DIAGONAL RICCATI STABILITY OF A CLASS OF TIME-DELAY SYSTEMS

A complex system describing interaction of subsystems of the second order with delay in connections between them is studied. Necessary and sufficient conditions of the existence of a diagonal Lyapunov– Krasovskii functional for the considered system are derived. The obtained results are applied for the stability analysis of a mechanical system and a model of population dynamics. In addition, it is shown that they can be used in a problem of formation control.


Introduction
Diagonal Lyapunov functions is a powerful tool for the stability analysis of wide classes of systems [Kaszkurewicz and Bhaya, 1999]. On the one hand, this is due to the fact that they possess a simple structure. On the other hand, for many types of nonlinear systems, the problem of constructing Lyapunov functions naturally results in choosing a Lyapunov candidate function in a diagonal form. It should be noted that diagonal Lyapunov functions are especially often used for the stability investigation of complex systems, neural networks and models of population dynamics [Hofbauer and Sigmund, 1998;Kaszkurewicz and Bhaya, 1999;Arcat and Sontag, 2006;Aleksandrov, Aleksandrova and Platonov, 2013;Talagaev, 2017;Alyshev, Dudarenko and Melnikov, 2018]. Moreover, in many cases, with the aid of such functions it is possible to derive not only sufficient, but also necessary stability conditions [Kaszkurewicz and Bhaya, 1999;Shorten and Narendra, 2009].
The problem of the existence of diagonal Lyapunov functions is well investigated for linear time-invariant systems of differential and difference equations. In [Kraaijevanger, 1991;Kaszkurewicz and Bhaya, 1999;Arcat and Sontag, 2006;Mason and Shorten, 2006;Shorten and Narendra, 2009], conditions are obtained under which quadratic Lyapunov functions with diagonal matrices can be constructed for these systems. A linear system possessing such a Lyapunov function is called diagonally Lyapunov stable [Mason and Shorten, 2006].
In [Mason, 2012], the problem of diagonal Riccati stability was stated. A linear positive differential system with a constant delay was considered, and conditions guaranteeing that the system admits a diagonal Lyapunov-Krasovskii functional were investigated. The results of [Mason, 2012] have got further development in [Aleksandrov and Mason, 2014;Aleksandrov and Mason, 2016;Aleksandrov and Mason, 2018]. In particular, in [Aleksandrov and Mason, 2016], a criterion of diagonal Riccati stability was obtained for linear time-invariant difference-differential systems of a general form (not necessary for positive ones). However, it is worth mentioning that the criterion is insufficiently constructive. Therefore, an interesting and important problem is that of finding classes of time-delay systems for which constructively verifiable conditions of diagonal Riccati stability can be derived. Some such classes both linear and nonlinear systems were determined in [Aleksandrov and Mason, 2016;Aleksandrov, Mason and Vorob'eva, 2017;Aleksandrov and Mason, 2018].
In the present contribution, a nonlinear time-delay system is studied. The system describes the interaction of subsystems of the second order and possesses a special structure of connections between the subsystems. We will look for conditions of the existence of a diagonal Lyapunov-Krasovskii functional of a prescribing form for the considered system. Moreover, we will show that such conditions can be used for the stability analysis of a mechanical system and a model of population dynamics and for the design of a protocol providing equidistant deployment of mobile agents on a line segment.
2 Statement of the Problem Consider the time-delay systeṁ (1) Here and belong to a sector-like constrained set defined as follows: . . , n, τ is a constant nonnegative delay.
The system (1) is well-known Persidskii type system, see [Kaszkurewicz and Bhaya, 1999]. Such systems are widely used for the modeling automatic control systems and neural networks. Let R n denote the n-dimensional Euclidean space, ∥·∥ be the Euclidean norm of a vector, and Ω H be the set of functions φ(θ) ∈ C([−τ, 0], R n ) such that ∥φ∥ τ < H. In addition, let x(t, t 0 , φ) stand for a solution of (1) with the initial conditions t 0 ≥ 0, φ(θ) ∈ Ω H , and x t (t 0 , φ) denote the restriction of the solution to the segment [t − τ, t], i.e., x t (t 0 , φ) : θ → x(t+θ, t 0 , φ), θ ∈ [−τ, 0]. When the initial conditions are not important, or are well defined from the context, we will write x(t) and x t , instead of x(t, t 0 , φ) and x t (t 0 , φ), respectively. Definition 1. The system (1) is called diagonally Riccati stable if there exist diagonal positive definite matrices P and Q such that the matrix is negative definite.
Remark 1. It is known, see [Aleksandrov and Mason, 2016], that if the system (1) is diagonally Riccati stable, then it admits a diagonal Lyapunov-Krasovskii functional of the form where p i and q i are diagonal entries of the matrices P and Q, respectively. It is worth mentioning that the existence of such a functional implies that the zero solution of the system (1) is asymptotically stable for an arbitrary constant nonnegative delay τ , see [Mason, 2012;Aleksandrov and Mason, 2016].
In the present paper, we consider the case where n is an even number (n = 2k and k is a positive integer), and the matrices A and B possess the following structures 0 0 0 0 · · · a n−1 n−1 a n−1 n 0 0 0 0 · · · a n n−1 a n n Thus, (1) can be treated as a closed-loop complex system describing interaction of subsystems of the second order with delay in the connections between the subsystems, see  (1) We will look for conditions of diagonal Riccati stability of the system (1) with matrices (3) and (4). Moreover, we will consider some applications of such conditions to problems of analysis and synthesis of time-delay systems.

