The Stability Cone for a Matrix Delay Difference Equation

Abstract

We construct a stability cone, which allows us to analyze the stability of the matrix delay difference equation 𝑥_𝑛=𝐴𝑥_𝑛−1+𝐵𝑥_𝑛−𝑘. We assume that 𝐴 and 𝐵 are 𝑚×𝑚 simultaneously triangularizable matrices. We construct 𝑚 points in ℝ³ which are functions of eigenvalues of matrices 𝐴,  𝐵 such that the equation is asymptotically stable if and only if all the points lie inside the stability cone.

1. Introduction

Parameters of linear systems are subject to time changes. That is why in order to construct such systems it is desirable to know if they are not only stable but also able to estimate the distance of the system from the boundary of the stability region in the parameter space. Therefore, it makes sense to investigate the geometry of the subset of stable polynomials in the space of characteristic polynomials of linear systems (in the canonical space [1]). This idea has already been applied to the investigation of geometry of the subset of stable polynomials in a two-dimensional subspace of the canonical space [2, 3], the stability simplex for general difference equations [4], connections of the convexity of the coefficients sequence with stability of difference equations [5], and stability ovals for matrix difference equations of the form 𝑥_𝑛=𝑥_𝑛−1+𝐵𝑥_𝑛−𝑘 with the delay 𝑘 [6].

Consider the matrix equation𝑥_𝑛=𝐴𝑥_𝑛−1+𝐵𝑥_𝑛−𝑘,𝑛=0,1,2…, (1.1) where 𝑘∈ℤ₊ is the delay. Equations of the form (1.1) have been used for investigations of a delayed discrete-time Hopfield neural network [7, 8]. The suitable representation of the solution of (1.1) with commutative matrices 𝐴,𝐵 and nonsingular 𝐴 is given in [9]. But [9] does not solve the stability problems of (1.1). The stability of (1.1) was investigated in [7, 8, 10, 11] with special 2×2 matrices 𝐴,𝐵. In [12], the stability of (1.1) was investigated without any restriction on the dimension but with the special matrix 𝐴=𝛼𝐼, 𝛼∈ℝ, 0⩽𝛼⩽1, where 𝐼 is the identity matrix.

In this paper we give a geometric solution to the problem of asymptotic stability of (1.1) in any dimension with simultaneously triangularizable matrices 𝐴,𝐵. It is known that commuting matrices are simultaneously triangularizable [13]. As usual, we say that (1.1) is stable if its zero solution is stable. Our solution is based on constructing the stability ovals which, in turn, form a stability cone. At the same time we give an algorithm for checking the stability of the scalar equation𝑥_𝑛=𝑎𝑥_𝑛−1+𝑏𝑥_𝑛−𝑘,𝑛=0,1,2… (1.2) with complex coefficients 𝑎,𝑏.

The paper is organized as follows. In the second section, we recall the results on the stability of the scalar equation (1.2) with real nonnegative 𝑎 and any real 𝑏 [14, 15]. Further in that section we construct the stability oval for (1.2) with real nonnegative coefficient 𝑎 and complex coefficient 𝑏. In Section 3, we consider a wider class of equations of the form (1.2) with 𝑎,𝑏 being complex numbers. In Section 4, we state a system of inequalities allowing us to check the stability of the scalar equation (1.2) with two complex coefficients. In Section 5, we give a geometrical criterion for the asymptotic stability of matrix equation (1.1) with simultaneously triangularizable matrices. Besides, we establish nongeometric necessary and sufficient conditions for the stability of matrix equation (1.1) in terms of inequalities. In Section 6, we use the stability ovals and cones for analysis of some numerical examples.

2. The Stability Oval for (1.2) with Real Nonnegative 𝑎 and Complex 𝑏

We start by stating the results from [14, 15] in the form which is suitable for us. Since the case 𝑘=1 is obvious, we consider only the case 𝑘>1.

Theorem 2.1 (see [14, 15]). In (1.2) let 𝑎 and 𝑏 be real, 𝑎⩾0, 𝑘>1. (1) If  𝑎⩾𝑘/(𝑘−1), then (1.2) is unstable.(2)If  0⩽𝑎<𝑘/(𝑘−1), then (1.2) is asymptotically stable if and only if −𝑎²+1−2𝑎cos𝜔₁<𝑏<1−𝑎, (2.1) where 𝜔₁∈[0,𝜋/𝑘] is the root of the equation𝑎=sin𝑘𝜔()𝜔.sin𝑘−1 (2.2)

Stability region of (1.2) is shown in Figure 1.

