State-Feedback Stabilization of Well-Posed Linear Systems

A finite-dimensional linear time-invariant system is output-stabilizable if and only if it satisfies the finite cost condition, i.e., if for each initial state there exists at least one L² input that produces an L² output. It is exponentially stabilizable if and only if for each initial state there exists at least one L² input that produces an L² state trajectory. We extend these results to well-posed linear systems with infinite-dimensional input, state and output spaces. Our main contribution is the fact that the stabilizing state feedback is well posed, i.e., the map from an exogenous input (or disturbance) to the feedback, state and output signals is continuous in L_loc² in both open-loop and closed-loop settings. The state feedback can be chosen in such a way that it also stabilizes the I/O map and induces a (quasi) right coprime factorization of the original transfer function. The solution of the LQR problem has these properties.     

Conditions for Admissibility of Observation Operators and Boundedness of Hankel Operators

We derive new necessary and sufficient conditions for admissibility of observation operators for certain C ₀-semigroups. We also prove a new sufficient criterion for admissibility for observation operators with infinite-dimensional output space on contraction semigroups. If the contraction semigroup is completely non-unitary and its co-generator has finite defect indices, then this criterion is also necessary. In the case of the right shift semigroup on L ²(0,), these conditions translate into conditions for the boundedness of Hankel operators.     

Finite-time boundedness and L2-gain analysis for switched delay systems with norm-bounded disturbance

Finite-time boundedness and finite-time weighted L₂-gain for a class of switched delay systems with time-varying exogenous disturbances are studied. Based on the average dwell-time technique, sufficient conditions which guarantee the switched linear system with time-delay is finite-time bounded and has finite-time weighted L₂-gain are given. These conditions are delay-dependent and are given in terms of linear matrix inequalities. Detail proofs are given by using multiple Lyapunov-like functions. An example is employed to verify the efficiency of the proposed method.     

A posteriori error estimates for a coupled wave system with a local damping

We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbR^d {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L ₂(L ₂) and L _∞(L ₂), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.     

Non-equilibrium entropy on stationary Markov processes

We study the evolution of probability measures under the action of stationary Markov processes by means of a non-equilibrium entropy defined in terms of a convex function . We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of for a wide class of convex functions, including ₀(t)=t log t. We also prove that this is equivalent to the convergence of all the densities of a finite norm to a uniform density, on the Orlicz spaces related to , which include the L ^p -spaces for p>1. By means of the quadratic function ₂(t)=t ²–1, we relate the non-equlibrium entropies defined by the past -algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.Partially supported by DIB Universidad de Chile, E19468412.     

Basis property of the eigenfunctions of a generalized gasdynamic Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

We find the eigenfunctions of a generalized Frankl problem with the use of Bessel functions. We prove that these eigenfunctions form a Riesz basis in the space L ₂(D ₊), where D ₊ is the elliptic part of the domain. In addition, we prove the Riesz basis property of a trigonometric function system and the completeness of this system in the space L ₂(0, π/2).     

Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]). Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L²(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L²-norm control. We illustrate the mathematical results with several numerical experiments. Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations". Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).     

Supports of Locally Linearly Independent M-Refinable Functions,Attractors of Iterated Function Systems and Tilings

This paper is devoted to a study of supports of locally linearly independent M-refinable functions by means of attractors of iterated function systems, where M is an integer greater than (or equal to) 2. For this purpose, the local linear independence of shifts of M-refinable functions is required. So we give a complete characterization for this local linear independence property by finite matrix products, strictly in terms of the mask. We do this in a more general setting, the vector refinement equations. A connection between self-affine tilings and L ₂ solutions of refinement equations without satisfying the basic sum rule is pointed out, which leads to many further problems. Several examples are provided to illustrate the general theory.     

Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization

Results of exponential/polynomial decay rates of the energy in L²-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, will be established in this paper. We use Kato's theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik's approach.     

The approximation of reachable sets of control systems with integral constraint on controls

In this paper an approximation method for the construction of reachable sets of control systems with integral constraints on the control is considered. It is assumed that the control system is non-linear with respect to the phase state vector and is linear with respect to the control vector. The admissible control functions are chosen from the ball centered at the origin with radius μ₀ in L_p, p > 1. The reachable set is replaced by the set which consists of finite number of points. The estimated accuracy of the Hausdorff distance between the reachable set and the set which is approximately constructed is obtained.     

