Nonsingular systems of generalized Sylvester equations: An algorithmic approach

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ?‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ?‐Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n³r) algorithm for computing the (unique) solution.     

A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(? ⁿ ) is a Banach space as well as an algebra, while B(? ⁿ , ? ^m ) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σ _r be the set of all operators of finite rank r in B(E, F) (or B(? ⁿ , ? ^m )). In fact, we have that 1) suppose Σ _r ∈ B(? ⁿ , ? ^m ), and then Σ _r is a smooth and path-connected submanifold of B(? ⁿ , ? ^m ) and dimΣ _r = (n + m)r ? r ², for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σ _r and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σ _r ∈ B(E, F), and dimF = ∞, and then Σ _r is a smooth and path-connected submanifold of B(E, F) with the tangent space T _A Σ _r = {B ∈ B(E, F): BN(A) ? R(A)} at each A ∈ Σ _r for 0 ? r ? ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(? ⁿ ), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.     

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.

     

Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

Suppose that ? ⁿ is the p-dimensional space with Euclidean norm ∥ ? ∥, K (? ^p ) is the set of nonempty compact sets in ? ^p , ?₊ = [0, +∞), D = ?₊ × ? ^m × ? ⁿ × [0, a], D ₀ = ?₊ × ? ^m , F ₀: D ₀ → K (? ^m ), and co F ₀ is the convex cover of the mapping F ₀. We consider the Cauchy problem for the system of differential inclusions $$\dot x \in \mu F(t,x,y,\mu ),\quad \dot y \in G(t,x,y,\mu ),\quad x(0) = x_0 ,\quad y(0) = y_0$$ with slow x and fast y variables; here F: D → K (? ^m ), G: D → K (? ⁿ ), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F ₀, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ₀ > 0 such that for any μ ∈ (0, μ₀] and any solution (x _μ(t), y _μ(t)) of the problem under consideration, there exists a solution u _μ(t) of the problem ${\dot u}$ ∈ μ co F ₀ (t, u), u(0) = x ₀ for which the inequality ∥x _μ(t) ? u _μ(t)∥ < ε holds for each t ∈ [0, 1/μ].     

Robust stabilization of some class of uncertain systems

Consider an uncertain system $$ \dot x = A( \cdot )x + B( \cdot )u, $$ where A(·) ∈ ? ^{n × n} , B(·) ∈ ? ^{n × m} , and the elements of matrices A(·) and B(·) are arbitrary functionals. It is assumed that all elements are uniformly bounded, and that the first r elements counted from above and situated on a certain fixed upper superdiagonal are alternating. It is also assumed that m = n ? r, and that a matrix formed by the last m rows of matrix B(·) is nonsingular. The control u = S(·)x is synthesized, and conditions on the admissible matrix B(·) ensuring the global asymptotical stability of the system are obtained. We consider the case when modulation of the components of the vector u is realized by means of synchronous amplitude-frequency pulse modulators of the first kind. A lower estimate for the pulse frequency under which the pulse system is globally asymptotically stable is obtained.     

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman-Morrison-Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver.The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ(m,n,k)=O(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

Modal stabilization of a certain class of uniformly controllable systems

The system $\dot x = A( \cdot )x + b( \cdot )u,$ where A(·) ∈ ? ^n×n and b(·) ∈ ? ^n×1, is considered. The elements of the matrix A(·) and the column b(·) are bounded by nonanticipating functionals of an arbitrary nature that satisfy the condition $\mathop {\inf }\limits_{( \cdot )} A^{n - 1} ( \cdot )b( \cdot ),...,A( \cdot )b( \cdot ),b( \cdot )| > 0$ . From a given constant spectrum contained in the left half-plane, a feedback u = (s(·), x) is constructed, the coefficients of which are expressed in terms of A(·) and b(·). Conditions for the closed system to be globally exponentially stable are found. A similar result is obtained for the system $x(k + 1) = A(k)x(k) + b(k)u(k)$ .     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

一类矩阵方程的广义Hermite问题

该文主要解决了如下两个问题 问题I 已知矩阵 M∈ C^n×e, A∈C^n×m, B∈ C^m×m, 求 X∈ HC_M,n使得 A^HXA=B, 其中 HC_M,n={ X∈ C^n×n}|α^H(X-X^H)=0, for all α∈ C(M) }. 问题II 任意给定矩阵 X^* ∈C^n×n, 求 $\hat{X}\in H_E$ 使得 ||\hat{X}-X^*||=\min\limits_{X∈ H_E}||X-X^*||, 这里 H_E 为问题I的解集. 利用广义奇异值分解定理,得到了问题I的可解条件及其通解表达式, 获得了问题II的解,并进行了相应的数值计算.     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

On the deformation method of study of global asymptotic stability

We consider the one-parameter family of systems $$x' = F(x,\lambda ), x \in \mathbb{R}^n , 0 \leqslant \lambda \leqslant 1,$$ where F: ? ⁿ × [0, 1] → ? ⁿ is a continuous vector field. The solution x(t) = φ(t, y, λ) is uniquely determined by the initial condition x(0) = y = φ(0, y, λ) and can be continued to the whole axis (?∞, +∞) for all λ ∈ [0, 1]. We obtain conditions ensuring the preservation of the property of global asymptotic stability of the stationary solution of such a system as the parameter λ varies.     