Figure 1 

The domain of asymptotic stability of (1.2), 𝑎⩾0, 𝑘=6.

Our first new result about the case of nonnegative real 𝑎 and complex 𝑏 in (1.2) is the following.

Theorem 2.2. Let 𝑎⩾0 be a real number and 𝑏 a complex number, 𝑘>1. (1) If  𝑎⩾𝑘/(𝑘−1), then (1.2) is unstable.(2) If  0⩽𝑎<𝑘/(𝑘−1), then (1.2) is asymptotically stable if and only if 𝑏 lies inside the oval bounded by 𝑏=exp(𝑖𝑘𝜔)−𝑎exp(𝑖(𝑘−1)𝜔),−𝜔₁⩽𝜔⩽𝜔₁, (2.3) where 𝜔₁∈[0,𝜋/𝑘] is the root of (2.2).(3) If  0⩽𝑎<𝑘/(𝑘−1) and 𝑏 is outside of the stability oval (2.3), then (1.2) is unstable.(4) If  0⩽𝑎<𝑘/(𝑘−1) and 𝑏 lies on the boundary (2.3) of the stability oval, then (1.2) is stable (nonasymptotically).

Proof. We will use the 𝐷-decomposition method (parameter plane method) [16, 17]. A characteristic polynomial for (1.2) is the following: 𝑓(𝑧)=𝑧^𝑘−𝑎𝑧^𝑘−1−𝑏. (2.4) For any fixed value of 𝑎, the complex plane of the parameter 𝑏 is divided into some regions by a curve 𝑓(exp(𝑖𝜔))=0, that is, exp(𝑖𝑘𝜔)−𝑎exp(𝑖(𝑘−1)𝜔)−𝑏=0,−𝜋⩽𝜔⩽𝜋. (2.5) Hence, 𝐷-decomposition occurs in the plane of the complex parameter 𝑏 by means of a curve 𝑏(𝜔)=exp(𝑖𝑘𝜔)−𝑎exp(𝑖(𝑘−1)𝜔),−𝜋⩽𝜔⩽𝜋. (2.6) The example of the 𝐷-decomposition for 𝑘=6, 𝑎=0.7 is shown in Figure 2. From (2.6) we obtain |𝑏(𝜔)|²=1+𝑎²−2𝑎cos𝜔. (2.7) Therefore, |𝑏| increases monotonically when 𝜔 runs from 0 to 𝜋. Similarly, |𝑏| increases monotonically when 𝜔 runs from 0 to (−𝜋). Let us construct an increasing sequence (𝜔_𝑖)^𝑟_𝑖=0 of all values 𝜔∈[0,𝜋], such that Im(𝑏(𝜔_𝑖))=Im(𝑏(−𝜔_𝑖))=0. Here 1⩽𝑟⩽𝑘, 𝜔₀=0, 𝜔_𝑟=𝜋. Each pair of the curves (2.6) formed by motion 𝜔 on intervals [𝜔_𝑖,𝜔_𝑖+1], [−𝜔_𝑖+1,−𝜔_𝑖] creates a new region of 𝐷-decomposition of the complex plane of parameter 𝑏. This region necessarily contains some real values of 𝑏 as the function |𝑏(𝜔)| is monotone on [0,𝜋] and on [−𝜋,0]. Due to the properties of the regions, if some inner point of this region is an asymptotically stable point, then the whole region consists of asymptotically stable points. If 𝑎⩾𝑘/(𝑘−1), then according to Theorem 2.1 there are no stable points on the axis Im(𝑏)=0. Hence, there are no stable points on the complex plane of parameter 𝑏. Let now 𝑎<𝑘/(𝑘−1). Let 𝜔₁∈[0,𝜋/𝑘] be the root of (2.2). Then the unique 𝐷-decomposition region containing the real straight line segment (2.1) is the oval with the boundary (2.3). Parts 1–3 of Theorem 2.2 are proved. Direct checking shows that the derivative of a characteristic polynomial (2.4) is not equal to zero on the boundary of the stability oval. Therefore, when the parameter 𝑏 runs along the boundary of the stability oval, all the corresponding roots 𝑧 of the characteristic equation, satisfying |𝑧|=1, are simple. Hence, (1.2) is stable (nonasymptotically). Theorem 2.2 is proved.

Figure 2 

𝐷-decomposition of the complex plane of the parameter 𝑏 for 𝑘=6, 𝑎=0.7.

