Showing papers on "Uniform boundedness published in 1982"

A note on approximate controllability for semilinear one-dimensional heat equation

Abstract: The one-dimensional semilinear heat equation\(\frac{{\partial y}}{{\partial t}} = \frac{{\partial ^2 y}}{{\partial x^2 }} + F(y) + bu(t)\) is considered. It is shown that if the nonlinear functionF(y) is uniformly bounded then the system is approximately controllable for every given terminal timeT>0 under some ordinary condition onb. The results may be extended to the general one-dimensional semilinear heat equation with one-dimensional control or to a boundary control heat system with semilinear boundary condition.

Characterizations of bounded solutions of linear complementarity problems

Abstract: A number of equivalent characterizations for the existence and boundedness of solutions of the linear complementarity problem: Mx+q≥0, x≥0, x T (Mx+q)=0 where M is an n×n real matrix and q is an n-vector, are given for the case when M is copositive plus. The special case when M is skew-symmetric covers the linear programming case. One useful characterization of existence and boundedness of solutions is given by solving a simple linear program. Other important characterizations are the Slater constraint qualification and the stability condition that for all arbitrary but sufficiently small perturbations of the data M and q which maintain copositivity plus, the perturbed linear complementarity problem is solvable and its solutions are uniformly bounded. An interesting sufficient condition for boundedness of solutions is that the linear complementarity problem have a nondegenerate vertex solution. Another result is that the subclass ™ of copositive plus matrices for which the linear complementarity problem has a solution for each q in R n , that is ™⊂Q, coincides with the subclass of copositive plus matrices for which the linear complementarity problem has a nonempty bounded solution set for each q in R n .

Lower Closure Problems with Weak Convergence Conditions in a New Perspective

Abstract: A transparent approach is taken towards the class of (lower) closure problems with weak convergence conditions for the “derivatives”. A deparametrization procedure is formulated for the abstract lower closure problem of this class, as well as for its variant in a control problem of Lagrange type. It is shown that, if one follows this procedure, the solution of the lower closure problem merely lies in proving that a certain “modified Lagrangian” is a normal integrand. A lower closure result is obtained that generalizes, in itself, all comparable results of [2a], [6e–f], [7a], [21]. For control problems of Lagrange type with uniform boundedness conditions on the “controls,” a special approximate Lagrangian can be formulated by which the deparametrization procedure yields results that are superior to those obtained previously [2b], [7b–c]. An important novelty in deriving the measurability properties of Lagrangians is also introduced here: it consists of the employment of the Kunugui–Novikov–Stchegolkov proj...

Local behaviour of hilbert space valued stochastic integrals and the continuity of mild solutions of stochastic evolution equations

Abstract: Levy's modulus of continuity is proved for infinite dimensional Wiener processes. Using the loglog law for a Banach space valued Wiener process in [7], we prove the loglog law for Hilbert space valued stochastic integrals, if the integrand is Holder continuous. From a corollary of Kolmogorov's law we derive the Holder continuity of Hilbert space valued stochastic integrals if the fourth moment of the integrand is uniformly bounded. As an application we show that the mild solution of a stochastic evolution equation has a continuous version if the semigroup governing this equation is analytic.

Quasi-continuous functions associated with a hunt process

Abstract: A stochastic characterization is given for quasi-continuous functions associated with Hunt processes. Introduction. Quasi-continuous functions have been studied in various frameworks of potential theory, especially Dirichlet spaces (cf. [1, 4, 5]). The main object of this paper is to establish the relation between the quasi-continuity and the quasi left continuity of processes. Under rather strong duality assumptions, it also appears that the quasi-continuous functions are the functions which are finely and cofinely continuous. 1. Definitions. Let Xt be a Hunt process defined on a compact space E containing a cemetery A with a reference measure m we may assume to be bounded. In the sequel we shall use mostly the notations of [2], and all functions and sets will be implicitly assumed nearly Borel. A bounded sequence of functions fn is said to be converging quasi-uniformly towards f iff for some a > 0 there exists a sequence gn of a-excessive functions decreasing to 0 m a.s. (and therefore quasi everywhere (cf. [2, Proposition 3.2, p. 280])), such that If fI is smaller than gn. This definition is independent of a. Indeed, for any 3 > 0, g9 + (a -,3)U13gn is ,3 excessive, decreases to 0 q.e. and dominates If fnl. One can give an alternate and more intuitive definition of the quasi-uniform convergence. PROPOSITION 1. A bounded sequence fn converges quasi-uniformly iff there exists a sequence Gm of open sets such that (a) fn converges uniformly on every set E Gm; (b) the capacitary potentials elmEx(e`TGm) are decreasing to 0 m a.s. (and therefore quasi everywhere), for a > 0. REMARK. The property (b) is independent of a and equivalent to (b') the stopping times TGm increase to +oo P. a.s. for every initial distribution 1i charging no polar set. PROOF. The sufficiency is easy to check. If fn is uniformly bounded by M, for each m, Ifn fI < 2-m + 2Me' for n large enough, and the sequence of a excessive functions 2-m + 2Mel decreases a.s. to 0. Gm Received by the editors May 9, 1980 and, in revised form, November 18, 1980 and November 1, 1981. 1980 Mathematics Subject Classification. Primary 60J40; Secondary 60J45.

Boundary stabilization of hyperbolic systems with no dissipative conditions.

Abstract: In this paper a theory of boundary stabilization of hyperbolic systems is given in such a way that the resulting system is not necessarily dissipative. Further, we obtain the uniform boundedness of a class of semigroups whose infinitesimal generator is not necessarily dissipative. The key point of this paper is that in contrast to most other works on stabilization of hyperbolic systems we do not require our generator to be dissipative.

Linear or nonlinear hyperbolic wave problems with input sets (Part I)

Abstract: In this first part, collections of linear hyperbolic initial boundary value problems are treated which are defined via sets of coefficient functions in the differential equations. If the solutions are oscillatory, the nonlinear dependency of the solutions on coefficients becomes more and more ill-conditioned as time progresses, unless there is a sufficiently strong damping term in the differential equation. For the problem of dynamic buckling, the theory of the Neumann series yields a sufficient condition for the uniform boundedness of the oscillatory solutions which are induced by arbitrary continuous transient perturbations whose range is restricted to a suitable interval. In this part I, there is an introductory discussion of the Taylor-representation of sets of solutions in terms of constant coefficients. Via such a Taylor-representation, it is shown that solutions of the ‘distortionless’ telephone line are insensitive to sufficiently small variations of the constant coefficients in this hyperbolic differential equation.

A Lower Bound for the Formula Size of Rational Functions

Abstract: In this paper we provide a method for bounding from below the formula size of rational functions over arbitrary fields. The basic operations allowed are addition, v subtraction, multiplication and division. The method is based on Neciporuk [4] (also followed in Savage [5]). It is quite easy to adapt his argument if the field is finite. Also if attention is restricted to polynomials over infinite fields and division is not allowed then there is again a fairly easy adaptation. As one would expect the general case presents some difficulties in connection with division. These are overcome by the use of formal power series.

Properties of Spline Projections

Abstract: Let X, Y be Hilbert spaces, T be a linear bounded operator that maps X into Y and A ⊂ X be a subset of X.

Output ε-trajectory controllability for a class of nonlinear distributed systems

Abstract: In this paper, with reference to the class of perturbed linear systems on Hilbert spaces, we study the input-output e-trajectory controllability property: i.e. the capability of the input to drive the output close to a fixed output trajectory. It is shown that under the uniform boundedness of the perturbation term the system has the desired property if the same is true for the associated unperturbed linear part.