, (3)

      with$f,g$and$q$being given functions, the state$x\; \in \;{R^n}$, the control$u\: \in \:R$, and$\gamma ( \cdot )$is a scalar function which maps$R$into$R$. The problem concerns a dynamic game problem in which$u$is the decision variable of one player called the Follower, and the function$\gamma $is up to the choice of the other player called the Leader. An essential feature of the problem is that the Leader’s profit (3) is a direct loss for the Follower in (2). The Leader lives as a parasite...

    • Problem 3.6 Does cheap control solve a singular nonlinear quadratic problem?
      (pp. 111-113)
      Yuri V. Orlov

      A standard control synthesis for affine systems

      $\dot x = f(x)\; + \;g(x)u,\;\;x\; \in \;{R^n},\;u\; \in \;{R^m}$(1)

      under degenerate perfomance criterion

      $J(u)\; = \;\int_0^\infty {{x^T}(t)\;Px(t)\,dt,\;\;P = {P^T} > \;0} $(2)

      depending on the state vector$x(t)$only, replaces this singular optimization problem by its regularization through$\varepsilon $-approximation

      ${J_\varepsilon }(u) = \int_0^\infty {[{x^T}(t)Px(t)} \; + \;\varepsilon {u^T}(t)\;Ru(t)]dt,\;\;\varepsilon \; > \;0,\;R = {R^T} > \;0$(3)

      of this criterion with small (cheap) penalty on the control input$u$. Hereafter, functions$f,g$, are assumed sufficiently smooth, and all quantities in (1)through(3) are assumed to have compatible dimensions.

      The optimal control synthesis corresponding to (2) is then obtained as a limit as$\varepsilon \to 0$of the optimal control law$u_\varepsilon ^0$corresponding to (3). Since only particular approximation is taken while other approximations...

    • Problem 3.7 Delta-Sigma modulator synthesis
      (pp. 114-116)
      Anders Rantzer

      Delta-Sigma modulators are among the key components in modern electronics. Their main purpose is to provide cheap conversion from analog to digital signals. In the figure below, the analog signal$r$with values in the interval [-1, 1] is supposed to be approximated by the digital signal$d$that takes only two values, -1 and 1. One cannot expect good approximation at all frequencies. Hence, the dynamic system$D$should be chosen to minimize the error$f$in a given frequency range$[{\omega _1},\;{\omega _2}]$.

      There is a rich literature on Delta-Sigma modulators. See [2, 1] and references therein. The purpose...

    • Problem 3.8 Determining of various asymptotics of solutions of nonlinear time-optimal problems via right ideals in the moment algebra
      (pp. 117-121)
      G. M. Sklyar and S. Yu. Ignatovich

      The time-optimal control problem is one of the most natural and at the same time difficult problems in the optimal control theory.

      For linear systems, the maximum principle allows to indicate a class of optimal controls. However, the explicit form of the solution can be given only in a number of particular cases [1-3]. At the same time [4], an arbitrary linear time-optimal problem with analytic coefficients can be approximated (in a neighborhood of the origin) by a certain linear problem of the form

      ${{\dot x}_i} = - {t^{qi}}u,\;\,i\; = \;1,\; \ldots ,\;n,\;\,{q_1} < \cdots < \;{q_n},\,\;x(0)\; = \;{x^0},\;x(\theta )\; = \;0$,

      $\,u\,|\; \leqslant \;1,\;\theta \; \to \;{\text{min}}$. (1)

      In the nonlinear case, the careful analysis is required for any particular system...

    • Problem 3.9 Dynamics of principal and minor component flows
      (pp. 122-128)
      U. Helmke, S. Yoshizawa, R. Evans, J. H. Manton and I. M. Y. Mareels

      Stochastic subspace tracking algorithms in signal processing and neural networks are often analyzed by studying the associated matrix differential equations. Such gradient-like nonlinear differential equations have an intricate convergence behavior that is reminiscent of matrix Riccati equations. In fact, these types of systems are closely related. We describe a number of open research problems concerning the dynamics of such flows for principal and minor component analysis.

      Principal component analysis is a widely used method in neural networks, signal processing, and statistics for extracting the dominant eigenvalues of the covariance matrix of a sequence of random vectors. In the literature, various...

    • Problem 4.1 ${\mathcal{L}_2}$-induced gains of switched linear systems
      (pp. 131-133)
      João P. Hespanha

      In the 1999 collection of Open Problems in Mathematical Systems and Control Theory, we proposed the problem of computing input-output gains of switched linear systems. Recent developments provided new insights into this problem leading to new questions.

      Aswitched linear systemis defined by a parameterized family of realizations$\{ ({A_p},\;{B_p},\;{C_p},\;{D_p})\;\;:\;\;p\; \in \;P\} $, together with a family of piecewise constantswitching signals${\text{S}}: = \;\left\{ {\sigma \;:\;\left[ {0,\;\infty } \right)\; \to \;P} \right\}$. Here we consider switched systems for which all the matrices${A_p},\;p\; \in \;P$are Hurwitz. The corresponding switched system is represented by

      $\dot x\; = \;{A_\sigma }x\; + \;{B_\sigma }u,\quad \quad y = {C_\sigma }x\; + \;{D_\sigma }u,\quad \quad \sigma \in {\text{S}}$(1)

      and by asolution to(1), we mean a pair$(x,\;\sigma )$for...

    • Problem 4.2 The state partitioning problem of quantized systems
      (pp. 134-139)
      Jan Lunze

      Consider a continuous system whose state can only be accessed through a quantizer. The quantizer is defined by a partition of the state space. The system generates an event if the system trajectory crosses the boundary between adjacent partitions.

      The problem concerns the prediction of the event sequence generated by the system for a given initial event. As the initial event does not define the initial system state unambiguously but only restricts the initial state to a partition boundary, when predicting the system behavior the bundle of all state trajectories have to be considered that start on this partition boundary....

    • Problem 4.3 Feedback control in flowshops
      (pp. 140-143)
      S. P. Sethi and Q. Zhang

      Consider a manufacturing system producing a single finished product using$m$machines in tandem that are subject to breakdown and repair. We are given a finite-state Markov chain$\alpha ( \cdot )\; = \;({\alpha _1}( \cdot ),\; \ldots ,\;{\alpha _m}( \cdot ))$on a probability space$(\Omega ,\;F,\;P)$, where${\alpha _i}(t),\;i\; = \;1,\; \ldots ,\;m$, is the capacity of the$i$-th machine at time$t$. We use${u_i}(t)$to denote the input rate to the$i$-th machine,$i\; = \;1,\; \ldots ,\;m$, and${x_i}(t)$to denote the number of parts in...