Example 2.3. Let 𝑘=6 and, also, (1) 𝑎=0.2, (2) 𝑎=0.75, and (3) 𝑎=1.1. Let 𝑏=0.33exp(𝑖𝛼), 0⩽𝛼<2𝜋. Let us analyze the asymptotic behavior of the solutions of (1.2) for all values of 𝛼. We construct three stability ovals for three values of 𝑎 and also the circle 𝑏=0.33exp(𝑖𝛼) (Figure 3). Theorem 2.2 and Figure 3 give the following result. (1) For 𝑎=0.2, the equation is asymptotically stable for any value of 𝛼. (2) For 𝑎=0.75, the equation is asymptotically stable for 2.0918≅𝛼₀<𝛼<2𝜋−𝛼₀≅4.1914, it is unstable for 𝛼∉[𝛼₀,2𝜋−𝛼₀], and it is stable (nonasymptotically) for 𝛼=𝛼₀ and 𝛼=2𝜋−𝛼₀. (3) Finally, for 𝑎=1.1, the equation is unstable for any value of 𝛼.

Figure 3 

Stability ovals for 𝑘=6,𝑎=0.2,and𝑎=0.75;𝑎=1.1. A circle 𝑏=0.33exp(𝑖𝛼) and a beam 𝑏=𝑟exp(𝑖19𝜋/20) are constructed for Examples 2.3 and 2.4.

Example 2.4. Let 𝑘=6 and, also, (1) 𝑎=0.2, (2) 𝑎=0.75, and (3) 𝑎=1.1. Let 𝑏=𝑟exp(𝑖19𝜋/20), 𝑟⩾0. Let us find the asymptotic behavior of (1.2) for all values of 𝑟. We construct the beam 𝑏=𝑟exp(𝑖19𝜋/20) together with three stability ovals (Figure 3). Theorem 2.2 and Figure 3 give the following result. (1)For 𝑎=0.2, the equation is asymptotically stable for 0⩽𝑟<𝑟₁≅0.8276, it is unstable for 𝑟>𝑟₁, and it is stable (nonasymptotically) for 𝑟=𝑟₁. (2)For 𝑎=0.75, the equation is asymptotically stable for 0⩽𝑟<𝑟₂≅0.4080, it is unstable for 𝑟>𝑟₂, and it is stable (nonasymptotically) for 𝑟=𝑟₂. (3)Finally, for 𝑎=1.1, the equation is stable for 0.1063≅𝑟₃₁<𝑟<𝑟₃₂≅0.1944, it is unstable for 𝑟∉[𝑟₃₁,𝑟₃₂], and it is stable (nonasymptotically) for 𝑟=𝑟₃₁, 𝑟=𝑟₃₂.

3. The Stability Cone for (1.2) with Complex Coefficients

The family of stability ovals depending on 𝑎∈𝑘/(𝑘−1) forms a surface which we call the stability cone.

Definition 3.1. The stability cone for delay 𝑘 is a surface in a three-dimensional space (Re(𝑏),Im(𝑏),𝑧) with 0⩽𝑧⩽𝑘/(𝑘−1), such that its intersection with any plane 𝑧=𝑎(0⩽𝑎⩽𝑘/(𝑘−1)) is the stability oval (2.3).

The stability cone is the image of the two-dimensional domain in the space (𝜔,𝑎)0⩽𝑎⩽sin𝑘𝜔()𝜔,−𝜋sin𝑘−1𝑘𝜋⩽𝜔⩽𝑘 (3.1)

under the mapping into ℝ³ by the functions(𝑏)()(𝑏)()Re=cos𝑘𝜔−𝑎cos𝑘−1𝜔,Im=sin𝑘𝜔−𝑎sin𝑘−1𝜔,𝑧=𝑎. (3.2)

The stability cone for 𝑘=6 is presented in Figure 4.

Figure 4 

Stability cone for 𝑘=6. The point on the cone is constructed for Example 3.3.

Let us now study the problem of the stability of scalar equation (1.2) with complex coefficients 𝑎,𝑏. We consider the equation𝑥_𝑛=𝜌₁exp𝑖𝛼₁𝑥_𝑛−1+𝜌₂exp𝑖𝛼₂𝑥_𝑛−𝑘, (3.3) with real nonnegative 𝜌₁,𝜌₂ and real 𝛼₁, 𝛼₂. We set 𝑥_𝑛=𝑦_𝑛exp(𝑖𝑛𝛼₁). Then (3.3) becomes𝑦_𝑛=𝜌₁𝑦_𝑛−1+𝜌₂exp𝑖𝛼₂−𝑘𝛼₁𝑦_𝑛−𝑘. (3.4) Obviously, (3.4) is stable (asymptotically stable) if and only if (3.3) is stable (asymptotically stable). The stability problem of (3.4) can be solved due to Theorem 2.2. Thus, we obtain the following theorem.

