Stability of uncertain discrete systems

The uncertain system

$x_{n + 1} = A_n x_n , n = 0,1,2, \ldots ,$

is considered, where the coefficients a _ij (n) of the m×m matrix A _n are functionals of any nature subject to the constraints

$\begin{array}{*{20}c} {\left| {a_{i,i} (n)} \right| \leqslant \alpha _ * < 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \alpha _0 for j \geqslant i + 1,} \\ {\left| {a_{i,j} (n)} \right| \leqslant \delta for j < i.} \\ \end{array} $

Such systems include, in particular, switched-type systems, whose matrix A can take values in a given finite set.By using a special Lyapunov function, a bound δ ≤ δ(α₀,α_*) ensuring the global asymptotic stability of the system is found. In particular, the system is stable if the last inequality is replaced by a _i,j (n) = 0 for j < i.It is shown that pulse-width modulated systems reduce to the uncertain systems under consideration; moreover, in the case of a pulse-width modulation of the first kind, the coefficients of the matrix A are functions of x(n), and in the case of a modulation of the second kind, they are functionals.     

Boundedness of Hausdorff operators on the power weighted Hardy spaces

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp.     

Spectrum and the trace formula for a two-dimensional Schrödinger operator in a homogeneous magnetic field

In the space L ₂(?²), we consider the operator

$H = \left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }} - x_2 } \right)^2 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }} + x_1 } \right)^2 + V,V = V(x) \in L_2 (\mathbb{R}^2 ).$

. We study the spectrum of H and, for V ∈ C ₀ ² (?²), prove the trace formula

$\sum\limits_{k = 0}^\infty {\left( {\sum\limits_{i = - k}^\infty {(4k + 2 - \mu _k^{(i)} ) + c_0 } } \right)} = \frac{1}{{8\pi }}\int\limits_{\mathbb{R}^2 } {V^2 (x)dx,} $

where c ₀ = π ^?1 \(\smallint _{\mathbb{R}^2 } \) V(x) dx and the µ _k ⁽ⁱ⁾ are the eigenvalues of H.

     

Acceleration of convergence of general linear sequences by the Shanks transformation

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A _n }. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {A _n } defined as solutions of the linear systems

$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$

where \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {A _n }, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζ _k  ≠ 1 are distinct and |ζ ₁| = ... = |ζ _m | = θ, and γ _k  ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A _n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems

$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$

have unique (integer) solutions for s ₁, . . . , s _m . A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with A _j ? A = O(θ ^j j ^α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.

     

Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

Multiple recurrence and convergence for hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for “typical” choices of Hardy field functions a(t) with polynomial growth, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[a(n)]}}x) \cdots {f_\ell }({T^{\ell [a(n)]}}x)} $

converge in mean and we determine their limit. For example, this is the case if a(t) = t ^3/2, t log t, or t ² + (log t)². Furthermore, if {a ₁(t), …, a _? (t)} is a “typical” family of logarithmico-exponential functions of polynomial growth, then for every ergodic system, the averages

${1 \over N}\sum\nolimits_{n = 1}^N {{f_1}({T^{[{a_1}(n)]}}x) \cdots {f_\ell }({T^{[{a_\ell }(n)]}}x)} $

converge in mean to the product of the integrals of the corresponding functions. For example, this is the case if the functions a _i (t) are given by different positive fractional powers of t. We deduce several results in combinatorics. We show that if a(t) is a non-polynomial Hardy field function with polynomial growth, then every set of integers with positive upper density contains arithmetic progressions of the form {m,m + [a(n)], …, m + ?[a(n)]}. Under suitable assumptions, we get a related result concerning patterns of the form {m,m + [a ₁(n)], …,m + [a _? (n)]}.

     

Hölder seminorm preserving linear bijections and isometries

Let (X, d) be a compact metric and 0 < α < 1. The space Lip ^α (X) of Hölder functions of order α is the Banach space of all functions ? from X into \(\mathbb{K}\) such that ∥?∥ = max{∥?∥_∞, L(?)} < ∞, where

$L(f) = sup\{ \left| {f(x) - f(y)} \right|/d^\alpha (x,y):x,y \in X, x \ne y\} $

is the Hölder seminorm of ?. The closed subspace of functions ? such that

$\mathop {\lim }\limits_{d(x,y) \to 0} \left| {f(x) - f(y)} \right|/d^\alpha (x,y) = 0$

is denoted by lip ^α (X). We determine the form of all bijective linear maps from lip ^α (X) onto lip ^α (Y) that preserve the Hölder seminorm.