    • Problem 4.4 Decentralized control with communication between controllers
      (pp. 144-150)
      Jan H. van Schuppen

      Consider a control system with inputs from$r$different controllers. Each controller has partial observations of the system and the partial observations of each pair of controllers is different. The controllers are allowed to exchange online information on their partial observations, state estimates, or input values, but there are constraints on the communication channels between each tuple of controllers. In addition, there is specified a control objective.

      The problem is to synthesize$r$controllers and a communication protocol for each directed tuple of controllers, such that when the controllers all use their received communications the control objective is met as...

    • Problem 5.1 Infinite dimensional backstepping for nonlinear parabolic PDEs
      (pp. 153-159)
      Andras Balogh and Miroslav Krstic

      This note explores an approach to global stabilization of boundary controlled nonlinear PDEs by a technique inspired by finite dimensional backstepping/feedback linearization. Solution of the problem presented herein would be of enormous significance because these are the only truly constructive and systematic techniques in finite dimension.

      We consider nonlinear parabolic PDEs of the form

      ${u_t}\;(x,\;t)\; = \;\varepsilon {u_{x\,x}}(x,\;t)\; + \;f(u\;(x,\;t))$(1)

      for$x\; \in \;(0,\;1),\;t\; > \;0$, with boundary conditions

      $u\;(0,\;t)\; = \;0$, (2)

      $u\;(1,\;t)\; = \;{\alpha _1}\,(u)$, (3)

      initial condition

      $u\,(x,\;0)\; = \;{u_0}(x),\quad \;\;x\; \in \;[0,\;1]$,

      and under the assumption

      $\varepsilon \; > \;0,\;\;f\; \in \;{C^\infty }(\mathbb{R})$

      The task is to derive a nonlinear (feedback) functional${\alpha _1}:\;C\;([0,\;1])\; \to \;\mathbb{R}$that stabilizes the trivial solution$u\,(x,\;t)\; \equiv \;0$...

    • Problem 5.2 The dynamical Lame system with boundary control: on the structure of reachable sets
      (pp. 160-162)
      M. I. Belishev

      The questions posed below come from dynamical inverse problems for the hyperbolic systems with boundary control. These questions arise in the framework of the BC–method, which is an approach to inverse problems based on their relations to the boundary control theory [1], [2].

      Let$\Omega \; \subset \;{{\mathbf{R}}^3}$be a bounded domain with the smooth (enough) boundary$\Gamma ;\;\lambda ,\;\mu ,\;\rho $smooth functions (Lame parameters) satisfying$\rho > 0,\;\mu \; > \;0,\;3\lambda \; + 2\mu > \;0$in$\Omega $

      The parameters determine two metrics in${\bar \Omega }$

      $dl_\alpha ^2 = \frac{{|\,dx\,{|^2}}} {{c_\alpha ^2}},\quad \quad \alpha = p,\;s$

      where${c_p}: = \;{\left( {\frac{{\lambda \; + \;2\mu }} {\rho }} \right)^{\tfrac{1} {2}}},\quad {c_s}: = {\left( {\frac{\mu } {\rho }} \right)^{\tfrac{1} {2}}}$

      are the velocities of p− (pressure) and s− (shear) waves; let...

    • Problem 5.3 Null-controllability of the heat equation in unbounded domains
      (pp. 163-168)
      Sorin Micu and Enrique Zuazua

      Let$\Omega $be a smooth domain of${\mathbb{R}^n}$with$n \geqslant 1$. Given T > 0 and${\Gamma _0} \subset \;\partial {\kern 1pt} \Omega $an open non-empty subset of the boundary of$\Omega $we consider the linear heat equation:

      $\left\{ {\begin{array}{*{20}{c}} {{u_t} - \Delta u = 0} & {{\text{in}}} & Q \\ {u = v{1_{{\Sigma _0}}}} & {{\text{on}}} & \Sigma \\ {u(x,\;0)\; = \;{u_0}(x)} & {{\text{in}}} & {\Omega ,} \\ \end{array} } \right.$(1)

      where$Q\; = \;\Omega \; \times \;(0,\;T),\;\Sigma \; = \;\partial {\kern 1pt} \Omega \; \times \;(0,\;T)$and${\Sigma _0} = {\Gamma _0}\; \times \;(0,\;T)$and where${1_{{\Sigma _0}}}$denotes the characteristic function of the subset${\Sigma _0}$of$\Sigma $.

      In (1)$v\; \in \;{L^2}(\Sigma )$is aboundary controlthat acts on the system through the subset${\Sigma _0}$of the boundary and$u\; = \;u(x,\;t)$is thestate.

      System (1) is said to benull-controllableat time$T$if for any${u_0}\; \in \;{L^2}(\Omega )$there exists a control$v\; \in \;{L^2}({\Sigma _0})$such that...

    • Problem 5.4 Is the conservative wave equation regular?
      (pp. 169-172)
      George Weiss

      We consider an infinite-dimensional system described by the wave equation on an$n$–dimensional domain, with mixed boundary control and mixed boundary observation, which has been analyzed (as an example for a certain class of conservative linear systems) in [13]. A somewhat simpler version of this system has appeared (also as an example) in the paper [11, section 7] and a related system has been discussed in [5].

      We assume that$\Omega \subset \;{\mathbb{R}^n}$is a bounded domain with Lipschitz boundary$\Gamma $, as defined in Grisvard [3]. This means that, locally, after a suitable rotation of the orthogonal coordinate system, the boundary...

    • Problem 5.5 Exact controllability of the semi-linear wave equation
      (pp. 173-178)
      Xu Zhang and Enrique Zuazua

      Let$T > 0$and$\Omega \; \subset \;{{\mathbf{R}}^n}\;(n\; \in \;{\mathbf{N}}{\text{)}}$be a bounded domain with a${C^{1,1}}$boundary$\partial {\kern 1pt} \Omega $. Let$\omega $be a proper subdomain of$\Omega $and denote the characteristic function of the set$\omega $by${\chi _\omega }$. Fix a nonlinear function$f\; \in \;{C^1}({\mathbf{R}}{\text{)}}$.

