Spectrum control problem for strongly irregular periodic vibrations of linear systems with zero mean of the coefficient matrix

We consider a linear periodic control system with zero mean of the coefficient matrix and with linear state feedback control periodic with the same period. We obtain necessary and sufficient conditions for the solvability of the frequency spectrum control problem with a given goal set for strongly irregular periodic vibrations. In this problem, one should find a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.     

Nonintegral criterion for oscillations of linear Hamiltonian matrix systems

In this paper we present nonintegral criteria for oscillation of linear Hamiltonian matrix system U^′=A(x)U+B(x)V, V^′=C(x)U−A^*(x)V under the hypothesis (H): A(x), B(x)=B^*(x)>0, and C(x)=C^*(x) are 2×2 matrices of real valued continuous functions on the interval I=[a,∞),(−∞<a). These criteria are conditions of algebraic type only. Our results are also useful for the detection of the oscillation of particular matrix differential systems.     

Closed irregularity sets of linear differential systems with a parameter multiplying the derivative

For any closed subset M of the real line that does not contain zero, we construct a linear differential system with bounded piecewise continuous coefficient matrix A(·) such that the corresponding system with coefficient matrix μA(·) linearly depending on a real parameter μ is Lyapunov irregular for all μ in M and Lyapunov regular for all other parameter values.     

A matrix Green's formula and optimal control of linear distributed-parameter systems

A matrix Green's formula is derived, and its application to the optimal control problem of distributed-parameter systems with both distributed and boundary inputs is presented.     

Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems

We present a new class of integer extended ABS algorithms for solving linear Diophantine systems. The proposed class contains the integer ABS (the so-called EMAS and our proposed MEMAS) algorithms and the generalized Rosser’s algorithm as its members. After an application of each member of the class a particular solution of the system and an integer basis for the null space of the coefficient matrix are at hand. We show that effective algorithms exist within this class by appropriately setting the parameters of the members of the new class to control the growth of intermediate results. Finally, we propose two effective heuristic rules for selecting certain parameters in the new class of integer extended ABS algorithms.      

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.     

On the spectrum of discrete time-varying linear systems

In this paper, we investigate the influence of small perturbations of the coefficients of discrete time-varying linear systems on the Lyapunov exponents. For that purpose we introduce the concepts of central exponents of the system and we show that these exponents describe the possible changes in the Lyapunov exponents under small perturbations. Finally, we present several formulas for the central exponents in terms of the transition matrix of the system and the so-called upper sequences. The results are illustrated by numerical examples.     

Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems

We consider quadratic matrix polynomials of the form L(l) = l²A + elB + CL(\lambda) = \lambda^{2}A + \epsilon\lambda B + C, where e\epsilon is a real parameter, A is positive definite and B and C are symmetric. The main results of the paper are the characterization of the class of symmetric matrices B for which the spectrum of the polynomial is symmetric with respect to the imaginary axis and solutions of the corresponding differential equation oscillate in time. We also extend the results in [2] to allow us to study the asymptotic behaviour of the eigenvalues for large e\epsilon.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Structure of stability and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: II

We consider families of linear differential systems depending on a real parameter that occurs only as a factor multiplying the matrix of the system. The asymptotic stability set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are asymptotically stable. We prove that a set on the real axis is the asymptotic stability set of such a family if and only if it is an F _σδ -set lying entirely on an open ray with origin at zero. In addition, for any set of this kind, the coefficient matrix of a family whose asymptotic stability set coincides with this set can be chosen to be infinitely differentiable and uniformly bounded on the time half-line.     

On the complexity of Boolean matrix ranks

We present a reduction which shows that the fooling set number, tropical and determinantal ranks of a Boolean matrix are NP-hard to compute.     

Inertias and ranks of some Hermitian matrix functions with applications

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications, we derive the necessary and sufficient conditions for the existence of maximal matrices of

Displacement ranks of matrices and linear equations

It takes of the order of N³ operations to solve a set of N linear equations in N unknowns or to invert the corresponding coefficient matrix. When the underlying physical problem has some time- or shift-invariance properties, the coefficient matrix is of Toeplitz (or difference or convolution) type and it is known that it can be inverted with O(N²) operations. However non-Toeplitz matrices often arise even in problems with some underlying time-invariance, e.g., as inverses or products or sums of products of possibly rectangular Toeplitz matrices. These non-Toeplitz matrices should be invertible with a complexity between O(N²) and O(N³). In this paper we provide some content for this feeling by introducing the concept of displacement ranks, which serve as a measure of how ‘close’ to Toeplitz a given matrix is.     

Control of asynchronous spectrum of linear systems with a two-sided dependence of blocks of complete column rank

We consider a linear periodic control system with a two-sided dependence of blocks of complete column rank in the nonstationary component of the coefficient matrix in the critical case. In this case, the nontrivial intersection of vector subspaces formed by linear spans of the columns in the blocks can be arbitrary. We assume that the control is given in the form of feedback linear in the state variables and is periodic with the period of the system. We derive necessary and sufficient conditions for the solvability of the control problem for the asynchronous spectrum, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies.