On a model of viscoelastic rod in unilateral contact with a rigid wall

^** Corresponding author. Email: atanackovic{at}uns.ns.ac.yu We study translatory motion of a body to which a viscoelasticrod with the constitutive equation with fractional derivativesis attached. The body with a rod impacts against a rigid wall.It is shown that the problem is described with a coupled systemof differential equations having integer and fractional derivativeshaving the form x⁽²⁾ = –f; f + af⁽⁾ = x + bx⁽⁾, x(0) =0, x⁽¹⁾(0) = 1. The unique solvability in S'₊ is proved andinterpretation of solutions is given. Also, some a priori estimatesof the solution are given. In particular, we showed that restrictionson coefficients that follow from the second law of thermodynamicsimply that the velocity after the impact is smaller than thevelocity before the impact.     

Invariant measures and maximal L2 regularity for nonautonomous Ornstein-Uhlenbeck equations

We characterize the domain of the realizations of a linear parabolicoperator defined in L² spaces with respect to a suitable measurethat is invariant for the associated evolution semigroup. Asa byproduct, we obtain optimal L² regularity results for evolutionequations with time-dependant Ornstein–Uhlenbeck operators.     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

Kato Class Potentials for Higher Order Elliptic Operators

Our goal in this paper is to determine conditions on a potentialV which ensure that an operator such as H:=(–)^m+V (1) acting on L²(R^N) defines a semigroup in L^p(R^N) for various valuesof p including p=1. The operator is defined as a quadratic formsum. That is, we put for (all integrals are on R^N and are with respect to Lebesgue measure), and note thatthe closure of the form is non-negative and has domain equalto the Sobolev space W^m,2. We then assume that the potentialhas quadratic form bound less than 1 with respect to Q₀, anddefine This form is closed and is associated with a semibounded self-adjointoperator H in L² (see [17, p. 348; 5, Theorem 4.23]). One canthen ask whether the semigroup e^–Ht defined on L² fort0 is extendable to a strongly continuous one-parameter semigroupon L^p for other values of p, and if so whether one can describethe domain and spectrum of its generator.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

Exponential stability for a contact problem in thermoelasticity

We consider the unilateral problem for the thermoelastic equationand we show that the solution decays exponentially to zero astime goes to infinity; that is, denoting by E(t) the first-orderenergy of the system, we show that positive constants C and exist which satisfy E(t)CE(0)e^–$$$.     

L1 Properties of Second order Elliptic Operators

We present a connected account of various spectral and regularityproperties of the semigroup associated with a fairly generalsymmetric second order elliptic operator on a Riemannian manifold.Our main goal is to relate the L² theory to the less well understoodL¹ theory, and hence to the approach via the theory of stochasticdifferential equations.     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

Second-order operators with degenerate coefficients

We consider properties of second-order operators on ^d with bounded real symmetric measurable coefficients.We assume that C = (c_ij) 0 almost everywhere, but allow forthe possibility that C is degenerate. We associate with H acanonical self-adjoint viscosity operator H₀ and examine propertiesof the viscosity semigroup S⁽⁰⁾ generated by H₀. The semigroupextends to a positive contraction semigroup on the L_p-spaceswith p [1, ]. We establish that it conserves probability andsatisfies L₂ off-diagonal bounds, and that the wave equationassociated with H₀ has finite speed of propagation. Nevertheless,S⁽⁰⁾ is not always strictly positive because separation of thesystem can occur even for subelliptic operators. This demonstratesthat subelliptic semigroups are not ergodic in general and theirkernels are neither strictly positive nor Hölder continuous.In particular, one can construct examples for which both upperand lower Gaussian bounds fail even with coefficients in C^2–(^d)with > 0.     

Systems of Inequalities and Numerical Semigroups

A one-to-one correspondence is described between the set S(m)of numerical semigroups with multiplicity m and the set of non-negativeinteger solutions of a system of linear Diophantine inequalities.This correspondence infers in S(m) a semigroup structure andthe resulting semigroup is isomorphic to a subsemigroup of N^m–1.Finally, this result is particularized to the symmetric case.     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Decay rates of solutions to thermoviscoelastic plates with memory

We consider the thermoelastic plate under the presence of along range memory. We find uniform rates of decay (in time)of the energy, provided that suitable assumptions on the relaxationfunctions are given. Namely, if the relaxation decays exponentiallythen the first order energy also decays exponentially. Whenthe relaxation g satisfies -c₁g(t)^1+1/_p g'(t) -c_og(t)^1+1/_p; and g,g^1-1/_p L¹ (R) withp > 2 then the energy decays as 1/(1+t)^p. A new Liapunov functionalis built for this problem.     

Pointwise Stabilization for a Chain of Coupled Vibrating Strings

For the system representing a chain of coupled vibrating strings,we show that the associated semigroup satisfies the assumptionof spectrum-determined growth, and hence obtain conditions forenergy to decay strongly or exponentially. We examine in detailthe three-string case, and our results include those obtainedby others for the two-string case. Permanent address: Beijing Institute of Information and Control,Beijing, China.     

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

A family of Hecke C^*-algebras can be realised as crossed productsby semigroups of endomorphisms. It is shown by dilating representationsof the semigroup crossed product that the category of representationsof the Hecke algebra is equivalent to the category of continuousunitary representations of a totally disconnected locally compactgroup.     

Right Cancellation in the LUC-Compactification of a Locally Compact Group

Write G^* = G^LUC\G where G^LUC is the largest semigroup compactificationof the locally compact group G. Then the set of points of G^*which are right cancellable in G^* = G^LUC is large; in fact ithas an interior in G^* which is dense in G^*. Corollaries aregiven about the number of left ideals in G^* = G^LUC and the sizeof right ideals in the algebra LUC(G)^*.     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.     

Continuity of Multiplication in the Largest Compactification of a Locally Compact Group

G^LUC is the largest semigroup compactification of the locallycompact group G. When G is not compact, given q G* = G^LUC \G, there is p G* such that x qx is discontinuous at p (Theorem2). If G is -compact, there is one p which will serve for allq (Theorem 1).     

On approximation theorems for controllability of non-linear parabolic problems

^** Email: anil{at}math.iitb.ac.in^*** Email: mcj{at}math.iitb.ac.in^**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u^*,y^*) is then obtained as the limit of the optimal pair sequence{(u_n^*, y_n^*)}, where u_n^* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and y_n^* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.