Jong Son Shin

On Stability and Robust Stability of Positive Linear Volterra Equations

We first introduce the notion of positive linear Volterra integrodifferential equations. Then we give some characterizations of these positive equations. An explicit criterion and a Perron-Frobenius-type theorem for positive linear Volterra integrodifferential equations are given. Then we offer a new criterion for uniformly asymptotic stability of positive equations. Finally, we study stability radii of positive linear Volterra integrodifferential equations. It is proved that complex, real, and positive stability radii of positive linear Volterra equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. To the best of our knowledge, most of the results of this paper are new.

Comparison theorems and uniqueness of mild solutions to semilinear functional differential equations in Banach spaces

Let E be a Banach space over the real field R with norm 1. IE and let 6~5 := CBE be the phase space satisfying the fundamental axioms introduced by Hale and Kato [l]. If X: (-00, (T + a) + E, O<asm, then for any t E (-00,~ + a) we define a mapping x,: (-oo,O] + E by ~~(8) = x(t + I??), --oo < 0 I 0. We denote by (T(t)],,, a linear semigroup on E of class (C,) (for brevity, C,-semigroup on E) and let A be the infinitesimal generator of (T(t)),,, . The purpose of this paper is concerned with the existence and uniqueness of mild solutions to the initial-value problem for a semilinear functional differential equation in E (for brevity, ZP(o, 9))

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae.

A characterization of solutions in linear differential equations with periodic forcing functions

We deal with periodic linear inhomogeneous differential equations of the form dx/dt=Ax(t)+f(t), where A is an m×m matrix and f a τ-periodic continuous function. The solutions of this equation will be characterized as a sum of τ-periodic functions and exponential-like functions in an explicit form. As applications of this result, we can obtain the complete classification of the set of initial values according to the behavior of solutions: bounded solutions on [0, ∞], τ-periodic solutions, quasi-periodic solutions, asymptotically periodic solutions and solutions with the growth order as t→∞, etc. The essential part of our method is to give a specific representation of the solutions of difference equations corresponding to the above equations.

Existence of solutions and Kamke's theorem for functional differential equations in Banach spaces

In this paper we shall consider the Cauchy problem for functional differential equations (FDEs) (for a simplicity, CP( 1) or CP(f, 6, cp)) where f: 52 + E, 52 c ( - co, co) x 99, is uniformly continuous. Here E is a Banach space with norm 1. IE, and g is an abstract phase space satisfying the fundamental axioms (see Section 1) introduced by Hale and Kato [6] (also refer to [8, 173). The purpose of this paper is to establish an existence theorem of solu- tions for CP(l) and Kamke’s theorem in FDEs. It is well known that the compactness condition which is described by means of the measure of non- compactness intrdoduced by Kuratowski is useful in showing the existence of solutions for ordinary and functional differential equations in Banach spaces (cf. [l, 3, 4, 10, 12, 13, 241). We note that these results on the existence of solutions are closely related to the property of the phase spaces Eand,%=C([--r,O],E),O

Delayed Feedback Control by Commutative Gain Matrices

The delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spectrum of a matrix which is a difference of commutative matrices. The relationship, moreover, allows us to design control gains of the DFC such that the unstable periodic solution is stabilized. In other words, the validity of the DFC is proved mathematically. As an application for the Rossler equation, we determine the best range of k such that the unstable periodic orbit is stabilized by taking a feedback gain K = kE.

On the Spectrum of Some Functional Differential Equations

The main result(Theorem 2) is that a left half plane is contained in the residual spectrum of the generator of a solution semigroup of evolution equations with infinite delay on the phase space of some continuous functions. As a result, the half plane is contained in the essential spectrum of the generator, and the growth bound of the semigroup is bounded below by the constant which is independent of the equation but depends on the phase space.