     

Irregular Wavelet Frames and Gabor Frames

Given g∈L²(R ⁿ ), we consider irregular wavelet for the form\(\left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j \) > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L²(R ⁿ ) are given. For a class of functions g∈L²²(R ⁿ ) we prove that certain growth conditions on {λ _j } will frames, and that some other types of sequences exclude the frame property. We also give a sufficient condition for a Gabor system\(\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} \)to be a frame.     

Regularized traces of singular differential operators with canonical boundary conditions

A self-adjoint differential operator \(\mathbb{L}\) of order 2m is considered in L ₂[0,∞) with the classic boundary conditions \(y^{(k_1 )} (0) = y^{(k_2 )} (0) = y^{(k_3 )} (0) = \ldots = y^{(k_m )} (0) = 0\), where 0 ≤ k ₁ < k ₂ < ... < k _m ≤ 2m ? 1 and {k _s } _s=1 ^m ∪ {2m ? 1 ? k _s } _s=1 ^m = {0, 1, 2, ..., 2m ? 1}. The operator \(\mathbb{L}\) is perturbed by the operator of multiplication by a real measurable bounded function q(x) with a compact support: \(\mathbb{P}\) f(x) = q(x)f(x), f ∈ L ₂[0,∞). The regularized trace of the operator \(\mathbb{L} + \mathbb{P}\) is calculated.     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of

$${\int}_{\Omega} f(x,u,\nabla u)\,dx $$

with f subject to the general structural conditions

$$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$

where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ L ^t (Ω) for some t > n/p ^?, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L ^∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L ^∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Hölder-continuous.

     

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS_) and their effect on spectral properties of the underlying Lax operator. The latter is a formallyJ (but nonself-adjoint) Diract-type differential expression of the form

$M(q) = i\left( {\begin{array}{*{20}c} {\frac{d}{{dx}}} &; { - q} \\ { - \bar q} &; { - \frac{d}{{dx}}} \\ \end{array} } \right)$

(1)

satisfying\({\mathcal{J}} M(q)\mathcal{J} = M(q)^* \), whereJ is defined byJ C, andC denotes the antilinear conjugation map in ?²,\({\mathcal{C}}(a,b)^{\rm T} = (\bar a,\bar b)^{\rm T} ,a,b \in \) ?. As one of our principla results, we prove that under the most general hypothesisq ∈ _loc ¹ (?) onq, the maximally defined operatorD(q) generated byM(q) is actually {itJ}-self-adjoint in inL ²(?)². Moreover, we establish the existence of Weyl-Titchmarsh-type solutions ψ+(z, ·) ?L ² ([R, ∞))² and ψ?(z, ·) ∈L ² ((?∞,R]) for allR∈? ofM(q)Ψ ± (z)=zΨ ± (z) forz in the resolvent set ofD(q).

The Darboux transformations considered in this paper are the analogue of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question guarantee that the resulting potentialsq are locally nonsingular. Moreover, we prove that the construction ofN-soliton NLS_potentialsq ^(N) with respect to a general NLS background potentialq ?L _loc ¹ (?), associated with the Dirac-type operatorsD(q ^(N) ) andD(q), respectively, amounts to the insertio ofN complex conjugate pairs ofL ²({?}²-eigenvalues\(\{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} \) into the spectrum σ(D(q)) ofD(q), leaving the rest of the spectrum (especially, the essential spectrum σ_e(itD)(q))) invariant, that is,

$\sigma (D(q^{(N)} )) = \sigma (D(q)) \cup \{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} ,$

(1)

$\sigma _e (D(q^{(N)} )) = \sigma _e (D(q))$

(1)

These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operatorD(q) and the Dirac operatorD(q ^(N) ) obtained afterN Darboux-type transformations.     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

Real submanifolds in complex spaces

Let(z_(11),..., z_(1N),..., z_(m1),..., z_(mN), w_(11),..., w_(mm)) be the coordinates in C~(mN) +m~2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows W = ZZ~t+ O(3),where W = {w_(ij)}_(1≤i,j≤m)and Z = {z_(ij) }_(1≤i≤m, 1≤j≤N). We prove that M is biholomorphically equivalent to the model W = ZZ~t if and only if is formally equivalent to it.     

Aubry–Mather Sets for Relativistic Oscillators with Anharmonic Potentials

In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models d/dt(x/(1-|x|~2~(1/2))+ |x|~(α-1)x=p(t),where p(t) ∈ C~0(R~1) is 1-periodic and α 0.     

On the exceptional sets in Sylvester expansions*

For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {d _j (x),?j?≥?1} is a sequence of positive integers satisfying d₁(x)?≥?2 and d_j?+?1(x)?≥?d _j (x)(d _j (x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?⁺ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set

$$ E\left(\phi \right)=\left\{x\kern0.5em \in \left(0,1\right]:\kern0.5em \underset{n\to \infty }{\lim}\frac{\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)}{\phi (n)}=1\right\} $$

and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$