Divergence Conditions for Investigation and Control of Nonautonomous Dynamical Systems^⋆

The method of Lyapunov functions is a powerful tool for studying the stability of solutions of differential equations without solving them. Depending on the problem being solved, Lyapunov function is also interpreted as a potential function [1], an energy function [2] or a storage function [3]. The main restriction of the method of Lyapunov functions is to find these functions.

Methods for stability study of dynamical systems based on the divergence of a vector field are alternative to the method of Lyapunov functions. The first fundamental results based on divergent stability conditions were proposed in [4, 5, 6]. The last important results for investigation of system stability were proposed by A. Rantzer, A.A. Shestakov, A.N. Stepanov and V.P. Zhukov. In [7] the instability problem of nonlinear systems using the divergence of a vector field is considered. In [8, 9] a necessary condition for stability of nonlinear systems in the form of non-positivity of the vector field divergence is proposed. First, an auxiliary scalar function is introduced in [8, 10] to study the instability of nonlinear systems. However, the similar scalar function is considered in [11] for stability and instability study of dynamical systems, but using the method of Lyapunov functions. In [8, 12] stability conditions for second-order systems are obtained. Then in [13, 14] the convergence of almost all solutions of arbitrary order nonlinear dynamical systems is considered. As in [8, 10, 12] the auxiliary scalar function (density function) is used for the stability study of dynamical models. Additionally, in [13, 14] the synthesis of the control law based on divergence conditions is proposed. The auxiliary functions in [8, 12, 13, 14] are similar except their properties at the equilibrium point. Currently, method from [13, 14] has been extended to various systems, see i.e. [15, 16, 17, 18].

However, in [4, 8, 12] the necessary condition is sufficiently rough and it is obtained only for autonomous systems. The sufficient condition stability is proposed only for second-order autonomous systems in [12]. Corollary 1 in [14] guarantees the convergence of almost all solutions, but not all solutions, for nonautonomous systems. Proposition 2 in [14] allows to study the asymptotic stability for autonomous systems, but proposition conditions have sufficient restriction. In the present paper new necessary and sufficient conditions will be obtained that will eliminate the above disadvantages and expand the class of investigated systems.

In this paper a new method for the stability study of nonautonomous systems using the flow and divergence of the vector field is proposed. The relation between the method of Lyapunov functions and the proposed method is established. The method for design the state feedback control law based on the new divergence conditions is proposed. Numerical examples illustrate the applicability of the proposed method and the methods from [4, 8, 12, 13, 14].

The paper is organized as follows. Section 2 contains new necessary and sufficient conditions, as well as, the numerical examples and comparisons with the methods from [4, 8, 12, 13, 14]. Section 3 describes methods for design the state feedback control law and numerical examples. Finally, Section 4 collects some conclusions.

Notations and definitions. In the paper the following notation are used: the superscript $\rm T$ stands for matrix transposition; $\mathbb{R}^{n}$ denotes the $n$ dimensional Euclidean space with vector norm $|\cdot|$; $\mathbb{R}^{n\times m}$ is the set of all $n\times m$ real matrices; $\nabla\{W(x,t)\}=\Big{[}\frac{\partial W}{\partial x_{1}},...,\frac{\partial W}{\partial x_{n}}\Big{]}^{\rm T}$ is the gradient of the scalar function $W(x,t)$, $\nabla\cdot\{h(x,t)\}=\frac{\partial h_{1}}{\partial x_{1}}+...+\frac{\partial h_{n}}{\partial x_{n}}$ is the divergence of the vector field $h(x,t)=[h_{1}(x,t),...,h(x,t)_{n}]^{\rm T}$, $|\cdot|$ is the Euclidean norm of the corresponding vector.

Additionally, in the paper we mean that the zero equilibrium point is stable if it is Lyapunov stable [19].

Consider the nonautonomous system

where $x=[x_{1},...,x_{n}]^{\rm T}$ is the state vector, $f=[f_{1},...,f_{n}]^{\rm T}:[0,\infty)\times D\to\mathbb{R}^{n}$ is piecewise continuous in $t$ and continuously differentiable in $x$ on $[0,\infty)\times D$. The open set $D\subset\mathbb{R}^{n}$ contains the origin $x=0$ and $f(t,0)=0$ for any $t\geq 0$. Denote by $\bar{D}$ a boundary of the domain $D$. Below, the structure of the set $D$ can be specified depending on the obtained result.

