Divergence Method for Exponential Stability Study of Autonomous Dynamical Systems

The recent proposed divergence method is used for investigation of an exponential stability of autonomous dynamical systems. New necessary conditions are derived by using Divergence theorem. Taking into account the relation between gradient and divergence of the vector field, novel sufficient conditions are obtained. The sufficient conditions are applied for design the state-feedback control laws ensuring exponential stability of the closed-loop system. Examples illustrate the efficiency of the proposed methods.


I. INTRODUCTION
T HE paper is devoted to the development of divergence method for stability study of dynamical systems. Interest of this method is associated with its relationship to many problems in physics and mechanics, in particular to processes that are described by the continuity equation [1]- [4].
First results of stability study of dynamical systems by using a divergence and flow of the vector field were proposed in [5]- [8] for second order dynamical systems. These results were improved in [9]- [13] for arbitrary order systems. An overview of these and others corresponding methods is described in more detail in [14]- [17]. Also, in [14]- [17] new necessary and sufficient stability conditions were proposed. Differently from [9]- [13] methods [14]- [17] allow one to extend the class of investigated systems. Additionally, in [14]- [17] the relation between necessary and sufficient conditions as well as continuity equation in electromagnetism, fluid dynamics, energy and heat, probability distributions, and quantum mechanics is established. However, all of these methods are proposed for investigation of stability or asymptotic stability, but not exponential one.
There are many tasks where stability and asymptotic stability is not enough. For example, the exponentially stable systems are more robust under parametric uncertainty and external disturbances, than stable or asymptotically stable systems [18], [19]. Also, if continuous system is an exponentially stable, then it remains stable under discrete control [20]. Currently, tasks with discrete controls are widespread, e.g., networked control [21]- [23], security control [24], eventtriggered control [25], [26], sampled-data control [27], etc. Therefore, the main contribution of the present paper is a generalization of results [14]- [17] to solution of the following problems: i) design new necessary and sufficient exponential stability conditions; ii) design new method of control law design ensuring exponential stability of the closed-loop systems.
The paper is organized as follows. Section II describes problem formulation. Section III proposes necessary and sufficient exponential stability conditions. Section IV describes application of the proposed results to linear systems. Section V proposes methods for designing the state feedback control laws. All proposed theoretical results are illustrated by numerical examples. Finally, Section VI collects some conclusions.
Notations. In the paper the superscript T stands for matrix or vector transposition; R n denotes the n dimensional Euclidean space with the vector norm | · |; R n×m is the set of all n × m real matrices; ∇{W (x)} = ∂W ∂x1 , ..., ∂W ∂xn T VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS is a gradient of the scalar function W (x); ∇ · {h(x)} = ∂h1 ∂x1 + ... + ∂hn ∂xn is a divergence of the vector field h(x) = [h 1 (x), ..., h n (x)] T ; int{Ῡ} is the set of interior points inῩ. We mean that the equilibrium point is stable if it is Lyapunov stable [28].

II. PROBLEM FORMULATION
In the paper we consider autonomous systems. An autonomous systems (or time-invariant systems) are described by autonomous differential equations which do not explicitly depend on time, i.e.ẋ (t) = f (x(t)). (1) It is required to design new exponential stability conditions in the divergence form. The proposed results will be obtained by using resent methods [14]- [17].