Theorem 3.2. Consider (3.3). Put 𝑎=𝜌₁,𝑏=𝜌₂exp𝑖𝛼₂−𝑘𝛼₁. (3.5) Construct the point 𝑀=(Re(𝑏),Im(𝑏),𝑎) in ℝ³.(1)Equation (3.3) is asymptotically stable if and only if the point 𝑀 lies inside the cone (3.2) (if 𝑎=0 and (Re(𝑏))²+(Im(𝑏))²<1, then the point 𝑀 is assumed to be the inner point of the cone).(2)If the point 𝑀 lies outside the cone (3.2) or on its top Re(𝑏)=−1/(𝑘−1),Im(𝑏)=0,𝑎=𝑘/(𝑘−1), then (3.3) is unstable.(3)If the point 𝑀 lies on the boundary of a cone (3.2), but not on its top, then (3.3) is stable (nonasymptotically).

Example 3.3. Let us test the stability of the equation with complex coefficients 𝑥_𝑛=𝜎exp𝑖𝜋5𝑥_𝑛−1+0.7exp𝑖𝜋3𝑥_𝑛−𝑘 (3.6) with a real parameter 𝜎 and a delay 𝑘=6. Put first 𝜎⩾0. By Theorem 3.2, according to (3.5) we calculate 𝑖𝜋𝑏=0.7exp3𝜋−6⋅5=0.7exp𝑖17𝜋.15 (3.7) The vertical line (Re(𝑏),Im(𝑏),𝜎), (0⩽𝜎<∞) in ℝ³ intersects the boundary of the stability cone at 𝑧=𝜎₀≅0.3442 (Figure 4). To study the negative values of 𝜎 we rewrite (3.6): 𝑥_𝑛=−𝜎exp𝑖6𝜋5𝑥_𝑛−1+0.7exp𝑖𝜋3𝑥_𝑛−𝑘. (3.8) By Theorem 3.2, according to (3.5) we calculate 𝑖𝜋𝑏=0.7exp3−6⋅6𝜋5=0.7exp𝑖17𝜋.15 (3.9) As the results (3.9), (3.7) coincide, we obtain the following answer: (3.6) is asymptotically stable for (−𝜎₀)<𝜎<𝜎₀≅0.3442, and it is unstable for 𝜎∉[−𝜎₀,𝜎₀]. According to part 2 of Theorem 3.2 for 𝜎=𝜎₀ or 𝜎=−𝜎₀, (3.6) is stable (nonasymptotically).

Example 3.4. We test (3.6) with a real parameter 𝜎 for stability. Unlike the previous example let now the delay be odd: 𝑘=7. For positive 𝜎 by Theorem 3.2, according to (3.5) we obtain 𝑖𝜋𝑏=0.7exp3𝜋−7⋅5=0.7exp𝑖14𝜋.15 (3.10) The vertical line (Re(𝑏),Im(𝑏),𝜎) in ℝ³ intersects the boundary of the stability cone at 𝑧=𝜎₀≅0.3377. To study the negative values of 𝜎 similarly to the previous example, we obtain 𝑖𝜋𝑏=0.7exp3−7⋅6𝜋5=0.7exp𝑖29𝜋.15 (3.11) The results (3.10), (3.11) do not coincide. For (3.11) the vertical line (Re(𝑏),Im(𝑏),𝜎), (0⩽𝜎<∞) intersects the boundary of the stability cone at 𝑧=𝜎₁≅0.3002. We obtain the following answer: (3.6) with 𝑘=7 is asymptotically stable if (−0.3002)≅−𝜎₁<𝜎<𝜎₀≅0.3377, it is unstable if 𝜎∉[−𝜎₁,𝜎₀], and it is stable (nonasymptotically) for 𝜎=𝜎₀ or 𝜎=−𝜎₁.

Comparison of Examples 3.3 and 3.4 reveals the difference in the behavior of (3.3) for even and odd values of the delay 𝑘. Let us compare the stability of (1.1) and the following equations:𝑥_𝑛=−𝐴𝑥_𝑛−1+𝐵𝑥_𝑛−𝑘,𝑥_𝑛=−𝐴𝑥_𝑛−1−𝐵𝑥_𝑛−𝑘. (3.12)(3.13) Substituting 𝑥_𝑛=(−1)^𝑛𝑦_𝑛 reduces (1.1) to𝑦_𝑛=−𝐴𝑦_𝑛−1+(−1)^𝑘𝐵𝑦_𝑛−𝑘. (3.14) Equations (1.1) and (3.14) are simultaneously stable or unstable. Therefore, we have the following symmetry property of the stability region for (1.1).

