Open Problems Related to the Hurwitz Stability of Polynomials Segments

1Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, 09340 Ciudad de México, Mexico 2SEPI-UPIICSA, Instituto Politécnico Nacional, 08400 Ciudad de México, Mexico 3División de Matemáticas Aplicadas, IPICyT, 78216 San Luis, SLP, Mexico 4Research Center on Information Technology and Systems, Hidalgo State University, Pachuca, HGO, Mexico 5Mathematics Department, University of Houston, Houston, TX 77204-3008, USA


Introduction
The design of control systems arises from the need of human beings to manipulate real plants with some degree of confidence and accuracy.However, this design requires mathematical models which are often very complex if all the dynamics that involve a real plant are taken into account.Thus, a complex model requires a complex controller design.Robust control theory intends to analyze a complex model by studying its linear approximation and always assuming that this approximation will incur with some degree of modeling errors.This error is regarded as uncertainty which is modeled and bounded to determine stability conditions of a system and control laws are obtained.
Problems that arise from robust control theory containing one uncertain parameter, such as gain or constants time, and the system stability should be determined for the entire uncertainty range.The study of criteria for deciding whether a convex combination of polynomials is stable can be applied to resolution of a wide variety of uncertainty systems.The problem to be raised is as follows: find conditions about the stable polynomials  0 () and  1 () such that the convex combination described by (, ) =  0 () + (1 − ) 1 () is stable (Hurwitz), for all  ∈ [0, 1].
The first result, where necessary and sufficient conditions were presented, is known as the Bialas Theorem [1][2][3].A different approach in terms of the frequency domain is known as the Segment Lemma and it was found by Bhattacharyya et al. [4,5].This result shows that the stability of (, ) is equivalent to certain conditions that must be met by even and odd part of the polynomials  0 () and  1 ().In complex polynomials case, Bose developed a method to determine the stability of the segment of complex polynomials [6].In regard to sufficient conditions, in [7,8], the well-known Rantzer's conditions were obtained.
Based on the above criteria computer algorithms have been developed to resolve this problem.The Segment Lemma has been used to design an algorithm in [9].On this same path, a method to verify the stability of convex combinations of polynomials in a finite number of calculus is obtained in [10].In [11] an algorithm to determine the stability of segment of complex polynomials is proposed; it is based on the work of Bose [12].In the present paper, we will discuss some of these criteria and propose some open problems in these topics.Applications of Hurwitz stability criterion to Engineering problems are huge, for example, for the stability of wave propagation in strain gradient [13] or mixture [14].
The paper is organized as follows.In Section 2, the problem statement is presented.Some results with necessary and sufficient condition on stability of segment of Hurwitz polynomials and its open problems are presented in Section 3. In Section 4, some open problems with sufficient condition and using Rantzer's conditions are postulated.In Section 5, a related problem with segments of stable polynomials is the calculating of the Minimum Left Extreme, which was studied by Bialas.In Section 6 some results and open problems related to Hadamardized Hurwitz polynomials are postulated.Finally, in Section 7, some concluding remarks are provided.

Problem Statement
In this section some definitions and one motivation example in the framework of the stability of polynomials are given.Definition 1.A polynomial with real coefficients, () =  0   +  1  −1 + ⋅ ⋅ ⋅ +  −1  +   , is Hurwitz if all its roots have negative real part.Definition 2. Suppose that  0 () and  1 () are polynomials (real or complex) of degree .Let and consider the following one parameter family of polynomials: This family will be referred to as a segment of polynomials.We shall say that the segment is stable if and only if every polynomial on the segment is stable.
The following example shows that a segment of polynomials is not necessarily a stable segment although the two extremes are Hurwitz polynomials.

Necessary and Sufficient Conditions
In this section, we present two of the most known necessary and sufficient conditions on the stability of segment of Hurwitz polynomials.
Open Problem 1.Is it possible to find necessary and sufficient conditions in a similar way of the Bialas Theorem for deg( 0 ) = deg( 1 ) case?
In order to establish the Segment Lemma we give some definitions.
Given a polynomial () =  0 +  1  +  2  2 + ⋅ ⋅ ⋅ +     , we define then () =  even () +  odd () and () can be written as () =   () +   (), where In the Bialas Theorem, it is necessary that the two extremes of the segment of polynomials have different degree.Now we present the Segment Lemma, where the two extremes of the segment of polynomials have equal degrees.

The Segment Lemma
Lemma 5 (see [4,5]).Let  1 () and  2 () be -degree Hurwitz polynomials with leading coefficients of the same sign.Then the segment of polynomials [ 1 (),  2 ()] is Hurwitz, if and only if there are not  > 0 that satisfies the three conditions: Open Problem 2. Is it possible to get a result, based on the approach of the Segment Lemma, to be applicable when polynomials  0 and  1 have different degrees?After we have discussed two necessary and sufficient conditions, now in the next section, we expose two sufficient conditions: Rantzer's conditions and one approach of matrix inequalities.

Sufficient Conditions
In this section, we present two sufficient conditions for the stability of segment that are known in the literature: Rantzer's conditions and one interesting approach with matrix inequalities.

Rantzer's Conditions.
Supposing that  0 () is a Hurwitz polynomial and  1 () is semistable (their roots have real part 0), then the segment of polynomials [ 0 (),  1 ()] consists of Hurwitz polynomials if you have one of the following four conditions.
(i) The difference  =  1 −  0 satisfies (iii) Each of the polynomials  0 and  1 has at least one root in C − and (iv) Each of the polynomials  0 and  1 has at least two roots in C − and The previous conditions are known as Rantzer's conditions; see [8].
Open Problem 3. Is there a mechanism to describe Rantzer's conditions in terms of the polynomial coefficients?
Open Problem 4. What should we add to Rantzer's conditions in order to obtain necessary and sufficient conditions?
In the next subsection, we present an interesting sufficient condition for checking the stability of segment of polynomials based on matrix inequalities.
Open Problem 5.With these approach matrix inequalities, what should we add to obtain necessary and sufficient conditions?
and the corresponding inequality is There is a relation between segments and rays of polynomials, which let us give applications in control theory.We explain it in the next two subsections.
The importance of the segments and rays of Hurwitz polynomials with the design of stabilizing controls is pointed out in the next subsection.
For the uncontrolled system ẋ = , the characteristic polynomial is  0 () =   +  1  −1 +  2  −2 + ⋅ ⋅ ⋅ +   .If into the controlled system ẋ =  +  we choose a control  as () = −   = −(  ,  −1 , . . .,  1 ), where  > 0 and  ∈ R  , then the controlled system takes the form: and its characteristic polynomial is that is, where Example 9. Consider the system ẋ = ( In this case, the matrix  is given by then we have the system Therefore, the system is stable ∀ ≥ 0. But this cannot be verified with the results presented by Aguirre et al. in [15]. ) , ( ) . (28) The Minimum Left Extreme is a subject very related to the study of segment of Hurwitz polynomials; hence in the next sections we present some ideas about the Minimum Left Extreme in order to propose some open problems.

The Minimum Left Extreme
A related problem with segment of Hurwitz polynomials is the calculating of the Minimum Left Extreme, which was studied by Bialas.We begin with some definitions.