      We are concerned with the exact controllability of the following semilinear wave equation:

      $\left\{ {\begin{array}{*{20}{c}} {{y_{tt}} - \Delta y + f(y) = {\chi _\omega }(x)\:\:u(t,x)} & {{\rm{in}}} & {(0,T) \times \Omega ,} \\ {y = 0} & {{\rm{on}}} & {(0,T) \times \partial \Omega } \\ {y(0) = {y_0},{y_t}(0) = {y_1}} & {{\rm{in}}} & {\Omega .} \\ \end{array}} \right.$, (1)

      In (1),$(y(t,\; \cdot ),\;{y_t}(t,\; \cdot ))$, is thestateand$u(t,\; \cdot )$is thecontrolthat acts on the system through the subset$\omega $of$\omega $.

      In what follows, we choose thestate spaceand thecontrol spaceas$H_0^1(\Omega )\; \times \;{L^2}(\Omega )$and${L^2}((0,\;T)\; \times \;\Omega )$, respectively. Of course, the choice of these spaces is not unique. But...

    • Problem 5.6 Some control problems in electromagnetics and fluid dynamics
      (pp. 179-186)
      Lorella Fatone, Maria Cristina Recchioni and Francesco Zirilli

      In recent years, as a consequence of the dramatic increases in computing power and of the continuing refinement of the numerical algorithms available, the numerical treatment of control problems for systems governed by partial differential equation; see, for example, [1], [3], [4], [5], [8]. The importance of these mathematical problems in many applications in science and technology cannot be overemphasized.

      The most common approach to a control problem for a system governed by partial differential equations is to see the problem as a constrained nonlinear optimization problem in infinite dimension. After discretization the problem becomes a finite dimensional constrained nonlinear...

    • Problem 6.1 Copositive Lyapunov functions
      (pp. 189-193)
      M. K. Çamlıbel and J. M. Schumacher

      The following notational conventions and terminology will be in force. Inequalities for vectors are understood component-wise. Given two matrices$M$and$N$with the same number of columns, the notation ($M$,$N$) denotes the matrix obtained by stacking$M$over$N$. Let$M$be a matrix. Thesubmatrix${M_{JK}}$of$M$is the matrix whose entries lie in the rows of$M$indexed by the set$J$and the columns indexed by the set$K$. For square matrices$M$,${M_{JJ}}$is called aprincipal submatrixof$M$. A symmetric matrix$M$is said to benon-negative (nonpositive) definiteif${x^T}Mx\; \geqslant \;0\;({x^T}Mx\; \leqslant \;0)$for all...

    • Problem 6.2 The strong stabilization problem for linear time-varying systems
      (pp. 194-196)
      Avraham Feintuch

      I will formulate the strong stabilization problem in the formalism of the operator theory of systems. In this framework, a linear system is a linear transformation$L$acting on a Hilbert space$H$that is equipped with a natural time structure, which satisfies the standard physical realizability condition known as causality. To simplify the formulation, we choose$H$to be the sequence space${l^2}[0,\;\infty )\; = \;\{ < {x_0},\;{x_1},\; \cdots > \;:\;{x_i}\; \in \;{\mathbb{C}^n},\;{\sum {\parallel \,{x_i}\,\parallel } ^2} < \;\infty \} $and denote by$P_n$the truncation projection onto the subspace generated by the first$n$vectors$\{ {e_0},\; \cdots ,\;{e_n}\} $of the standard orthonormal basis on$H$. Causality of$L$is expressed as${P_n}L = {P_n}L{\kern 1pt} {P_n}$for all non-negative integers$n$....

    • Problem 6.3 Robustness of transient behavior
      (pp. 197-202)
      Diederich Hinrichsen, Elmar Plischke and Fabian Wirth

      By definition, a system of the form

      $\dot x(t)\; = \;Ax(t),\quad t \geqslant 0$(1)

      $(A\; \in \;{\mathbb{K}^{n\, \times \,n}},\;\mathbb{K}\; = \;\mathbb{R},\;\mathbb{C})$is exponentially stable if and only if there are constants$M \geqslant \;1,\;\beta \; < \;0$such that

      $\parallel \,{e^{At}}\,\parallel \; \leqslant \;M{e^{\beta t}},\quad t \geqslant 0$. (2)

      The respective roles of the two constants in this estimate are quite different. The exponent$\beta < 0$determines the long-term behavior of the system, whereas the factor$M \geqslant 1$bounds its short-term or transient behavior. In applications large transients may be unacceptable. This leads us to the following stricter stability concept.

      Definition 1: Let$M \geqslant 1,\;\beta < 0$. A matrix$A\; \in \;{\mathbb{K}^{n\, \times \,n}}$is called$(M,\;\beta )$-stableif (2) holds.

      Here$\beta < 0$can be chosen in such a way that$M \geqslant 1$-stability guarantees both an...

    • Problem 6.4 Lie algebras and stability of switched nonlinear systems
      (pp. 203-207)
      Daniel Liberzon

      Suppose that we are given a family${f_p},\;p\; \in \;P$of continuously differentiable functions from$R^n$to$R^n$, parameterized by some index set$P$. This gives rise to theswitched system

      $\dot x = {f_\sigma }(x),\quad \quad x\; \in \;{R^n}$(1)

      where$\sigma :\;[0,\;\infty )\; \to \;P$is a piecewise constant function of time, called aswitching signal. Impulse effects (state jumps), infinitely fast switching (chattering), and Zeno behavior are not considered here. We are interested in the following problem: find conditions on the functions${f_p},\;p\; \in \;P$which guarantee that the switched system (1) is asymptotically stable, uniformly over the set of all possible switching signals. If this property holds, we will refer to the...

