Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.     

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H^∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô’s dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis.     

Well-Posed State/Signal Systems in Continuous Time

We introduce a new class of linear systems, the L ^p -well-posed state/signal systems in continuous time, we establish the foundations of their theory and we develop some tools for their study. The principal feature of a state/signal system is that the external signals of the system are not a priori divided into inputs and outputs. We relate state/signal systems to the better-known class of well-posed input/state/output systems, showing that state/signal systems are more flexible than input/state/output systems but still have enough structure to provide a meaningful theory. We also give some examples which point to possibilities for further study.     

The Nehari problem for infinite-dimensional linear systems of parabolic type

A complete solution is obtained to the suboptimal Nehari extension problem for transfer functions of parabolic systems with Dirichlet boundary control and smooth observations. The solutions are given in terms of the realization (–A, B, C), whereA is a uniformly strongly elliptic operator of order two with smooth coefficients defined on a bounded open domain ofR _d ,B=AB _D andB _D is the Dirichlet map associated with Dirichlet boundary conditions andC is a bounded observation map fromL ₂() to the output spaceY. The approach is to solve an equivalentJ-spectral factorization problem for this particular realization.     

On ε‐biased generators in NC0

Cryan and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272–284) recently considered the question of whether there can be a pseudorandom generator in NC⁰, that is, a pseudorandom generator that maps n‐bit strings to m‐bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC⁰ generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC⁰ generator can sample an ε‐biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2. For large values of k, we construct generators that map n bits to bits such that every XOR of outputs has bias . We also present a polynomial‐time distinguisher for k = 4,m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2^kn^?k/2?). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^?Ω(n) and which maps n bits to Ω(n²) bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Finite L 2-gain with nondifferentiable storage functions

We consider affine control systems with the finite L₂-gain property in the case the storage function is nondifferentiable. We generalize some classical results concerning the connection of the finite L₂-gain property with the stability properties of the unforced system, the characterization of finite L₂-gain by means of partial differential inequalities of the Hamilton-Jacobi type and the problem of giving to a system the finite L₂-gain property by means of a feedback law. Moreover, we introduce and study the apparently new notion of exact storage function.     

Well-posedness and sharp uniform decay rates at the L 2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schr?dinger equation defined on a bounded open domain of and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L₂(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L₂(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L₂(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L₂(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H¹(Ω)-level energy estimate to the L₂(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.     

Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data

This paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in L^p and the velocity u converges to the steady state velocity in W^1,p for any 1p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity.     

Identifiability at the boundary for first-order terms

Let Ω be a domain in R ⁿ whose boundary is C ¹ if n≥3 or C ^1,β if n=2. We consider a magnetic Schrödinger operator L _{W , q} in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L _{W , q} . We also consider a steady state heat equation with convection term Δ+2W·? and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.     

Sufficient conditions for viability under imperfect measurement

Viability theory gives a necessary and sufficient condition for the existence of a (set-valued) state feedback control such that all trajectories of the closed-loop system starting from the graph of a given tube in the state space remain in the tube. Here we investigate the same problem in the case where only incomplete and inexact measurement of the state is available. In the time-invariant case, we give a sufficient condition for the existence of anoutput feedback regulation map. The condition is shown to be equivalent to Haddad's viability condition if the measurement is perfect.     

A uniformly differentiable approximation scheme for delay systems using splines

A new spline-based scheme is developed for linear retarded functional differential equations within the framework of semigroups on the Hilbert spaceR ⁿ ×L ². The approximating semigroups inherit in a uniform way the characterization for differentiable semigroups from the solution semigroup of the delay system (e.g., among other things the logarithmic sectorial property for the spectrum). We prove convergence of the scheme in the state spacesR ⁿ ×L ² andH ¹. The uniform differentiability of the approximating semigroups enables us to establish error estimates including quadratic convergence for certain classes of initial data. We also apply the scheme for computing the feedback solutions to linear quadratic optimal control problems.Work done by K. Ito was supported by AFOSR under Contract No. F-49620-86-C-0111, by NASA under Grant No. NAG-1-517, and by NSF under Grant No. UINT-8521208. Work done by F. Kappel was supported by AFOSR under Grant No. 84-0398 and by FWF(Austria) under Grants S3206 and P6005.     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.     

Disjunctive Rado numbers

If L₁ and L₂ are linear equations, then the disjunctive Rado number of the set {L₁,L₂} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to either L₁ or L₂. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers and b1, the disjunctive Rado number for the equations x₁+a=x₂ and x₁+b=x₂ is a+b+1-gcd(a,b) if is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a>1 and b>1, the disjunctive Rado number for the equations ax₁=x₂ and bx₁=x₂ is c^s+t-1 if there exist natural numbers c,s, and t such that a=c^s and b=c^t and s+t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

The four-block Adamjan-Arov-Krein problem

This paper is concerned with the mixed sensitivity H^∞ design in the most general, i.e., the four-block, case. This problem involves in a crucial manner the so-called four-block operator Γ, the norm of which is the achievable feedback tolerance. Our objective in this paper is to provide an interpretation for all singular values of Γ. These singular values are given an Adamjan-Arov-Krein interpretation in terms of the L^∞ distance of an L^∞ function to H^∞(l). Intuitively, the singular values of Γ are the various tolerance levels that can be achieved if we allow a various number of unstable poles in the closed loop. We finally provide an upper bound on the number of singular values.     

A new cryptosystem based on chaotic map and operations algebraic

Based on the study of some existing chaotic encryption algorithms, a new block cipher is proposed. The proposed cipher encrypts 128-bit plaintext to 128-bit ciphertext blocks, using a 128-bit key K and the initial value x₀ and the control parameter mu of logistic map. It consists of an initial permutation and eight computationally identical rounds followed by an output transformation. Round r uses a 128-bit roundkey K^(r) to transform a 128-bit input C^(r-1), which is fed to the next round. The output after round 8 enters the output transformation to produce the final ciphertext. All roundkeys are derived from K and a 128-bit random binary sequence generated from a chaotic map. Analysis shows that the proposed block cipher does not suffer from the flaws of pure chaotic cryptosystems and possesses high security.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

An LMI description for the cone of Lorentz-positive maps II

Let L _n be the n-dimensional second-order cone. A linear map from ? ^m to ? ⁿ is called positive if the image of L _m under this map is contained in L _n . For any pair (n,?m) of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality of size (n???1)(m???1) that describes this cone.