Remark 1:
The conditions (i T1 ) and (ii T1 ) are presented in the integral forms. However, it can be rewritten in the corresponding differential forms if Illustrate the result of Theorem 1 in the following example.
Example 1. Consider the systeṁ with the equilibrium point (0, 0). The phase portrait of (6) is shown in Fig. 1. Here and below, phase trajectory has its own colour for appropriate initial point. Investigate the equilibrium point of (6). Introduce where β is a positive integer. Calculate the following auxiliary expressions Using the last relations, verify the conditions (i T1 ) and (ii T1 ): As a result, the equilibrium point x = 0 is an exponentially stable.
In the next theorem we introduce new auxiliary function ξ(x) that allows one to extend the class of investigated systems in comparison with Theorem 1. Theorem 2: If the equilibrium point x = 0 of (1) is an exponentially stable, then there exists twice differentiable function µ(x) > 0 and differentiable function ξ(x) > 0 in x ∈ D \ {0}, µ(0) ≥ 0 and |∇{µ(x)}| ̸ = 0 for any x ∈ D \ {0} such that one of the following conditions holds: where α > 0.
The next theorem generalises the sufficient stability condition [14]- [17] to exponential one. Theorem 3: Let µ(x) be a positive definite continuously differentiable function in D, α > 0 and x = 0 be an equilibrium point of the system (1). If one of the following inequalities Taking into account well-known relation [3] ∇{µ represent the left-hand side of (8) in the form (2), i.e., x = 0 is an exponentially stable. From (ii T2 ) we have Taking into account the first expression in (9), rewrite (10) as follows Considering or in the form Taking into account ∇{µ( (9), we get (2) from (13). Theorem 3 is proved. Example 3. Consider the systeṁ VOLUME 4, 2016 with the equilibrium point (0, 0). The phase portrait of (14) is shown in Fig. 2.
Consider Lyapunov function V = a[1−cos(x 1 )]+0.5x 2 2 in the form of total system energy [28]. Taking time derivative of V along the trajectories of (25), one getsV = −abx 2 2 [28]. Thus, only asymptotic stability can be established by using Lyapunov method for the same V and µ.

Consider the linear systemẋ
where x ∈ R n , A ∈ R n×n . Introduce µ(x) = (x T Qx) β with Q = Q T > 0 and β is a positive integer. Calculate the following relations According to condition (i T2 ) and (30), the linear system The inequality (31) is the well-known linear matrix inequality for exponential stability of linear system (29) [29]. According to condition (ii T2 ) and (30), the linear system is exponential The result (32) is similar to (31). According to condition (iii T2 ) and (30), the linear system is exponential The results (33) are similar to (31).

Consider a dynamical system in the forṁ
where x ∈ D ⊂ R n , u(x) ∈ R m is the control signal, ψ(x, u) = ξ(x) + g(x)u(x), the functions ξ(x), g(x) and u(x) are continuously differentiable in D, ψ(0, u(0)) = 0 and the system (34) is stabilizable in D. Theorem 4: Let µ(x) be a positive definite continuously differentiable function in x ∈ D. The closed-loop system is an exponentially stable if the control law u(x) is chosen such that one of the following conditions holds for any x ∈ D \ {0}, where α > 0.
Since the system (34) is stabilizable in D, then the proof of Theorem 4 is similar to the proof of Theorem 3 (denoting by f (x) = ψ(x, u(x)).
Example 6. Consider the systeṁ The phase trajectories under u = 0 are presented in Fig. 7. It is required to design the control law ensures the exponential stability of the closed-loop system. Choose µ(x) in the form (7). Find the following relations rewrite (36) as follows Thus, the conditions (i T3 ) and (ii T3 ) hold. Additionally estimating we have fulfilment of the condition (iii T3 ).

VI. CONCLUSIONS
In this paper the recent divergence method [14]- [17] is used for investigation of an exponential stability of autonomous dynamical systems. The property of exponential stability makes it possible to increase system robustness to certain types of perturbations and uncertainties. New necessary and sufficient exponential stability conditions are derived from [14]- [17]. Examples illustrate the efficiency of the proposed method by the simulations. The advantage of the proposed method is that the formulated necessary and sufficient conditions have the form of a continuity equation, which is widely used in mechanics and physics, as well as in some industrial tasks [30]. The difficulties are in the selection of the density function (auxiliary function µ(x)) such that the obtained conditions hold. This problem is equivalent to the problem of finding the Lyapunov function.