    • Problem 6.5 Robust stability test for interval fractional order linear systems
      (pp. 208-211)
      Ivo Petráš, YangQuan Chen and Blas M. Vinagre

      Recently, a robust stability test procedure is proposed for linear time-invariant fractional order systems (LTI FOS) of commensurate orders with parametric interval uncertainties [6]. The proposed robust stability test method is based on the combination of the argument principle method [2] for LTI FOS and the celebrated Kharitonov’s edge theorem. In general, an LTI FOS can be described by the differential equation or the corresponding transfer function of non commensurate real orders[7]of the following form:

      $G(s) = \frac{{{b_m}{s^{{\beta _m}}} + \; \ldots + {b_1}{s^{{\beta _1}}} + {b_0}{s^{{\beta _0}}}}} {{{a_n}{s^{{\alpha _n}}} + \ldots + {a_1}{s^{{\alpha _1}}} + {a_0}{s^{{\alpha _0}}}}} = \frac{{Q({s^{{\beta _k}}})}} {{P({s^{{\alpha _k}}})}}$, (1)

      where${\alpha _k},\;{\beta _k}\;(k\; = \;0,\;1,\;2,\; \ldots )$are real numbers and without loss of generality they can be arranged as${\alpha _n} > \; \ldots > \;{\alpha _1} > \;{\alpha _0},\;{\beta _m} > \; \ldots > \;{\beta _1} > {\beta _0}$. The coefficients${\alpha _k},\;{\beta _k}\;(k\; = \;0,\;1,\;2,\; \ldots )$are uncertain...

    • Problem 6.6 Delay-independent and delay-dependent Aizerman problem
      (pp. 212-220)
      Vladimir Răsvan

      The half-century old problem of Aizerman consists in a comparison of the absolute stability sector with the Hurwitz sector of stability for the linearized system. While the first has been shown to be, generally speaking, smaller than the second one, this comparison still serves as a test for the sharpness of sufficient stability criteria as Liapunov function or Popov inequality. On the other hand, there are now very popular for linear time delay systems two types of sufficient stability criteria: delay-independent and delay-dependent. The present paper suggests a comparison of these criteria with the corresponding ones for nonlinear systems with...

    • Problem 6.7 Open problems in control of linear discrete multidimensional systems
      (pp. 221-228)
      Li Xu, Zhiping Lin, Jiang-Qian Ying, Osami Saito and Yoshihisa Anazawa

      This chapter summarizes several open problems closely related to the following control problems in linear discrete multidimensional$(nD,{\text{ }}n \leqslant 2)$systems:

      output feedback stabilizability and stabilization,

      strong stabilizability and stabilization, or, equivalently, simultaneous stabilizability and stabilization of two given$n$D systems,

      regulation and tracking control.

      Though some of the open problems presented here have been scattered in the literature (see e.g., [7, 13, 24, 26] and the references therein), it seems that they have not received sufficient attention, and were even occasionally mistaken as known results. The purpose of this chapter is twofold: first,

      to clear up such confusions...

    • Problem 6.8 An open problem in adaptative nonlinear control theory
      (pp. 229-232)
      Leonid S. Zhiteckij

      We deal with the problem of globally stable adaptive control for discretetime, time-invariant, nonlinear, but linearly parameterized (LP) systems described by the difference equation

      ${y_t} = {\theta ^{\text{T}}}\varphi ({x_{i - 1}})\; + \;b{u_{t - 1}} + {v_t}$, (1)

      where${y_t}:\;{\mathbb{Z}^ + } \to \mathbb{R}$and${u_t}:\;{\mathbb{Z}^ + } \to \mathbb{R}$are the measurable output and control input, respectively, and${v_t}:\;{\mathbb{Z}^ + } \to \mathbb{R}$is the unmeasured disturbance (the integer$t$denotes the discrete time).$\theta \; \in \;{\mathbb{R}^d}$and$b\; \in \;\mathbb{R}$are the unknown parameter vector and scalar$(d\; \geqslant \;1)$.$f( \cdot )\;:\;{\mathbb{R}^N} \to {\mathbb{R}^d}$represents a known nonlinear vector function depending on the vector$x_{t - 1}^{\text{T}} = [{y_{t - 1}},\; \ldots ,\;{y_{t - N}}]$of$N$past outputs. Its growth is given by

      $\parallel \,\varphi (x)\,\parallel \; = \;O(\parallel \,x\,{\parallel ^\beta })\;{\text{ }}\;\;{\text{as}}\;\;\;\parallel \,x\,\parallel \; \to \infty $. (2)

      Assume that${v_t}$is upper bounded by some finite$η$, i.e.,

      $\parallel \,{v_t}\,{\parallel _\infty }\; \leqslant \;\eta \; < \;\infty $, (3)...

    • Problem 6.9 Generalized Lyapunov theory and its omega-transformable regions
      (pp. 233-238)
      Sheng-Guo Wang

      The open problem discussed here is a Generalized Lyapunov Theory and its$\Omega $-transformable regions. First, we provide the definition of the$\Omega $-transformable regions and its degrees. Then the open problem is presented and discussed.

      Definition 1: (Gutman & Jury 1981) A region .

      ${\Omega _v} = \{ (x,\;y)\;|\;f(\lambda ,\;\lambda *)\; = \;f(x\; + \;jy,\;x\; - \;jy)\; = \;{f_{xy}}(x,\;y)\; < \;0\} $(1)

      is$\Omega $-transformable if any two points$\alpha ,\;\beta \; \in \;{\Omega _v}$imply$Re[f(\alpha ,\;\beta *)] < \;0$, where function$f(\lambda ,\;\lambda *)\; = \;{f_{xy}}(x,\;y)\; = \;0$is the boundary function of the region${\Omega _v}$and$v$is the degree of the function$f$. Otherwise, the region${\Omega _v}$is non-$\Omega $-transformable.

      It is noticed that a region on one side of a line and a region within a circle in the...

    • Problem 6.10 Smooth Lyapunov characterization of measurement to error stability
      (pp. 239-244)
      Brian P. Ingalls and Eduardo D. Sontag

      Consider the system

      $\dot x(t)\; = \;f(x(t),\;u(t))$(1)

      with two output maps

      $y(t)\; = \;h(x(t)),\quad \;w(t)\; = \;g(x(t))$,

      with states$x(t)\; \in \;\mathbb{R}$and controls$u$measurable essentially bounded functions into${\mathbb{R}^m}$. Assume that the function$f:\;{\mathbb{R}^n}\; \times \;{\mathbb{R}^m} \to \;{\mathbb{R}^n}$is locally Lipschitz, and that the system is forward complete. Assume that the output maps$h:\;{\mathbb{R}^n} \to \;{\mathbb{R}^{{p_y}}}$and$g:\;{\mathbb{R}^n} \to \;{\mathbb{R}^{{p_w}}}$are locally Lipschitz.