A finiteness criterion and asymptotics for codimensions of generalized identities

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc _n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim _F I < ∞ and J is nilpotent. In the latter case, there exist numbers n ₀ ∈ ?, C ∈ ?₊, and t ∈ ?₊ for which gc _n (A) < +∞ if n ≥ n ₀ and gc _n (A) ～ Cn ^t d ⁿ as n → ∞, where d = PIexp(A) ∈ ?₊. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.     

Linear operators preserving unitarily invariant norms of matrices

Let F _{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n (F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on F m ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U _m (F). and V ∈ U_n (F). We characterize those linear operators T F _{m × n} → F _{m × n} which satisfy N (T(A)) = N(A)for all A ∈ F _{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F _{m × n} To develop the theory we prove some results concerning unitary operators on F _{m × n} which are of independent interest.     

Simultaneous approximation by algebraic polynomials

Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C ^r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP _n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ω^υ(f^(k),δ) denotes the usual vth modulus of smoothness off ^(k), and Moreover, for no 0≤k≤r can (1?x ²)( ^{r?k+1)/(r?k+m)}(1/n²)^{(m?1)/(r?k+m)} be replaced by (1-x²)^αkn^2αk-2, with α_k>(r-k+a)/(r-k+m).     

An artificial boundary condition for an advection–diffusion equation

Consider the advection–diffusion equation: u₁ + au_x1 ? vδu = 0 in ?ⁿ × ?⁺ with initial data u⁰; the Support of u⁰ is contained in ?(x₁ < 0) and a: ?ⁿ → ? is positive. In order to approximate the full space solution by the solution of a problem in ? × ?⁺, we propose the artificial boundary condition: u₁ + au_x1 = 0 on ∑. We study this by means of a transmission problem: the error is an O(v²) for small values of the viscosity v.     

Partial integrability of Hamiltonian systems with homogeneous potential

In this paper we consider systems with n degrees of freedom given by the natural Hamiltonian function of the form $$ H = \frac{1} {2}p^T Mp + V(q), $$ where q = (q ₁, …, q _n ) ∈ ? ⁿ , p = (p ₁, …, p _n ) ∈ ? ⁿ , are the canonical coordinates and momenta, M is a symmetric non-singular matrix, and V (q) is a homogeneous function of degree k ∈ ?*. We assume that the system admits 1 ? m < n independent and commuting first integrals F ₁, … F _m . Our main results give easily computable and effective necessary conditions for the existence of one more additional first integral F _m+1 such that all integrals F ₁, … F _m+1 are independent and pairwise commute. These conditions are derived from an analysis of the differential Galois group of variational equations along a particular solution of the system. We apply our result analysing the partial integrability of a certain n body problem on a line and the planar three body problem.     

Dissipativity of some class of uncertain systems

We consider the system $$ \dot x = A\left( \cdot \right)x + B\left( \cdot \right)u, u = S\left( \cdot \right)x, t \geqslant t_0 , $$ where A(·) ∈ ? ^n×n , B(·) ? ^n×p , and S(·) ∈ ? ^p×n . The entries of matrices A(·), B(·), and S(·) are arbitrary bounded functionals. We consider the problem of constructing a matrix H > 0 and finding relations between the entries of the matrices B(·) and S(·) such that for a given constant matrix R the inequality $$ V\left( {x\left( t \right)} \right) < V\left( {x\left( {t_0 } \right)} \right) + \int\limits_{t_0 }^t {x*\left( \tau \right)Rx\left( \tau \right)d\tau ,} $$ where V(x) = x*Hx, is satisfied. This problem is solved for the cases where matrix A(·) has p sign-definite entries on the upper part of some subdiagonal or on the lower part of some superdiagonal. It is assumed also that all entries located to the left (or to the right) of the sign-definite entries are equal to zero.     

Phragmén-Lindelöf conditions for graph varieties

For P_m ∈ ?[z₁, …, z_n], homogeneous of degree m we investigate when the graph of P_m in ?ⁿ⁺¹ satisfies the Phragmén-Lindelöf condition PL(?ⁿ⁺¹, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C^∞(?ⁿ⁺¹). This is shown to happen if the varieties V_+- ? {z ∈ ?ⁿ: P_m(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V_+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V_+- and u(x) ≤ 0 on V_+- ∩ ?ⁿ also satisfies u(z) A¦ Im z¦ on V_+-. Necessary as well as sufficient conditions for V_+- to satisfy (SPL) are derived and several examples are given.     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.