On approximation of stability radius for an infinite-dimensional feedback control system

Hideki Sano

Displaying similar documents to “On approximation of stability radius for an infinite-dimensional feedback control system”

Quantitative stability for sumsets in $ℝ^{n}$

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

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Given a measurable set $A \subset ℝ^{n}$ of positive measure, it is not difficult to show that $| A + A | = | 2 A |$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $(| A + A | - | 2 A |) / | A |$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $(| A + A | - | 2 A |) / | A |$ .

On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

Chia-chi Tung (2013)

Annales Polonici Mathematici

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Conditions characterizing the membership of the ideal of a subvariety arising from (effective) divisors in a product complex space Y × X are given. For the algebra $_{Y} [V]$ of relative regular functions on an algebraic variety V, the strict stability is proved, in the case where Y is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in $Y \times ℂ^{N}$ , respectively,...

Linearization techniques for $𝕃^{\infty}$ See PDF-control problems and dynamic programming principles in classical and $𝕃^{\infty}$ See PDF-control problems

Dan Goreac, Oana-Silvia Serea (2012)

ESAIM: Control, Optimisation and Calculus of Variations

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The aim of the paper is to provide a linearization approach to the $𝕃^{\infty}$ See PDF-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $𝕃^{p}$ See PDF approach and the associated linear formulations. This seems to be the most appropriate tool for treating $𝕃^{\infty}$ See PDF problems in continuous and...

Small and large time stability of the time taken for a Lévy process to cross curved boundaries

Philip S. Griffin, Ross A. Maller (2013)

Annales de l'I.H.P. Probabilités et statistiques

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This paper is concerned with the small time behaviour of a Lévy process $X$ . In particular, we investigate theof the times, ${\bar{T}}_{b} (r)$ and $T_{b}^{*} (r)$ , at which $X$ , started with $X_{0} = 0$ , first leaves the space-time regions ${(t, y) \in ℝ^{2} : y \leq r t^{b}, t \geq 0}$ (one-sided exit), or ${(t, y) \in ℝ^{2} : | y | \leq r t^{b}, t \geq 0}$ (two-sided exit), $0 \leq b l t; 1$ , as $r ↓ 0$ . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^{p}$ . In many instances these are...

Improving the performance of semiglobal output controllers for nonlinear systems

Abdallah Benabdallah, Walid Hdidi (2017)

Kybernetika

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For a large class of nonlinear control systems, the main drawback of a semiglobal stabilizing output feedback controllers ${(𝒰_{R})}_{R > 0}$ with increasing regions of attraction ${(Ω_{R})}_{R > 0}$ is that, when the region of attraction $Ω_{R}$ is large, the convergence of solutions of the closed-loop system to the origin becomes slow. To improve the performance of a semiglobal controller, we look for a new feedback control law that preserves the semiglobal stability of the nonlinear system under consideration and that is...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

Steady state in a biological system: global asymptotic stability

Maria Adelaide Sneider (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A suitable Liapunov function is constructed for proving that the unique critical point of a non-linear system of ordinary differential equations, considered in a well determined polyhedron $K$ , is globally asymptotically stable in $K$ . The analytic problem arises from an investigation concerning a steady state in a particular macromolecular system: the visual system represented by the pigment rhodopsin in the presence of light.

The covering semigroup of invariant control systems on Lie groups

Víctor Ayala, Eyüp Kizil (2016)

Kybernetika

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It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $Σ \subset 𝔤$ a cone in the Lie algebra $𝔤$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on...

Steady state in a biological system: global asymptotic stability

Maria Adelaide Sneider (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Stabilization of monomial maps in higher codimension

Jan-Li Lin, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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A monomial self-map $f$ on a complex toric variety is said to be $k$ -stable if the action induced on the $2 k$ -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$ , we can find a toric model with at worst quotient singularities where $f$ is $k$ -stable. If $f$ is replaced by an iterate one can find a $k$ -stable model as soon as the dynamical degrees $λ_{k}$ of $f$ satisfy $λ_{k}^{2} > λ_{k - 1} λ_{k + 1}$ . On the other hand, we give examples of monomial maps $f$ , where...

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

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We consider the damped wave equation $α u_{t t} + u_{t} = u_{x x} - V^{'} (u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u (x, t) = h (x - s t)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$ . We show that, if the initial data are sufficiently close to the profile of a front for large $| x |$ , the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t \to + \infty$ . The proof of this...

Stability of nonlinear $h$ -difference systems with $n$ fractional orders

Małgorzata Wyrwas, Ewa Pawluszewicz, Ewa Girejko (2015)

Kybernetika

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In the paper we study the subject of stability of systems with $h$ -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with $n$ fractional orders. The equivalent descriptions of fractional $h$ -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with $n$ -orders.

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

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Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup $S_{t} : = \int_{0}^{t} T_{s} d s$ belongs to . For analytic semigroups, $S_{t} \in$ implies $T_{t} \in$ , and in this case we give precise estimates for the growth of the -norm of $T_{t}$ (e.g. the trace of $T_{t}$ ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

Second order elliptic operators with complex bounded measurable coefficients in $L^{p}$ , Sobolev and Hardy spaces

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$ , such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^{p}$ , Sobolev, and some new Hardy spaces naturally associated to $L$ . First, we show...

L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

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Let Ω be an open subset of $ℝ^{d}$ with 0 ∈ Ω. Furthermore, let $H_{Ω} = - \sum_{i, j = 1}^{d} \partial_{i} c_{i j} \partial_{j}$ be a second-order partial differential operator with domain $C_{c}^{\infty} (Ω)$ where the coefficients $c_{i j} \in W_{l o c}^{1, \infty} (Ω ̅)$ are real, $c_{i j} = c_{j i}$ and the coefficient matrix $C = (c_{i j})$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If $\int_{0}^{\infty} d s s^{d / 2} e^{- λ μ (s) ²} < \infty$ for some λ > 0 where $μ (s) = \int_{0}^{s} d t c {(t)}^{- 1 / 2}$ then we establish that $H_{Ω}$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup....

On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control $u(\cdot)$ . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control $u=u(\cdot)$ , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system $\Sigma$ , referred to a system $\chi$ of Lagrangian co-ordinates, and to try and write some handy...

$H^{\infty}$ calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the $H^{\infty}$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $- A$ generates a bounded analytic $C_{0}$ semigroup $(T_{t})$ on a UMD space, then the $H^{\infty}$ calculus of $A$ is bounded if and only if $(T_{t})$ has a dilation to a bounded group on $L^{2} ([0, 1], X)$ . This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^{p}$ space we can choose another $L^{p}$ space in place of $L^{2} ([0, 1], X)$ .

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(SBaw)$ , $(SBab)$ , $(SBw)$ and $(SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(SBab)$ , then $S \oplus T$ has the property $(SBab)$ if and only if $σ_{{SBF}_{+}^{-}} (S \oplus T) = σ_{{SBF}_{+}^{-}} (S) \cup σ_{{SBF}_{+}^{-}} (T)$ , where $σ_{{SBF}_{+}^{-}} (T)$ is the upper semi-B-Weyl spectrum of $T$ . We obtain analogous preservation results for the properties $(SBaw)$ ,...

Hydrodynamical behavior of symmetric exclusion with slow bonds

Tertuliano Franco, Patrícia Gonçalves, Adriana Neumann (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{- β}$ , with $β \in [0, \infty)$ . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$ . If $β \in [0, 1)$ , the hydrodynamic limit is given by the usual heat equation. If $β = 1$ , it is given by a parabolic equation involving an operator...