      The Euclidean norm in a space${\mathbb{R}^k}$denoted simply by |·|. If$z$is a function defined on a real interval containing$[0,\;t],\;\parallel \,z\,{\parallel _{[0,\;t]}}$is the sup norm of the restriction of$z$to$[0,t]$, that is$\parallel \,z\,{\parallel _{[0,\;t]}} = \;{\text{ess}}\;{\text{sup}}\;{\text{\{ |}}\,z(t)\,|\;:\;t\; \in \;[0,\;t]\} $.

      A function$\gamma :\;{\mathbb{R}_{ \geqslant 0}} \to {\mathbb{R}_{ \geqslant 0}}$is of class$k$(denoted$\gamma \; \in \;K$) if it is continuous, positive...

    • Problem 7.1 Time for local controllability of a 1-D tank containing a fluid modeled by the shallow water equations
      (pp. 247-250)
      Jean-Michel Coron

      We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to one-dimensional horizontal moves. We assume that the horizontal acceleration of the tank is small compared to the gravity constant and that the height of the fluid is small compared to the length of the tank. This motivates the use of the Saint-Venant equations [5] (also called shallow water equations) to describe the motion of the fluid; see, e.g., [2, Sec. 4.2]. After suitable scaling arguments, the length of the tank and the gravity constant can be taken to be equal to 1; see [1]....

    • Problem 7.2 A Hautus test for infinite-dimensional systems
      (pp. 251-255)
      Birgit Jacob and Hans Zwart

      We consider the abstract system

      $\dot x(t)\; = \;Ax(t),\;x(0) = {x_0},\quad t\; \geqslant \;0$, (1)

      $y(t)\; = \;Cx(t),\quad t \geqslant 0$, (2)

      on a Hilbert space$H$. Here$A$is the infinitesimal generator of an exponentially stable$C_0$-semigroup${{\text{(}}T(t))_{t \geqslant 0}}$and by the solution of (1) we mean$x(t)\; = \;T(t){x_0}$, which is the weak solution. If$C$is a bounded linear operator from$H$to a second Hilbert space$Y$, then it is straightforward to see that$y( \cdot )$in (2) is well-defined and continuous. However, in many PDE’s, rewritten in the form (1)-(2),$C$is only a bounded operator from$D(A)$to${\text{Y (}}D(A)$denotes the domain of$A$), although the output is a well-defined (locally)...

    • Problem 7.3 Three problems in the field of observability
      (pp. 256-259)
      Philippe Jouan

      Let$X$be a${C^\infty }{\text{ (resp}}{\text{. }}{C^\omega }{\text{)}}$, connected manifold. We consider on$X$the system

      $\sum = \left\{ {\begin{array}{*{20}{c}} {\dot x = f(x,\;u)} \\ {y = h(x)} \\ \end{array} } \right.$(1)

      where$x\; \in \;X,\;u\; \in \;U\; = \;{[0,\;1]^m}$, and$y\; \in \;{\mathbb{R}^p}$. The parametrized vector field$f$and the output function$h$are assumed to be${C^\infty }{\text{ (resp}}{\text{. }}{C^\omega }{\text{)}}$. In order to avoid certain complications, the state space$X$is assumed to be compact, but this assumption is not crucial (we can for instance assume that the vector field$f$vanishes out of a relatively compact open subset of$X$).

      The three problems addressed herein concern observability and the existence of observers for such systems.

      We first consider an uncontrolled system:

      ${\sum _u} = \left\{ {\begin{array}{*{20}{c}} {\dot x = f(x)} \\ {y = h(x)} \\ \end{array} } \right.$. (2)

      This system is assumed to be observable (in...

    • Problem 7.4 Control of the KdV equation
      (pp. 260-264)
      Lionel Rosier

      The Korteweg-de Vries (KdV) equation is the simplest model for unidirectional propagation of small amplitude long waves in nonlinear dispersive systems. It occurs in various physical contexts (e.g., water waves, plasma physics, nonlinear optics). It reads

      ${y_t} + {y_{xxx}} + {y_x} + y\,{y_x} = 0$,$t > 0,\;\;x\; \in \;\Omega $. (1)

      the subscripts denoting partial derivatives (e.g.,${y_t} = \tfrac{{\partial y}} {{\partial t}}$). The KdV equation has been intensively studied since the 1960s because of its fascinating properties (infinite set of conserved integral quantities, integrability, Kato smoothing effect, etc.). (See [5] and the references therein.)

      Here, we are concerned with the boundary controllability of the KdV equation in the domain$\Omega \; = \;(0,\; + \infty )$. For any pair$(a, b)$with...

    • Problem 8.1 ${H_\infty }$-norm approximation
      (pp. 267-270)
      A. C. Antoulas and A. Astolfi

      Let$RH_\infty ^m$be the (Hardy) space of real-rational scalar¹ transfer functions of order$m$, bounded on the imaginary axisand analytic into the right-half complex plane. The optimal approximation problem in the${H_\infty }$norm can be statedas follows.

      $A^*$(Optimal Approximation in the${H_\infty }$norm) Given$G(s)\; \in \;RH_\infty ^N$and an integer$n < N$find,² if possible,$A*(s)\; \in \;RH_\infty ^n$such that

      $A*(s)\; = \;{\text{arg}}\,{\text{mi}}{{\text{n}}_{A(s)\; \in \;RH_\infty ^n}}\parallel \,G(s)\; - \;A(s)\,{\parallel _\infty }$.(1)

      For such a problem, let

      $\gamma _n^* = {\text{mi}}{{\text{n}}_{A(s) \in RH_\infty ^n}}\parallel \,G(s)\; - \;A(s)\,{\parallel _\infty }$

      then two further problems can be posed.

      (D) (Optimal Distance problem in the${H_\infty }$norm) Given$G(s)\; \in \;RH_\infty ^N$and an integer$n < N$find$\gamma _n^*$.

      (A) (Sub-optimal Approximation in the${H_\infty }$norm) Given$G(s)\; \in \;RH_\infty ^N$, an integer$n < N$and$\gamma > \gamma _n^*$find$\tilde A(s)\; \in \;RH_\infty ^n$...

