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”

Practical $h$ -stability behavior of time-varying nonlinear systems

Abir Kicha, Hanen Damak, Mohamed Ali Hammami (2023)

Commentationes Mathematicae Universitatis Carolinae

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We deal with the problem of practical uniform $h$ -stability for nonlinear time-varying perturbed differential equations. The main aim is to give sufficient conditions on the linear and perturbed terms to guarantee the global existence and the practical uniform $h$ -stability of the solutions based on Gronwall’s type integral inequalities. Several numerical examples and an application to control systems with simulations are presented to illustrate the applicability of the obtained results. ...

Numerical stability of the intrinsic equations for beams in time domain

Klesa, Jan

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Intrinsic equations represent promising approach for the description of rotor blade dynamics. They are the system of non-linear partial differential equations. Stability of numeric solution by the finite difference method is described. The stability is studied for various numerical schemes with different methods for the computation of spatial derivatives from time level $n + 0.5$ (i.e., mean values of old and new time step) to $n + 1$ (i.e., only from new time step). Stable solution was obtained only...

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 $(| A + A | - | 2 A |) / | A |$ .

Delay-dependent stability conditions for fundamental characteristic functions

Hideaki Matsunaga (2023)

Archivum Mathematicum

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This paper is devoted to the investigation on the stability for two characteristic functions $f_{1} (z) = z^{2} + p e^{- z τ} + q$ and $f_{2} (z) = z^{2} + p z e^{- z τ} + q$ , where $p$ and $q$ are real numbers and $τ > 0$ . The obtained theorems describe the explicit stability dependence on the changing delay $τ$ . Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space

Michael Gil' (2023)

Czechoslovak Mathematical Journal

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We consider the equation $d y (t) / d t = (A + B (t)) y (t)$ $(t \geq 0)$ , where $A$ is the generator of an analytic semigroup ${(e^{A t})}_{t \geq 0}$ on a Banach space $𝒳$ , $B (t)$ is a variable bounded operator in $𝒳$ . It is assumed that the commutator $K (t) = A B (t) - B (t) A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_{l}^{- 1}$ such that $∥ S e^{A t} ∥$ is integrable and the operator $K (t) S_{l}^{- 1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

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,...

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_{t} (t) + A u (t) = f (t, u (t))$ , $0 \leq t < τ$ with $u (τ) = φ$ , where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $φ$ and $f$ , that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$ , which is also ill-posed. Spectral truncation...

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...

The covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías, José Carlos Rosales (2024)

Czechoslovak Mathematical Journal

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Let $S$ be a numerical semigroup. We say that $h \in ℕ ∖ S$ is an isolated gap of $S$ if ${h - 1, h + 1} \subseteq S .$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $m (S)$ the multiplicity of a numerical semigroup $S$ . A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$ , and $S ∖ {m (S)} \in 𝒞$ for all $S \in 𝒞$ such that $S \neq min (𝒞) .$ We prove that the set $𝒫 (F) = {S : S$ is a perfect numerical semigroup...

Tykhonov well-posedness of a heat transfer problem with unilateral constraints

Mircea Sofonea, Domingo A. Tarzia (2022)

Applications of Mathematics

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We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D \subset ℝ^{d}$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $𝒫$ . We associate to Problem $𝒫$ an optimal control problem, denoted by $𝒬$ . Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\tilde{K}$ , we provide results concerning the well-posedness...

On the continuity of the elements of the Ellis semigroup and other properties

Salvador García-Ferreira, Yackelin Rodríguez-López, Carlos Uzcátegui (2021)

Commentationes Mathematicae Universitatis Carolinae

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We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a \in X^{'}$ in order that all functions of the Ellis semigroup $E (X, f)$ be continuous at the given point $a$ . In the second part, we consider transitive...

On left $ϕ$ -biflat Banach algebras

Amir Sahami, Mehdi Rostami, Abdolrasoul Pourabbas (2020)

Commentationes Mathematicae Universitatis Carolinae

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We study the notion of left $ϕ$ -biflatness for Segal algebras and semigroup algebras. We show that the Segal algebra $S (G)$ is left $ϕ$ -biflat if and only if $G$ is amenable. Also we characterize left $ϕ$ -biflatness of semigroup algebra $l^{1} (S)$ in terms of biflatness, when $S$ is a Clifford semigroup.

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.