A Criterion of Diagonal Riccati Stability
Let P and Q be positive definite diagonal martices with diagonal entries p 1 , . . . , p n and q 1 , . . . , q n , respectively. If matrices A and B are defined by the formulae (3) and (4), then the matrix (2) can be rewritten as follows and q 0 = q n . Hence, the system (1) is diagonally Riccati stable if and only if there exist positive numbers p 1 , . . . , p n and q 2 , q 4 , . . . , q n such that the inequalities hold.

(6)
Assume that conditions (6) are fulfilled. Choose some s ∈ {1, . . . , k} and consider the following cases: (I) If a 2s 2s−1 = 0, then, for any positive numbers q 2s−2 and q 2s , the corresponding inequalities from (5) will be valid for sufficiently small values of p 2s−1 and for sufficiently large values of p 2s .
Thus, we arrive at the following theorem.
Theorem 1. Let the matrices A and B in the system (1) be of the form (3) and (4), respectively. Then the system is diagonally Riccati stable if and only if the conditions (6) and are valid.
Remark 2. If there exists a number r ∈ {1, . . . , k} such that a 2r 2r−1 = 0 or b r = 0, then for the diagonal Riccati stability of (1) it is necessary and sufficient the fulfilment of the conditions (6).
4 Stability Analysis of a Model of Population Dynamics In this section, we will show how the results described above can be applied to a generalized Lotka-Volterra model of population dynamics. Let the systeṁ be given. The system is a generalized Lotka-Volterra model describing interaction of species in a biological community, see [Hofbauer and Sigmund, 1998;Kaszkurewicz and Bhaya, 1999;Fan and Wang, 2000;Aleksandrov, Aleksandrova and Platonov, 2013]. Here x i (t) is the population density of the ith species, c i , a ij , b ij are constant coefficients, τ is a constant nonnegative delay. The coefficients c i characterize the intrinsic growth rate of the ith population, the selfinteraction terms a ii x 2 Let R n + be the nonnegative cone of the space R n : R n + := {x ∈ R n | x ≥ 0}, and int R n + be the interior of R n + . It should be noted that int R n + is an invariant set for (8). For biological reasons, we will consider this system with respect to the state space int R n + . Let A and B denote the matrices A = {a ij } n i,j=1 , B = {b ij } n i,j=1 . We consider the case where n is an even number (n = 2k, k is a positive integer) and the matrices A and B have the form (3) and (4), respectively. Thus, interactions between populations with numbers 2s − 1 and 2s are competition, predation or symbiosis type, whereas interactions between populations with numbers 2s and 2s + 1 are commensalism or amensalism type (see [Begon, Harper and Townsend, 1996;Hofbauer and Sigmund, 1998]), s = 1, . . . , k, and x n+1 (t) = x 1 (t). Moreover, we assume that there is a delay in interactions of commensalism and amensalism type.
Theorem 2. If the system (8) admits an equilibrium positionx ∈ int R n + , then, under the conditions (6) and (7), the equilibrium position is globally asymptotically stable in int R n + for any value of the delay τ .
Proof. Ifx = (x 1 , . . . ,x n ) T ∈ int R n + is an equilibrium position of (8), then the system can be rewritten as followṡ Assume that the inequalities (6) and (7) are fulfilled. Hence (see Theorem 1), there exist diagonal positive definite matrices P and Q such the matrix (2) is negative definite. Choose a Lyapunov-Krasovskii functional for (9) in the form where positive coefficients p i and q i are diagonal entries of the matrices P and Q.
Taking into account positive definiteness of the matrix (2) and using the Schur complement, see Theorem 7.7.6 in [Horn and Johnson, 1985], we obtain that the derivative of V (x t ) with respect to (9) satisfies the estimatė Here α is a positive constant. This completes the proof.
Then inequalities (6) are fulfilled for entries of the matrix A.
Applying Theorem 1, we obtain that, under the conditions (12) and the system (11) is diagonally Riccati stable. Here and β s = λ s (a s − λ s ), s = 1, . . . , k. From (12) it follows that β s ∈ (0, a 2 s /4), s = 1, . . . , k. Find β 1 , . . . , β k for which the condition (13) defines the largest domain of values for the parameters a s , b s , c s . It is easy to verify that we should take β s = c s /2 for a 2 s > 2c s , and β s → a 2 s /4 − 0 for a 2 s ≤ 2c s . Thus, the following theorem is valid.
Remark 3. The condition (14) determines, how small the coefficients characterizing the connections between the subsystems should be compared with the parameters of the isolated subsystems, in order to guarantee the delay-independent asymptotic stability of (10).
Remark 4. If, for some r ∈ {1, . . . , k}, the corresponding damping coefficient a r is sufficiently large (a 2 r ≥ 2c r ), then the condition (14) is independent of the coefficient.
In particular, we obtain the following corollary.