    • Problem 8.2 Noniterative computation of optimal value in ${H_\infty }$ control
      (pp. 271-275)
      Ben M. Chen

      We consider an$n$-th order generalized linear system$\Sigma $characterized by the following state-space equations:

      $\Sigma \;:\;\left\{ {\begin{array}{*{20}{c}} {\dot x\; = \;A\,\;x\; + \;\;\,B\,\,\;u\; + \;\;\;E\,\;w} \\ {y\; = \;{C_1}\;x\; + \;{D_{11}}\;u\; + \;\;{D_1}\,\;w} \\ {h\; = \;{C_2}\;x\; + \;\,{D_2}\;u\; + \;{D_{22}}\;w} \\ \end{array} } \right.$(1)

      where$x$is the state,$u$is the control input,$w$is the disturbance input,$y$is the measurement output, and$h$is the controlled output of$\Sigma $. For simplicity, we assume that${D_{11}} = 0$and${D_{22}} = 0$. We also let$\Sigma_P$be the subsystem characterized by the matrix quadruple$(A, B, C_2, D_2)$, and$\Sigma_Q$be the subsystem characterized by$(A, E, C_1, D_1)$.

      The standard${H_\infty }$optimal control problem is to find an internally stabilizing proper measurement feedback control law,

      ${\Sigma _{{\text{cmp}}}}\;:\;\left\{ {\begin{array}{*{20}{c}} {\dot v\; = \;{A_{{\text{cmp}}}}\;v\; + \;{B_{{\text{cmp}}}}\;y} \\ {u\; = \;{C_{{\text{cmp}}}}\;v\; + \;{D_{{\text{cmp}}}}\;y} \\ \end{array} } \right.$(2)

      such that when it is applied to...

    • Problem 8.3 Determining the least upper bound on the achievable delay margin
      (pp. 276-279)
      Daniel E. Davison and Daniel E. Miller

      Control engineers have had to deal with time delays in control processes for decades and, consequently, there is a huge literature on the topic, e.g., see [1] or [2] for collections of recent results. Delays arise from a variety of sources, including physical transport delay (e.g., in a rolling mill or in a chemical plant), signal transmission delay (e.g., in an earth-based satellite control system or in a system controlled over a network), and computational delay (e.g., in a system which uses image processing). The problems posed here are concerned in particular with systems where the time delay is not...

    • Problem 8.4 Stable controller coefficient perturbation in floating point implementation
      (pp. 280-284)
      Jun Wu and Sheng Chen

      For real matrix${\mathbf{X}} = [{x_{ij}}]$, denote

      $\parallel \,{\mathbf{X}}\,{\parallel _{{\text{max}}}}\; = \mathop {{\text{max}}}\limits_{i,j} \;|\,{x_{ij}}\,|$. (1)

      For real matrices${\mathbf{X}} = [{x_{ij}}]$and${\mathbf{Y}} = [{y_{ij}}]$of the same dimension, denote the Hadamard product of$X$and$Y$as

      ${\mathbf{X}}\; \circ {\mathbf{Y}} = [{x_{ij}}{y_{ij}}]$. (2)

      A square real matrix is said to be stable if its eigenvalues are all in the interior of the unit disc.

      Consider a stable discrete-time closed-loop control system, consisting of a linear time invariant plant$P(z)$and a digital controller$C(z)$. The plant model$P(z)$is assumed to be strictly proper with a state-space description

      $\left\{ {\begin{array}{*{20}{c}} {{{\mathbf{x}}_P}(k\; + \;1)\; = \;{{\mathbf{A}}_P}{{\mathbf{x}}_P}(k)\; + \;{{\mathbf{B}}_P}{\mathbf{u}}(k)} \\ {{\mathbf{y}}(k) = {{\mathbf{C}}_P}{{\mathbf{x}}_P}(k)} \\ \end{array} } \right.$(3)

      where${{\mathbf{A}}_P}\; \in \;{R^{m\, \times \,m}},\;{{\mathbf{B}}_P}\; \in \;{R^{m\, \times \,l}}$and${{\mathbf{C}}_P}\; \in \;{R^{q\, \times \,m}}$. The controller$C(z)$is described by

      $\left\{ {\begin{array}{*{20}{c}} {{{\mathbf{x}}_c}(k\; + \;1)\; = \;{{\mathbf{A}}_c}{{\mathbf{x}}_c}(k)\; + \;{{\mathbf{B}}_c}y(k)} \\ {u(k) = {{\mathbf{C}}_c}{{\mathbf{x}}_c}(k) + {D_c}y(k)} \\ \end{array} } \right.$(4)...

    • Problem 9.1 A conjecture on Lyapunov equations and principal angles in subspace identification
      (pp. 287-292)
      Katrien De Cock and Bart De Moor

      The following conjecture relates the eigenvalues of certain matrices that are derived from the solution of a Lyapunov equation that occurred in the analysis of stochastic subspace identification algorithms [3]. First, we formulate the conjecture as a pure matrix algebraic problem. In Section 2, we will describe its system theoretic consequences and interpretation.

      Conjecture: Let$A\; \in \;{\mathbb{R}^{n\, \times \,n}}$be a real matrix and$v,\;w\; \in \;{\mathbb{R}^n}$be real vectors so that there are no two eigenvalues${\lambda _i}$and${\lambda _j}$of$\left( {\begin{array}{*{20}{c}} A & 0 \\ 0 & {A\; + \;v{w^T}} \\ \end{array} } \right)$for which${\lambda _i}{\lambda _j} = 1\;(i,\;j = 1,\; \ldots ,\;2n)$. If the$n \times n$matrices$P, Q$and$R$satisfy the

      Lyapunov equation

      $\left( {\begin{array}{*{20}{c}} P & R \\ {{R^T}} & Q \\ \end{array} } \right)\; = \;\left( {\begin{array}{*{20}{c}} A & 0 \\ 0 & {{{(A\; + \;v{w^T})}^T}} \\ \end{array} } \right)\;\left( {\begin{array}{*{20}{c}} P & R \\ {{R^T}} & Q \\ \end{array} } \right)\;\left( {\begin{array}{*{20}{c}} {{A^T}} & 0 \\ 0 & {A\; + \;v{w^T}} \\ \end{array} } \right)\; + \;\left( {\begin{array}{*{20}{c}} v \\ w \\ \end{array} } \right)\;\left( {\begin{array}{*{20}{c}} {{v^T}} \hfill & {{w^T}} \hfill \\ \end{array} } \right)$(1)

      and$P, Q$and${({I_n} + PQ)^{ - 1}}$are nonsingular,² then the matrices...

    • Problem 9.2 Stability of a nonlinear adaptive system for filtering and parameter estimation
      (pp. 293-296)
      Masoud Karimi-Ghartemani and Alireza K. Ziarani

      We are concerned about the mathematical properties of the dynamical system presented by the following three differential equations:

      $\left\{ {\begin{array}{*{20}{c}} {\tfrac{{dA}} {{dt}}} & = & { - 2{\mu _1}A\;{\text{si}}{{\text{n}}^2}\phi \; + \;2{\mu _1}\;{\text{sin}}\,\phi \;f(t),} \\ {\tfrac{{d\omega }} {{dt}}} & = & { - {\mu _2}{A^2}\,{\text{sin(2}}\phi {\text{)}}\; + \;2{\mu _2}A\,{\text{cos}}\phi \;f(t),} \\ {\tfrac{{d\phi }} {{dt}}} & = & {\omega \; + \;{\mu _3}\tfrac{{d\omega }} {{dt}}} \\ \end{array} } \right.$(1)

      where parameters${\mu _i},\;i\; = \;1,\;2,\;3$are positive real constants and$f(t)$is a function of time having a general form of

      $f(t) = {A_o}\,{\text{sin(}}{\omega _o}t\; + \;{\delta _o})\; + \;{f_1}(t)$. (2)

      ${A_o},\;{\omega _o}$and${\delta _o}$are fixed quantities and it is assumed that${f_1}(t)$has no frequency component at${\omega _o}$. Variables$A$and$\omega $are in${\mathbb{R}^1}$and$\varphi $varies on the one-dimensional circle${\mathbb{S}^1}$with radius 2$\pi $.

      The dynamical system presented by (1) is designed to (i) take the signal$f(t)$as its input signal and extract the component${f_o}(t)\; = \;{A_o}\,{\text{sin(}}{\omega _o}t\; + \;{\delta _o})$...

    • Problem 10.1 Root-clustering for multivariate polynomials and robust stability analysis
      (pp. 299-303)
      Pierre-Alexandre Bliman

      Given the$(m + 1)$complex matrices${A_0},...,{A_m}$of size$n \times n$and denoting$\bar \mathbb{D}{\text{ }}({\text{resp}}{\text{. }}\overline {{\mathbb{C}^ + }} )$the closed unit ball in$\mathbb{C}$(resp. the closed right-half plane), let us consider the following problem: determine whether

      $\forall s\; \in \;\overline {{\mathbb{C}^ + }} ,\;\forall z\;\mathop = \limits^{{\text{def}}} \;({z_1},\; \ldots ,\;{z_m})\; \in \;{{\bar \mathbb{D}}^m}$,${\text{det(}}s{I_n} - {A_0} - {z_1}{A_1} - \cdots - {z_m}{A_m})\; \ne \;0$.(1)

      We have proved in [1] that property (1) is equivalent to the existence of$k\; \in \;\mathbb{N}$and$(m + 1)$matrices$p,{\text{ }}{Q_1}{\text{ }} \in {H^{{k^m}n}},{\text{ }}{Q_2}{\text{ }} \in {H^{{k^{m - 1}}(k + 1)n}},...,{Q_m} \in {\text{ }}{H^{k{{(k + 1)}^{m - 1}}n}}$, such that

      $P > {0_{{k^m}n}}\;{\text{and}}\;R(P,\;{Q_1},\; \ldots ,\;{Q_m})\; < {0_{{{(k + 1)}^m}n}}$. (2)

      Here,$H^n$represents the space of$n \times n$hermitian matrices, and$R$is a linear application taking values in${H^{{{(k + 1)}^m}n}}$, defined as follows. Let${{\hat J}_k}\mathop = \limits^{{\text{def}}} \;({I_k}\;\;{0_{k \times 1}})$,${{\hat J}_k}\mathop = \limits^{{\text{def}}} \;({0_{k \times 1}}{\text{ }}{I_k})$, then (using the power of Kronecker product with the natural meaning):

      $\begin{gathered} R\mathop = \limits^{{\text{def}}} {\left( {\left( {\hat J_k^{m \otimes } \otimes \;{A_0}} \right) + \sum\limits_{i = 1}^m {\left( {\hat J_k^{(m - i) \otimes } \otimes {{\mathop J\limits^ \vee }_k} \otimes \hat J_k^{(i - 1) \otimes } \otimes {A_i}} \right)} } \right)^H}P\left( {\hat J_k^{m \otimes } \otimes \;{I_n}} \right) \ \\ + \;{\left( {\hat J_k^{m \otimes } \otimes \;{I_n}} \right)^T}P\;\left( {\left( {\hat J_k^{m \otimes } \otimes \;{A_0}} \right)\; + \;\sum\limits_{i = 1}^m {\left( {\hat J_k^{(m - i) \otimes } \otimes {{\hat J}_k} \otimes \mathop J\limits^ \vee \,_k^{(i - 1) \otimes } \otimes {A_i}} \right)} } \right) \ \\ + \sum\limits_{i = 1}^m {\;{{\left( {\hat J_k^{(m - i + 1 \otimes } \otimes \;{I_{{{(k + 1)}^{i - 1}}n}}} \right)}^T}{Q_i}\left( {\hat J_k^{(m - i + 1 \otimes } \otimes {I_{{{(k + 1)}^{i - 1}}n}}} \right)} \ \\ - \sum\limits_{i = 1}^m {\;{{\left( {\hat J_k^{(m - i \otimes } \otimes \;{{\mathop J\limits^ \vee }_k} \otimes {I_{{{(k + 1)}^{i - 1}}n}}} \right)}^T}{Q_i}\left( {\hat J_k^{(m - i \otimes } \otimes {{\mathop J\limits^ \vee }_k} \otimes {I_{{{(k + 1)}^{i - 1}}n}}} \right)} \ \\ \end{gathered} $(3)...