Theorem 3.5. For even delays 𝑘 the (asymptotic) stability of (1.1) implies the (asymptotic) stability of (3.12) and vice versa. For odd 𝑘 the (asymptotic) stability of (1.1) implies the (asymptotic) stability of (3.13) and vice versa.

Similar properties of symmetry have been specified in [11, 15] for the scalar equation (1.2) with real 𝑎,𝑏.

4. A System of Inequalities for Checking the Stability of (3.3)

In the previous sections we used some geometric procedures. In this section we construct a system of inequalities in order to check the stability of (3.3). Henceforth we assume 0⩽arg(𝑧)<2𝜋 for a complex variable 𝑧.

Theorem 4.1. (1) If 𝜌₁<1−𝜌₂, then (3.3) is asymptotically stable.(2)If1−𝜌₂⩽𝜌₁<min(1+𝜌₂,𝑘/(𝑘−1)), then for the asymptotic stability of (3.3) it is necessary and sufficient to fulfill simultaneously the following conditions ((𝐻1)), ((𝐻2)):𝜌₂<𝜌²₁+1−2𝜌₁cos𝜔₁,𝐻1 () where 𝜔₁∈[0,𝜋/𝑘] is the root of the equation𝜌₁=sin𝑘𝜔()𝜔,𝑖𝛼sin𝑘−1𝜋−argexp₂−𝑘𝛼₁<𝜋−(𝑘−1)arccos1+𝜌²₁−𝜌²₂2𝜌₁−arccos1−𝜌²₁−𝜌²₂2𝜌₁𝜌₂.𝐻2 ()(3)If𝜌₁⩾min(1+𝜌₂,𝑘/(𝑘−1)), then (3.3) is not asymptotically stable.

Proof. The stability of (3.3) is equivalent to the stability of (3.4), so we will work with (3.4).(1)For 𝜌₁<1−𝜌₂ by Theorem 2.2 the stability oval exists and the circle of radius 𝜌₂ lies completely inside the oval. Therefore, Theorem 2.2 implies the asymptotic stability of (3.4). Part 1 of Theorem is proved.(2)Let 1−𝜌₂⩽𝜌₁<min(1+𝜌₂,𝑘/(𝑘−1)). Let us consider two cases.
Case 1 ((1−𝜌₂)⩽𝜌₁<1). In this case, the stability oval exists by Theorem 2.2, and the origin of the coordinates lies inside the oval. For (3.4) to be asymptotically stable, it is necessary and sufficient to satisfy the two following conditions. The first one is that the circle of radius 𝜌₂ should intersect the stability oval. It is equivalent to ((𝐻1)). The second condition is that the argument of a point exp(𝑖(𝛼₂−𝑘𝛼₁)) should be between the arguments of the two crosspoints 𝑀₁,𝑀₂ of a circle of radius 𝜌₂ with the stability oval (2.3). Let us assume that Im(𝑀₁)>0, and let the parameter 𝜔 correspond to the point 𝑀₁. We obtain arg𝑀₁=argexp(𝑖𝑘𝜔)−𝜌₁exp(𝑖(𝑘−1)𝜔)=(𝑘−1)𝜔+argexp(𝑖𝜔)−𝜌₁ (4.1) from (2.3). But we also obtain cos𝜔=1+𝜌²₁−𝜌²₂2𝜌₁ (4.2) from (2.7). Equalities (4.1), (4.2) give arg𝑀₁=(𝑘−1)arccos1+𝜌²₁−𝜌²₂2𝜌₁+arccos1−𝜌²₁−𝜌²₂2𝜌₁𝜌₂. (4.3) It follows from (4.3) that the second requirement is equivalent to ((𝐻2)). Part 2 of Theorem 4.1 is proved in Case 1.
Case 2 (1⩽𝜌₁<min(1+𝜌₂,𝑘/(𝑘−1))). By virtue of the inequality 𝜌₁<𝑘/(𝑘−1), the stability oval exists, and, since 𝜌₁⩾1, the origin does not lie inside the oval. The same requirements as in the previous case lead to the same conditions ((𝐻1)), ((𝐻2)). Part 2 of the Theorem is proved.(3)Let 𝜌₁⩾min(1+𝜌₂,𝑘/(𝑘−1)). We consider two cases.
Case 1 (1+𝜌₂⩽𝜌₁<𝑘/(𝑘−1)). In this case the stability oval exists by Theorem 2.2, and, by virtue of the inequality 𝜌₁⩾1, the origin does not lie inside the oval. Due to the inequality 𝜌₂⩽𝜌₁−1, no point of the circle of radius 𝜌₂ lies inside the oval and so (3.4) is not asymptotically stable by Theorem 2.2.
Case 2 (𝜌₁⩾𝑘/(𝑘−1)). In this case, (3.4) is unstable by Theorem 2.2.
Theorem 4.1 is proved.

