We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg–Krein–Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg–Krein–Bojarsky theorem.

The continuous models are considered in the most works on optimal advertising. Articles on the discrete-time models are more rare because in this case it is dicult to obtain an explicit solution. In this paper a new discrete model of optimal advertising for a monopolist-seller of a new goods is proposed. In the model, the dynamics is given by a nonlinear dierence equation. The non-linearity depends on a parameter σ, 0 < σ < 1, i.e. a continuous family of the models is considered. The discrete versions of the Vidale Wolfe model and the Sethi model are particular cases of this model. The seller's problem is to maximize its prot up to the nite horizon T by the optimal advertising expenditure. This problem is a discrete multistep optimal control problem, where an advertising expenditure is a control variable. For our model the optimal control problem can be solved explicitly. The Bellman method of dynaming programming is used to study the problem. Explicit recurrence relations for the optimal control and the market share up to the step t, t = 1,. .. , T , are obtained under the assumption that the dierence equation of the model has a solution. Sucient conditions on the parameters of the model, which ensure the existence of a solution, are found. The proposed algorithm is implemented as the procedure OptimalAdvertising in the package Maple. Numerical experiments with the procedure were carried out.

We consider the Wiener–Hopf factorization of two matrix functions A(t) and B(t) that are quite close in the norm of the Wiener algebra. The aim of this work is to study the question when the factorization factors of A(t), B(t) will be close enough to each other. This problem is of considerable interest in connection with the development of methods for approximate factorization of matrix functions. There are two main obstacles in the study of this problem: the instability of the partial indices of matrix functions and the non-uniqueness of their factorization factors. The problem was previously studied by M.A. Shubin, who showed that the stability of factorization factors takes place only in the case when A(t) and B(t) have the same partial indices. Then there is a factorization B(t) for which the factorization factors are sufficiently close to the factors of A(t). Theorem M.A. Shubin is non-constructive since it is not known when the partial indices of two close matrix functions will be the same, and the method for choosing the required Wiener–Hopf factorization of the matrix function B(t) is not indicated. To overcome these shortcomings, in the present paper we study the problem of normalization of the factorization in the stable case, describe all possible types of normalizations, and prove their stability under a small perturbation A(t). Now it is possible to find a constructive way of choosing the factorization of the perturbed matrix function, which guarantees the stability of the factorization factors.

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