Let us formulate the necessary stability condition for system (1).

Proof 1 According to [19, Theorem 3.13], if Jacobian matrix $[\partial f/\partial x]$ is bounded on $D=\{x\in\mathbb{R}^{n}:\|x\|<r\}$ and uniformly in $t$, trajectories of system (1) satisfies $\|x(t)\|\leq\beta(\|x(t_{0})\|,t-t_{0})$ for any $x(t_{0})\in D_{0}$ and $t\geq t_{0}\geq 0$, then there exists a continuously differentiable function $S(x,t):[0,\infty)\times D_{0}\to\mathbb{R}$ that satisfies the inequality $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}f(x,t)\leq-\alpha(\|x\|)$. Here the function $\alpha(\|\cdot\|)$ is a class $\mathcal{K}$ functions on $[0,r_{0}]$. Next, we consider two cases separately which correspond to the functions $S(x,t)$ and $S^{-1}(x,t)$.

1. If $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}f(x,t)<0$ for any $x\in D_{0}\setminus\{0\}$ and $t\geq 0$, then $\frac{\partial S(x,t)}{\partial t}+\frac{1}{|\nabla\{S(x,t)\}|}\nabla\{S(x,t)\}^{\rm T}|\nabla\{S(x,t)\}|f(x,t)<0$. Therefore, the following expression holds

Using Divergence theorem (or Gauss theorem), we get $F_{1}=\int_{V}\Big{[}\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{|\nabla\{S(x,t)\}|f(x,t)\}\Big{]}dV<0$.

2. If $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}f(x,t)<0$ for any $x\in D_{0}\setminus\{0\}$ and $t\geq 0$, then $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\{S^{-1}(x,t)\}^{\rm T}f(x,t)=-S^{-2}(x,t)\big{[}\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{S(x,t)\}^{\rm T}f(x,t)\big{]}>0$. On the other hand, $\nabla\{S^{-1}(x,t)\}^{\rm T}f(x,t)=\frac{1}{|\nabla\{S^{-1}(x,t)\}|}\nabla\{S^{-1}(x)\}^{\rm T}\times\\ |\nabla\{S^{-1}(x)\}|f(x,t).$ Therefore, the following relation is satisfied

According to Divergence theorem, we get $F_{2}=\int_{V_{inv}}\Big{[}\frac{\partial S^{-1}(x,t)}{\partial t}+\\ \nabla\cdot\{|\nabla\{S^{-1}(x,t)\}|f(x,t)\}\Big{]}dV_{inv}>0.$ Theorem 1 is proved.

The next theorem extends results of Theorem 1 to the case of introducing new auxiliary function in the integrand. It allows to simplify the investigation of stability of system (1).

Proof 2 According to [19, Theorem 3.13], if the Jacobian matrix $[\partial f/\partial x]$ is bounded on $D=\{x\in\mathbb{R}^{n}:\|x\|<r\}$ and uniformly in $t$, trajectories of the system (1) satisfies $\|x(t)\|\leq\beta(\|x(t_{0})\|,t-t_{0})$ for any $x(t_{0})\in D_{0}$ and $t\geq t_{0}\geq 0$, then there exists a continuously differentiable function $S(x,t):[0,\infty)\times D_{0}\to\mathbb{R}$ that satisfies the inequalities

Here $\alpha_{1}(x)$, $\alpha_{2}(x)$ and $\alpha_{3}(x)$ are class $\mathcal{K}$ functions on $[0,r_{0}]$. If $\alpha_{1}(x)=c_{3}\|x\|^{\chi}$, $\chi>0$, then $\alpha_{2}(x)=c_{4}\|x\|^{\chi+\gamma-1}$ and $\alpha_{3}(x)=c_{5}\|x\|^{\chi-1}$ Consider the following relations

Choosing $r=\frac{c_{4}}{c_{0}c_{2}c_{5}}$ and $r_{0}<r$, one gets $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}\mu(x,t)f(x,t)\leq 0$. Next, we consider two cases separately which correspond to the functions $S(x,t)$ and $S^{-1}(x,t)$.

1. Since $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}\mu(x,t)f(x,t)<0$ for any $x\in D_{0}\setminus\{0\}$ and $t\geq 0$, then

Using Divergence theorem, we get $F_{1}=\int_{V}\Big{[}\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{|\nabla\{S(x,t)\}|\mu(x,t)\\ f(x,t)\}\Big{]}dV<0$.

2. Since $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}\mu(x,t)f(x,t)<0$ for any $x\in D_{0}\setminus\{0\}$ and $t\geq 0$, then $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\{S^{-1}(x,t)\}^{\rm T}\mu(x,t)f(x,t)=-S^{-2}(x,t)\Big{[}\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}\mu(x,t)f(x,t)\Big{]}>0$. On the other hand, $\nabla\{S^{-1}(x,t)\}^{\rm T}\mu(x,t)f(x,t)\\ =\frac{1}{|\nabla\{S^{-1}(x,t)\}|}\nabla\{S^{-1}(x)\}^{\rm T}|\nabla\{S^{-1}(x)\}|\mu(x,t)f(x,t).$ Therefore, the following relation is satisfied

Considering Divergence theorem, we get $F_{2}=\int_{V_{inv}}\Big{[}\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{|\nabla\{S^{-1}(x,t)\}\\ |\mu(x,t)f(x,t)\}\Big{]}dV_{inv}>0.$ Theorem 2 is proved.

Remark 1 There are various physical interpretations of the integral conditions in Theorem 2. Rewriting the integral inequalities in Theorem 2 as $\int_{V}\Big{[}\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S(x,t)\}|f(x,t)\}\Big{]}dV=-\Sigma$ or $\int_{V_{inv}}\Big{[}\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S^{-1}(x,t)\}|f(x,t)\}\Big{]}dV_{inv}=\Sigma$, $\Sigma\geq 0$, one gets the integral forms of continuity equation with the sources (the flux is directed inward) located in the equilibrium points in the domain $V$ (or $V_{inf}$), see [23].

1. (i)

   Choosing $\mu(x,t)$ such that $|\nabla\{S(x,t)\}|\mu(x,t)=S(x,t)$ or $|\nabla\{S^{-1}(x,t)\}|\\ \times\mu(x,t)=S^{-1}(x,t)$, one has the continuity equation in fluid dynamics [20], where $S$ is fluid density and $f$ is the flow velocity of a vector field.

2. (ii)

   In electromagnetic theory [21], $\mu(x,t)|\nabla\{S(x,t)\}|f(x,t)$ or $\mu(x,t)|\times\\ \nabla\{S^{-1}(x,t)\}|f(x,t)$ means the current density and $S$ is the charge density.

3. (iii)

   Due to conservation of energy [20], $\mu(x,t)|\nabla\{S(x,t)\}|f(x,t)$ or $\mu(x,t)|\times\\ \nabla\{S^{-1}(x,t)\}|f(x,t)$ is the vector energy flux and $S$ is local energy density.

4. (iv)

   In quantum mechanics [22], $S$ is the probability density function and $\mu(x,t)|\nabla\{S(x,t)\}|f(x,t)$ or $\mu(x,t)|\nabla\{S^{-1}(x,t)\}|f(x,t)$ is the probability current.

Remark 2 If $\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S(x,t)\}|f(x,t)\}$ or $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S^{-1}(x,t)\}|f(x,t)\}$ are integrable and $\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S(x,t)\}\\ |f(x,t)\}=-\sigma$ or $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{\mu(x,t)|\nabla\{S^{-1}(x,t)\}|f(x,t)\}=\sigma$, $\sigma\geq 0$ holds for any $x\in D_{0}\setminus\{0\}$ anf $t\geq 0$, then the corresponding integral relations in Theorem 2 are satisfied. According to [20, 21, 22, 23] and Remark 2, one gets the appropriate differential forms of continuity equations with the sources (the flux is directed inward) in the domain $V$ (or $V_{inf}$).