As we see, the text of Theorem 4.1 does not contain any geometric terms. However, Theorem 3.2 has a considerable advantage over Theorem 4.1 because of its simplicity and geometric visualization. That is why in the future examples we prefer describing the stability of matrix equation (1.1) in geometric terms.

5. The Stability Cone for the Matrix Equation with Simultaneously Triangularizable Matrices

In this section we consider (1.1) with simultaneously triangularizable matrices 𝐴, 𝐵.

Theorem 5.1. Let 𝐴,𝐵,𝑆∈ℝ^𝑚×𝑚 and 𝑆⁻¹𝐴𝑆=𝐴_𝑇 and 𝑆⁻¹𝐵𝑆=𝐵_𝑇, where 𝐴_𝑇,𝐵_𝑇 are the lower triangular matrices with elements, respectively, 𝜆_𝑗𝑠,𝜇_𝑗𝑠(1⩽𝑗,𝑠⩽𝑚). Let 𝑏_𝑗=𝜇_𝑗𝑗exp𝑖arg𝜇_𝑗𝑗−𝑘arg𝜆_𝑗𝑗,𝑎_𝑗=𝜆_𝑗𝑗1⩽𝑗⩽𝑚, (5.1) and let the points 𝑀_𝑗 in ℝ³ be constructed by 𝑀_𝑗=Re𝑏_𝑗,Im𝑏_𝑗,𝑎_𝑗1⩽𝑗⩽𝑚. (5.2) Then (1.1) is asymptotically stable if and only if for any 𝑗(1⩽𝑗⩽𝑚) the point 𝑀_𝑗 lies inside cone (3.2).
If for some 𝑗(1⩽𝑗⩽𝑚) the point 𝑀_𝑗 lies outside cone (3.2), then (1.1) is unstable.

Proof. In (1.1) we substitute 𝑥_𝑛=𝑆𝑦_𝑛 and multiply the equation by 𝑆⁻¹. We obtain 𝑦_𝑛=𝐴_𝑇𝑦_𝑛−1+𝐵_𝑇𝑦_𝑛−𝑘 (5.3) with lower triangular matrices 𝐴_𝑇,𝐵_𝑇. By virtue of the nondegeneracy of matrix 𝑆, the stability of (5.3) is equivalent to the stability of (1.1). Let us assume that 𝑦_𝑛=(𝑦_𝑛⁽¹⁾,…,𝑦_𝑛^(𝑚))^𝑇. The system (5.3) consists of 𝑚 scalar equations 𝑦_𝑛^(𝑗)=𝜆_𝑗𝑗𝑦^(𝑗)_𝑛−1+𝜇_𝑗𝑗𝑦^(𝑗)_𝑛−𝑘+^𝑗−1_𝑠=1𝜆_𝑗𝑠𝑦^(𝑠)_𝑛−1+^𝑗−1_𝑠=1𝜇_𝑗𝑠𝑦^(𝑠)_𝑛−𝑘1⩽𝑗⩽𝑚. (5.4) As usual, ∑⁰_𝑠=1=0. Equation (5.4) is called exponentially stable if there are real 𝐶>0,𝑞∈(0,1), such that for any solution 𝑦_𝑛^(𝑗) the estimate 𝑦_𝑛^(𝑗)⩽𝐶𝑞^𝑛max_{()−𝑘⩽𝑢⩽1,1⩽𝑠⩽𝑗}𝑦_𝑢^(𝑠) (5.5) holds. The exponential stability is equivalent to the asymptotic stability for the equations under consideration. It is more convenient to prove the exponential stability. Let the points (5.2) (1⩽𝑗⩽𝑚) lie inside the cone (3.2). Due to Theorem 3.2, all equations of the form 𝑦_𝑛^(𝑗)=𝜆_𝑗𝑗𝑦^(𝑗)_𝑛−1+𝜇_𝑗𝑗𝑦^(𝑗)_𝑛−𝑘1⩽𝑗⩽𝑚 (5.6) are exponentially stable. Let us prove by induction on 𝑗 that (5.4) are exponentially stable. For 𝑗=1, (5.4) coincides with (5.6), so it is exponentially stable. Let, for any 𝑟<𝑗, (5.4) with 𝑟 instead of 𝑗 be exponentially stable. Then (5.4) is represented in the form 𝑦_𝑛^(𝑗)=𝜆_𝑗𝑗𝑦^(𝑗)_𝑛−1+𝜇_𝑗𝑗𝑦^(𝑗)_𝑛−𝑘+𝑔_𝑛^(𝑗), (5.7) where |𝑔_𝑛^(𝑗)| has an estimate of the form (5.5) by the induction assumption. Assuming 𝑧_𝑛=(𝑦_𝑛^(𝑗),𝑦^(𝑗)_𝑛−1,…,𝑦^(𝑗)_𝑛−𝑘)^𝑇, we represent (5.7) in the form 𝑧_𝑛=𝐺𝑧_𝑛−1+ℎ_𝑛, (5.8) where 𝐺∈ℝ^𝑘×𝑘, 𝐺 is a stable matrix, and |ℎ_𝑛| has an estimate of the form (5.5). From (5.8) we obtain 𝑧_𝑛=𝐺^𝑛𝑧₀+∑^𝑛_𝑟=1𝐺^𝑛−𝑟ℎ_𝑟, which implies the exponential stability of (5.4). The induction is finished, and asymptotic stability of (1.1) is proved.
Let us assume that some point (5.2) does not lie strictly inside the cone. Then, for the initial data in (5.4), we assume that 𝑦^(𝑠)_−𝑛=0 for any 𝑠,𝑛, such that 1⩽𝑠⩽𝑗, 1⩽𝑛⩽𝑘. Thus, (5.4) becomes (5.6). If the point (5.2) lies on the cone boundary, then (5.6) has a trajectory which does not tend to zero because the characteristic polynomial of (5.6) has a root 𝑧 such that |𝑧|=1. If some point (5.2) lies outside the cone, then by Theorem 3.2 the equation has unlimited trajectories. Theorem 5.1 is proved.

