On exponential stability of infinite dimensional linear systems with bounded or unbounded perturbations

In this paper, we study the question of stabilization of uncertain systems governed by evolution equations in Ililbert space. It is shown that exponential stability can be achieved by choice of suitable state feedback controls even in the presence of bounded or relatively bounded (uncertain) perturbations. Our proof is based on perturbation theory of semigroups. The results are illustrated by two examples involving heat equation and wave equation.     

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.     

Stability radii of positive linear systems under affine parameter perturbations in infinite dimensional spaces

In this paper we study the stability radii of positive linear discrete system under arbitrary affine parameter perturbations in infinite dimensional spaces. It is shown that complex, real, and positive stability radii of positive systems coincide. More importantly, estimates and computable formulas of these stability radii are also derived. The results are then illustrated by a simple example. The obtained results are extensions of the recent results in [3].     

Topological stability of linear semi-infinite inequality systems

In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.This work has been supported partially by the DGICYT of Spain, Grant PB93-0943, by Generalitat Valenciana, Grant GV-2219/94, and by IVEI, Grant 003/026.The authors would like to thank J. E. Martínez Legaz for his valuable comments.     

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

Mathematical Programming - This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in $${mathbb {R}}^{n}$$ . To start with, we...     

Correction to: Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

Mathematical Programming -     

On the stability of infinite-dimensional linear inequality systems

The principal aim of this paper is to study the stability of the solution set mapping of a system composed by an arbitrary set of linear inequalities in an infinite-dimensional space. The unknowns space is assumed to be metrizable, which allows us to measure the size of any possible perturbation. Conditions guaranteeing the closedness, the lower semicontinuity and the upper semicontmuity of this mapping, at a particular nominal system, are given in the paper. The more significant differences with respect to the finite dimensional case, previously approached in the context of the so-called semi-infinite optimization, are illustrated by means of convenient examples.     

Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory

The asymptotic log-Harnack inequality is established for several kinds of models on stochastic differential systems with infinite memory: non-degenerate SDEs, neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate, uniqueness of the invariant probability measure, asymptotic gradient estimate (hence, asymptotically strong Feller property), as well as asymptotic irreducibility.     

Exact controllability of infinite dimensional systems persists under small perturbations

This paper studies the concept of controllability for infinite-dimensional linear control systems in Banach spaces. First, we prove that the set of admissible control operators for the semigroup generator is unchanged if we perturb the generator by the Desch–Schappacher perturbations. Second we show that exact controllability is not changed by such perturbations.     

Quantitative stability analysis of stochastic quasi-variational inequality problems and applications

We consider a parametric stochastic quasi-variational inequality problem (SQVIP for short) where the underlying normal cone is defined over the solution set of a parametric stochastic cone system. We investigate the impact of variation of the probability measure and the parameter on the solution of the SQVIP. By reformulating the SQVIP as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem, we effectively convert stability of the SQVIP into that of a one stage stochastic program with stochastic cone constraints. Under some moderate conditions, we derive Hölder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter. The stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with SQVIP constraints.     

Optimal stabilizing compensator for linear systems under white noise perturbations

For time varying linear stochastic systems it is proved that the best stabilizing compensator is the one constructed by using stabilizing solutions for the standard control and estimation Riccati equations     

A property of convex piecewise linear functions with applications to mathematical programming

Summary It is proved a lemma on convex piecewise linear functions and there are given some applications of this result to non-linear and parametric programming.

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Zusammenfassung Es wird ein Hilfssatz über konvexe stückweise lineare Funktionen bewiesen und einige Anwendungen des erlangten Ergebnisses bei nichtlinearer und parametrischer Programmierung werden gegeben.

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Vorgel. v.:H. P. Künzi     

Delay-dependent stability criterion for neutral time-delay systems via linear matrix inequality approach

In this paper, the asymptotic stability problem for a class of neutral systems with discrete and distributed delays is considered. Based on linear matrix inequality, a delay-dependent criterion is proposed to guarantee asymptotic stability for such systems. Some numerical examples are given to illustrate our main result.     

Exponential stability of nonautonomous linear differential equations with linear perturbations by Liao methods

In this paper, the author considers, by Liao methods, the stability of Lyapunov exponents of a nonautonomous linear differential equations: with linear small perturbations. It is proved that, if A(t) is a upper-triangular real n by n matrix-valued function on R₊, continuous and uniformly bounded, and if there is a relatively dense sequence in R₊, say 0=T₀<T₁<?<Ti<?, such that

     

On perturbations of controllable implicit linear systems

Various notions of controllability for general (i.e. possiblynon-regular) implicit linear systems are considered. Some genericityproperties of the corresponding controllable systems are established.