A Problem of Formation Control
In recent years, the problem of formation control of multiagent systems has attracted considerable attention due to its broad applications, see [Bullo, Cortes and Martinez, 2009;Ren and Cao, 2011;Parsegov, Polyakov and Shcherbakov, 2012;Proskurnikov and Parsegov, 2016;Frolov, Koronovskii, Makarov, Maksimenko, Goremyko and Hramov, 2017]. One of the simplest formation control problems is related to equidistant deployment of agents on a line segment. Some approaches to the solution of the problem were proposed in [Wagner and Bruckstein, 1997;Kvinto and Parsegov, 2012;Parsegov, Polyakov and Shcherbakov, 2012;Proskurnikov and Parsegov, 2016]. In the papers [Wagner and Bruckstein, 1997;Parsegov, Polyakov and Shcherbakov, 2012] the protocols were designed providing equidistant distribution for agents modeled as the first order integrators. In particular, in [Parsegov, Polyakov and Shcherbakov, 2012], conditions of fixed-time stability of such systems were derived. In [Kvinto and Parsegov, 2012;Proskurnikov and Parsegov, 2016], the case was considered where agent dynamics are described by the double integrators. However, it should be noted that in [Wagner and Bruckstein, 1997;Kvinto and Parsegov, 2012;Parsegov, Polyakov and Shcherbakov, 2012;Proskurnikov and Parsegov, 2016] it was assumed that there are no communication delays in the investigated multiagent systems. Let us show that the results of the present paper can be used in the problem of equidistant deployment of agents on a line segment under protocols with communication delays. Consider a group of k mobile agents on the line. Let x s (t) ∈ R be the position of the sth agent at time t ≥ 0, s = 1, . . . , k. Assume that the dynamics of agents are described by the double integrators x s (t) + aẋ s (t) = u s , s = 1, . . . , k.
Here u s ∈ R is the control input, a is a constant positive damping coefficient. Let a segment [x b , x e ] be given. The problem is to design a feedback control protocol providing the equidistant distribution of the agents on the segment for t → ∞ and any initial conditions. Such a problem was studied in [Kvinto and Parsegov, 2012;Proskurnikov and Parsegov, 2016]. It was assumed that each agent receives information about the distances between itself and its nearest left and right neighbors, i.e., the agent x s receives information about the distances In this section, we will study the problem of equidistant deployment of agents under the following assumptions.
Assumption 1. The agent x 1 knows the total number of agents in the system, but it doesn't know the length of the interval [x b , x e ]. In addition, it receives information about the distances Assumption 2. For each j ∈ {2, . . . , k}, the agent x j knows the desired final distance ∆ between agents (∆ = (x e − x b )/(k + 1)), but it knows neither the total number of agents in the system nor the length of the Under Assumptions 1 and 2, one can use the protocol Substituting (16) into (15), we obtain the closed-loop system It is easy to verify that (17) admits the equilibrium positionx = (x 1 , . . . ,x k ) T , wherex s = x b + s∆, s = 1, . . . , k. Hence, the equilibrium position corresponds to the equidistant distribution of agents on the segment With the aid of the transformation y(t) = x(t) −x, we arrive at the system y 1 (t) + aẏ 1 (t) = y k (t − τ ) − y 1 (t) k − 1 − y 1 (t), y j (t) + aẏ j (t) = y j−1 (t − τ ) − y j (t), j = 2, . . . , k.
Applying Theorem 3, we obtain that the following theorem is valid.
Theorem 4. If one of the conditions (i) a ≥ √ 2; (ii) 0 < a < √ 2 and holds, then the equilibrium positionx of (17) is asymptotically stable for any nonnegative delay τ .

Conclusion
In the present paper, necessary and sufficient conditions of the diagonal Riccati stability are derived for a complex time-delay system with a special structure of connections. The fulfilment of the conditions implies delay-independent asymptotic stability of the zero solution of the considered system. Some applications of the obtained results are presented. Future research aims at constructing diagonal Lyapunov-Krasovskii functionals for complex systems with delay and switched connections.