Remark 5.2. If no points (5.2) lie outside the stability cone, but some of them lie on the cone boundary, then (1.1) can be stable (nonasymptotically) or unstable.

Remark 5.3. The stability cones (Figure 5) are constructed for each delay 𝑘 independently of the dimension 𝑚 in (1.1). If 𝑘→∞, then the intersection of all stability cones is the right circular cone with the base radius 1 and the height 1. The interior of this cone is the “absolute stability domain,” that is, the stability domain for any delay.

(a)

(b)

(c)

(d)

Figure 5 

Stability cones for 𝑘=2,3,4,5.

The next theorem, which is the evident consequence of Theorems 4.1 and 5.1, will establish the asymptotic stability criterion in the form of inequalities for matrix equation (1.1).

Theorem 5.4. Let 𝐴,𝐵,𝑆∈ℝ^𝑚×𝑚 and 𝑆⁻¹𝐴𝑆=𝐴_𝑇 and 𝑆⁻¹𝐵𝑆=𝐵_𝑇, where 𝐴_𝑇,𝐵_𝑇 are the lower triangular matrices with elements 𝜆_𝑗𝑠,𝜇_𝑗𝑠(1⩽𝑗,𝑠⩽𝑚), respectively. Let 𝜌_1𝑗=𝜆_𝑗𝑗,𝛼_1𝑗=arg𝜆_𝑗𝑗,𝜌_2𝑗=𝜇_𝑗𝑗,𝛼_2𝑗=arg𝜇_𝑗𝑗1⩽𝑗⩽𝑚. (5.9) Construct a set 𝐴𝑆⊆𝑃={𝑗∈ℤ₊∶1⩽𝑗⩽𝑚} by the following rules (cf. Theorem 4.1).(1) If  𝜌_1𝑗<1−𝜌_2𝑗, then 𝑗∈𝐴𝑆. (2) If  1−𝜌_2𝑗⩽𝜌_1𝑗<min(1+𝜌_2𝑗,𝑘/(𝑘−1)), then for 𝑗∈𝐴𝑆 it is necessary and sufficient to fulfill simultaneously the following conditions (𝐻1𝑗), (𝐻2𝑗): 𝜌_2𝑗<𝜌²_1𝑗+1−2𝜌_1𝑗cos𝜔_1𝑗,𝐻1𝑗 where 𝜔_1𝑗∈[0,𝜋/𝑘] is the root of the equation𝜌_1𝑗=sin𝑘𝜔()𝜔,𝑖𝛼sin𝑘−1𝜋−argexp_2𝑗−𝑘𝛼_1𝑗<𝜋−(𝑘−1)arccos1+𝜌²_1𝑗−𝜌²_2𝑗2𝜌_1𝑗−arccos1−𝜌²_1𝑗−𝜌²_2𝑗2𝜌_1𝑗𝜌_2𝑗.𝐻2𝑗(3)If 𝜌_1𝑗⩾min(1+𝜌_2𝑗,𝑘/(𝑘−1)), then 𝑗∉𝐴𝑆. Equation (1.1) is asymptotically stable if and only if 𝐴𝑆=𝑃.