    • Problem 10.2 When is a pair of matrices stable?
      (pp. 304-308)
      Vincent D. Blondel, Jacques Theys and John N. Tsitsiklis

      We consider problems related to the stability of sets of matrices. LetΣbe a finite set of$n \times n$real matrices. Given a system of the form

      ${x_{t + 1}} = {A_t}{x_t}\;\;\;t = 0,\;1,\; \ldots $

      suppose that it is known that${A_t}\; \in \;\Sigma $for each$t$, but that the exact value of$A_t$is not a priori known because of exogenous conditions or changes in the operating point of the system. Such systems can also be thought of as a time-varying systems. We say that such a system isstableif

      $\mathop {{\text{lim}}}\limits_{t \to \infty } \;{x_t} = 0$

      for all initial stated$x_0$and all sequences of matrix products. This condition is equivalent to...

    • Problem 10.3 Freeness of multiplicative matrix semigroups
      (pp. 309-314)
      Vincent D. Blondel, Julien Cassaigne and Juhani Karhumäki

      Matrices play a major role in control theory. In this note, we consider a decidability question for finitely generated multiplicative matrix semigroups. Such semigroups arise, for example, when considering switched linear systems. We consider embeddings of the free semigroup${\Sigma ^ + } = {\{ {a_0},\; \ldots ,\;{a_{k - 1}}\} ^ + }$into the multiplicative semigroup of 2×2 matrices over nonnegative integers N:

      $\varphi :\;{\Sigma ^ + } \to {M_{2 \times 2}}({\mathbf{N}}{\text{)}}$.

      For a two generator semigroup, i.e.,${\Sigma ^ + } = {\{ a,\;b\} ^ + }$, such embeddings are defined, for example, by mappings:

      ${\varphi _1}:\;\begin{array}{*{20}{c}} a & \mapsto & {\left( {\begin{array}{*{20}{c}} 1 & 1 \\ 0 & 1 \\ \end{array} } \right)} \\ b & \mapsto & {\left( {\begin{array}{*{20}{c}} 1 & 0 \\ 1 & 1 \\ \end{array} } \right)} \\ \end{array} {\text{ and }}{\varphi _2}:\;\begin{array}{*{20}{c}} a & \mapsto & {\left( {\begin{array}{*{20}{c}} 2 & 0 \\ 0 & 1 \\ \end{array} } \right)} \\ b & \mapsto & {\left( {\begin{array}{*{20}{c}} 2 & 1 \\ 0 & 1 \\ \end{array} } \right)} \\ \end{array} \;$. (1)

      Actually,$\varphi_1$provides an embedding of the two generator free group into the multiplicative semigroup of unimodular matrices, e.g., into$SL(2,{\text{ N}})$. The embedding$\varphi_2$, in turn, directly extends to all...

    • Problem 10.4 Vector-valued quadratic forms in control theory
      (pp. 315-320)
      Francesco Bullo, Jorge Cortés, Andrew D. Lewis and Sonia Martínez

      For finite dimensional$\mathbb{R}$-vector spaces$U$and$V$, we consider a symmetric bilinear map$B:\;U\; \times \;U\; \to \;V$. This then defines a quadratic map${Q_B}:\;U\; \to \;V$by${Q_B}(u) = B(u,\;u)$. Corresponding to each$\lambda \; \in \;{V^*}$is a$\mathbb{R}$-valued quadratic form$\lambda {Q_B}$on$U$defined by$\lambda {Q_B}(u)\; = \;\lambda \; \cdot \;{Q_B}(u)$.$B$isdefiniteif there exists$\lambda \; \in \;{V^*}$so that$\lambda {Q_B}$is positive-definite.$B$isindefiniteif for each$\lambda \; \in \;{V^*}\backslash {\text{ann(image(}}{Q_B})),{\text{ }}\lambda {Q_B}$is neither positive nor negative-semidefinite, where ann denotes the annihilator.

      Given a symmetric bilinear map$B:\;U\; \times \;U \to V$, the problems we consider are as follows.

      i. Find necessary and sufficient conditions characterizing when$Q_B$is surjective.

      ii. If$Q_B$...

    • Problem 10.5 Nilpotent bases of distributions
      (pp. 321-325)
      Henry G. Hermes and Matthias Kawski

      When modeling controlled dynamical systems one commonly chooses individual control variables${u_1},\; \ldots {u_m}$that appearnaturalfrom a physical, or practical point of view. In the case of nonlinear models evolving on${{\mathbf{R}}^n}$(or more generally, an analytic manifold${M^n}$) that are affine in the control, such a choice corresponds to selecting vector fields${f_0},\;{f_1},\; \ldots {f_m}:\;M \mapsto \;TM$, and the system takes the form

      $\dot x = {f_0}(x)\; + \;\sum\limits_{k = 1}^m {{u_k}} {f_k}(x)$. (1)

      From a geometric point of view such a choice appears arbitrary, and the natural objects are not the vector fields themselves but their linear span. Formally, for a set$F = \{ {v_1},\; \ldots \;{v_m}\} $of vector fields define thedistribution spanned by$F$as...

    • Problem 10.6 What is the characteristic polynomial of a signal flow graph?
      (pp. 326-329)
      Andrew D. Lewis

      Suppose one is given signal flow graph$G$with$n$nodes whose branches have gains that are real rational functions (the open loop transfer functions). The gain of the branch connecting node$i$to node$j$is denoted${R_{ji}}$and we write${R_{ji}} = \tfrac{{{N_{ji}}}} {{{D_{ji}}}}$as a coprime fraction. The closed-loop transfer function from node$i$to node$j$for the closed-loop system is denoted${T_{ji}}$.

      The problem can then be stated as follows:

      Is there an algorithmic procedure that takes a signal flow graph G and returns a “characteristic polynomial”${P_G}$with the following properties:

      $i.$${P_G}$is formed by products and sums of the polynomials...

    • Problem 10.7 Open problems in randomized µ analysis
      (pp. 330-334)
      Onur Toker

      In this chapter, we review the current status of the problem reported in [5], and discuss some open problems related to randomized$\mu $analysis. Basically, what remains still unknown after Treil’s result [6] are the growth rate of the$\bar \mu \,/\,\mu $ratio, and how likely it is to observe a high conservatism. In the context of randomized$\mu $analysis, we discuss two open problems (i) Existence of polynomial time Las Vegas type randomized algorithms for robust stability against structured LTI uncertainties, and (ii) The minimum sample size to guarantee that$\mu \,/\,\hat \mu $conservatism will be bounded by$G$, with a confidence level...