New filter design for linear time-delay systems

This paper develops new robust delay-dependent filter design for a class of linear systems with time-varying delays and convex-bounded parameter uncertainties. The design procedure hinges upon the constructive use of an appropriate Lyapunov functional plus a free-weighting matrices in order to exhibit the delay-dependent dynamics. The developed approach utilizes smaller number of LMI decision variables thereby leading to less conservative solutions to the delay-dependent stability and filtering problems. Subsequently, linear matrix inequalities (LMIs)-based conditions are characterized such that the linear delay system is robustly asymptotically stable with an γ-level L₂-gain. All the developed results are tested on representative examples.     

Fejér means of functions on noncommutative Vilenkin groups with respect to the character system

Summary It is known that the Fejér means - with respect to the character system of the Walsh, and bounded Vilenkin groups - of an integrable function converge to the function a.e. In this work we discuss analogous problems on the complete product of finite, not necessarily Abelian groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case differs from the commutative case. We prove the a.e. convergence of the (C,1) means of the Fourier series of square integrable functions. We also prove the existence of a function f∊L^q for some q >1 such that sup |σ_n f | = + ∞ a.e. This is a sharp contrast between the Abelian and the nonabelian cases.     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

Laplacian Operators and Radon Transforms on Grassmann Graphs

Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra End_G(L ₂(X)) of intertwining maps on L ₂(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon transforms.     

On a problem involving bounded epimorphisms of lattices

A latticeL satisfies thebounded epimorphism condition if wheneverM is a lattce and ϕ:M →L is a bounded epimorphism, there exists a homomorphismι:L →M such that ιϕ=id _L . we show that the class of finite lattices satisfying the bounded epimorphism condition is properly contained in the class of finite lattices satisfying Whitman's condition (W). We also introduce a property defined for finite lattices that is sufficient to imply the bounded epimorphism condition. Presented by B. Jónsson.     

Observer-embedded L2-gain control

This paper demonstrates that the H_∞ controller of Doyle . [1] can be reinterpreted as an observer-embedded L₂-gain controller. The embedded observer is similar to the conventional Luenberger observer except having an additional state-dependent calibration term that compensates for exogenous inputs and model uncertainties. Based on this fact, we derive mixed Linear Matrix Inequalities, Hamiltonian Matrix, and Linear Parameterization to provide solutions to feasible observer-embedded L₂-gain controllers.     

Insensitizing controls with one vanishing component for the Navier–Stokes system

In this paper we prove the existence of insensitizing controls, having one vanishing component, for the local L²$L^2$-norm of the solutions of the Navier–Stokes system. This problem can be recast as a null controllability problem for a nonlinear cascade system. We first prove a controllability result, with controls having one vanishing component, for a linear problem. Then, by means of an inverse mapping theorem, we deduce the controllability for the cascade system.     

Subspaces and Orthogonal Decompositions Generated by Bounded Orthogonal Systems

We investigate properties of subspaces of L₂ spanned by subsets of a finite orthonormal system bounded in the L_∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L₁ and the L₂ norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L₂ⁿ, complementary to each other and each of dimension roughly n/2, spanned by ± 1 vectors (i.e. Kashin’s splitting) and in logarithmic distance to the Euclidean space. The same method applies for p > 2, and, in connection with the Λ_p problem (solved by Bourgain), we study large subsets of this orthonormal system on which the L₂ and the L_p norms are close (again, up to a logarithmic factor). Partially supported by an Australian Research Council Discovery grant. This author holds the Canada Research Chair in Geometric Analysis.     

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

We study linear systems, described by operators A, B, C for which the state space X is a Banach space.We suppose that − A generates a bounded analytic semigroup and give conditions for admissibility of B and C corresponding to those in G. Weiss’ conjecture. The crucial assumptions on A are boundedness of an H^∞-calculus or suitable square function estimates, allowing to use techniques recently developed by N. Kalton and L. Weis. For observation spaces Y or control spaces U that are not Hilbert spaces we are led to a notion of admissibility extending previous considerations by C. Le Merdy. We also obtain a characterisation of wellposedness for the full system. We give several examples for admissible operators including point observation and point control. At the end we study a heat equation in X = L^p(Ω), 1 < p < ∞, with boundary observation and control and prove its wellposedness for several function spaces Y and U on the boundary ∂Ω.     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.