6. Examples of the Stability Oval and the Cone for Matrix Equations

Example 6.1. Consider the equation 𝑥_𝑛=1.0309𝐴𝑥_𝑛−1+(0.9680𝐵)^𝑠𝑥_𝑛−6, (6.1) where 𝐴=cos𝛼−sin𝛼sin𝛼cos𝛼,𝐵=cos𝛽−sin𝛽sin𝛽cos𝛽 (6.2) with 𝛼=0.0314, 𝛽=0.1745. Let us find out for what values of 𝑠∈ℤ₊ (6.1) is stable. Matrices 𝐴,𝐵 are commuting; therefore, they are simultaneously triangularizable. The eigenvalues of matrices 𝐴,𝐵 are 𝜆_1,2=exp(±0.0314𝑖) and 𝜇_1,2=exp(±0.1745𝑖) correspondingly. In Figure 6 the stability oval is shown, which is the section of the stability cone (3.2) on the level 𝑧=1.0309. By Theorem 5.1 we have to know whether the points 𝑀^(𝑠)=0.9680^𝑠cos(0.1745𝑠−0.1884),0.9680^𝑠sin(0.1745𝑠−0.1884) (6.3) lie inside the cone. Figure 6 illustrates that points 𝑀^(𝑠) enter the oval of stability twice (𝑠=53,𝑠=87) and leave it twice (𝑠=58,𝑠=94). The conclusion is that the system (6.1), (6.2) is stable for 53⩽𝑠⩽58 and for 87⩽𝑠⩽94 and is unstable for all the other values of 𝑠.

Figure 6 

The stability oval and points 𝑀(𝑠) for Example 6.1.

Example 6.2. Consider the equation 𝑥_𝑛=13𝐴^𝑠𝑥_𝑛−1+𝐴𝑥_𝑛−6, (6.4) where 𝐴=01.0150−1.01502.0300. (6.5)
Let us find out for what values of 𝑠∈ℤ₊ (6.4) is stable. The eigenvalues of 𝐴 are 𝜆_1,2=1.0150exp(±0.0374𝑖). Stability ovals are symmetric about the real axis. Therefore, by Theorem 5.1, only points 𝑀^(𝑠)=0.3383cos(0.0374(1−6𝑠)),0.3383sin(0.0374(1−6𝑠)),1.0150^𝑠3 (6.6) (see Figure 7) should be checked. Figure 7 displays that points 𝑀^(𝑠) for 1⩽𝑠⩽49 are inside the stability cone (3.2) and for 𝑠⩾50 points are outside of the stability cone. The conclusion is that (6.4) is stable for 1⩽𝑠⩽49 and it is unstable for 𝑠⩾50.

Figure 7 

The stability cone and points 𝑀(𝑠) for Example 6.2. Grey points 𝑀(𝑠) (1⩽𝑠⩽49) are located inside the cone therefore (6.4) is stable. Dark points 𝑀(𝑠) (𝑠⩾50) are located outside therefore (6.4) is unstable.

7. Conclusion

The stability analysis for (1.1) in ℝ^𝑚 can be reduced to the pole placement problem for a polynomial of degree 𝑘𝑚. Our geometric approach allows us to reduce the dimension. To use the approach, we need to know the eigenvalues of 𝐴,𝐵. This is the problem of finding the roots of a polynomial of degree 𝑚. Using these eigenvalues, we get a finite sequence of points in ℝ³ such that their position with respect to the stability cone allows us to make a conclusion about the stability of (1.1).

In our future work we intend to analyze the stability of equation 𝑥_𝑛=𝐴𝑥_𝑛−𝑚+𝐵𝑥_𝑛−𝑘 with two delays 𝑚,𝑘 with simultaneously triangularizable matrices 𝐴,𝐵. The scalar version of this equation was examined in [2, 3, 18]. The stability cone for the matrix differential equation 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏) was introduced in [19].

Acknowledgment

The authors are indebted to K. Chudinov, I. Goldsheid, and D. Sheglov for very useful comments.