Remark 3 According to [14], if the relation $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{S^{-1}(x,t)f(x,t)\}>0$ holds, then only almost all solutions of (1) tend to equlibrium. Thus, the results of [14] are special case in Theorem 2 for $|\nabla\{S^{-1}(x,t)\}|\mu(x,t)=S^{-1}(x,t)$.

Now let us formulate a sufficient condition for stability of (1).

Proof 3 Consider the proof for each case separately. The proof of asymptotic stability is omitted because it is similar to the proof of stability, but taking into account the sign of a strict inequality.

1. From the relation $\nabla\cdot\{S(x,t)f(x,t)\}-\nabla\cdot\{f(x,t)\}S(x,t)=\nabla\{S(x,t)\}^{\rm T}\times\\ f(x,t)$ implies that if $\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\ \{S(x,t)f(x,t)\}\leq\nabla\cdot\{f(x,t)\}S(x,t)$, then $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}f(x,t)\leq 0$ for any $x\in D\setminus\{0\}$ and $t\geq 0$. Therefore, according to Lyapunov theorem [19], system (1) is stable.

2. From the expression $S^{-2}(x,t)\nabla\{S(x,t)\}^{\rm T}f(x,t)=S^{-1}(x,t)\nabla\cdot\{f(x,t)\}\\ -\nabla\cdot\{S^{-1}(x,t)f(x,t)\}$ follows that if $\frac{\partial S^{-1}(x,t)}{\partial t}+\nabla\cdot\{S^{-1}(x,t)f(x,t)\}\geq 0$ and $\nabla\cdot\{f(x,t)\}\leq 0$, then $\frac{\partial S(x,t)}{\partial t}+\nabla\{S(x,t)\}^{\rm T}f(x,t)\leq 0$ for any $x\in D\setminus\{0\}$ and $t\geq 0$.

3. Condition 3 is a combination of conditions 1 and 2. Introduce $\beta(x,t)\geq 1$ for any $x\in D\setminus\{0\}$ and $t\geq 0$. Summing $\beta(x,t)\nabla\{S(x,t)\}^{\rm T}f(x,t)=\beta(x,t)S(x,t)\nabla\cdot\{f(x,t)\}-\beta(x,t)S^{2}(x,t)\nabla\cdot\ \{S^{-1}(x,t)f(x,t)\}$ and $\nabla\{S(x,t)\}^{\rm T}\\ \times f(x,t)=\nabla\cdot\{S(x,t)f(x,t)\}-\nabla\cdot\{f(x,t)\}S(x,t)$, we get

If

then $\nabla\{S(x,t)\}^{\rm T}f(x,t)\leq 0$. Let $\beta(x,t)=1$. If $2\frac{\partial S(x,t)}{\partial t}+\nabla\cdot\{S(x,t)f(x)\}\leq 0$ and $\nabla\cdot\{S^{-1}(x)f(x,t)\}\geq 0$, then $\nabla\{S(x,t)\}^{\rm T}f(x,t)\leq 0$ holds for any $x\in D\setminus\{0\}$ and $t\geq 0$. Theorem 3 is proved.

It is noted in Introduction that the result of [4, 8, 12] is applicable only to second-order autonomous systems. Next, we consider an illustration of the proposed results for third-order nonautonomous systems and compare the results with ones from [14].

Example 1. Consider the system

which has an equilibrium point $(0,0,0)$. The function $g(t)>0$ is continuously differentiable and bounded, $\dot{g}(t)<0$ for any $t\geq 0$. The functions $\varphi_{1}(t)>0$, $\varphi_{2}(t)>0$, and $\varphi_{3}(t)>0$ are continuous and bounded for any $t\geq 0$.

Choose $S(x,t)=(x_{1}^{2}+g(t)x_{2}^{2}+x_{3}^{2})^{\alpha}$, where $\alpha$ is a positive integer. Verify the conditions of Theorem 2, where $\mu(x,t)$ is chosen such that $|\nabla\{S(x,t)\}|\mu(x,t)\\ =S(x,t)$ or $|\nabla\{S^{-1}(x,t)\}|\mu(x,t)=S^{-1}(x,t)$. The condition

holds for any $\alpha$, $x_{2}\neq 0$, $x_{3}\neq 0$, and $t\geq 0$. The relation