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ON A METHOD FOR CONSTRUCTING LYAPUNOV FUNCTIONS FOR LINEAR SYSTEMS WITH VARIABLE COEFFICIENTS
M. K. YAKOVLEV
The paper describes a certain method for constructing Lyapunov functions for linear systems
\[ \frac{d x_i}{d t}=p_{i1}(t)x_1+p_{i2}(t)x_2+\cdots+p_{in}(t)x_n \tag{1} \]
\[ (i=1,2,\ldots,n), \]
where \(p_{ij}(t)\) \((i,j=1,2,\ldots,n)\) are continuous bounded functions and the coefficients \(p_{ii}(t)\) \((i=1,2,\ldots,n)\) satisfy the conditions
\[ F_i(t)=2\int_0^t p_{ii}(t)\,dt=F_{1i}(t)+F_{2i}(t), \tag{*} \]
where \(F_{2i}\) is a bounded function, and
\[ \frac{dF_{1i}}{dt}=F'_{1i} \]
is bounded and negative. In the works [2, 3], E. G. Ivanov studied the asymptotic stability of systems (1) of a special form, namely, \(p_{ij}(t)\) are periodic functions and \(F'_{1i}(t)\) is a negative constant. For such systems, paper [2] gives an algorithm for constructing a Lyapunov function satisfying the conditions of theorem [1] on asymptotic stability.
In the present paper Ivanov’s algorithm is generalized to systems (1), and is also extended to the construction of Lyapunov functions satisfying the other fundamental theorems of A. M. Lyapunov on stability and instability.
First of all, we formulate obvious conditions ensuring sign-definiteness, for each fixed value of the parameter \(t\in[0,\infty]\), of the quadratic form
\[ V(t,x)=\sum_{i,j=1}^{n} B_{ij}(t)x_i x_j,\qquad B_{ij}=B_{ji}, \tag{2} \]
sign-definite in the sense of Lyapunov. As is known, the function \(V(t,x)\) is called positive definite in the sense of Lyapunov if there exists some function \(W(x)\), independent of \(t\), such that
\[ V(t,x)\ge W(x)>0 \quad \text{for } x\ne0,\qquad V(t,0)=W(0)=0. \]
Fulfillment of Sylvester’s conditions
\[ B_{11}(t)>0,\quad \left|\begin{array}{cc} B_{11}(t)&B_{12}(t)\\ B_{21}(t)&B_{22}(t) \end{array}\right|>0,\ \ldots,\ \left|\begin{array}{ccc} B_{11}(t)&\ldots&B_{1n}(t)\\ \cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\\ B_{n1}(t)&\ldots&B_{nn}(t) \end{array}\right|>0 \tag{3} \]
for \(t \geqslant 0\) still does not ensure positive definiteness, in the sense of Lyapunov, of the form (2). For example, \(V(t,x_1)=e^{-t}x_1^2\), which for fixed \(t \geqslant 0\) satisfies condition (3), is not positive definite in the sense of Lyapunov.
Lemma 1. The form (2) will be positive definite in the sense of Lyapunov if, simultaneously with conditions (3), the conditions are satisfied
\[ \operatorname{Sp}(B_{ii})=\sum_{i=1}^{n} B_{ii}(t)\text{ is bounded for }t\geqslant 0, \]
and
\[ \inf_{t>0}\{\det(B_{ij})\}>0. \]
We shall construct a Lyapunov function satisfying the conditions of A. M. Lyapunov’s theorem on asymptotic stability in the form
\[ V(t,x)=\sum_{i=1}^{n} A_i(t)x_i^2, \tag{4} \]
whose coefficients \(A_i(t)\) are as yet undetermined. Suppose that the time derivative, in virtue of system (1), of the function (4) is the quadratic form
\[ \frac{dV}{dt}=-\sum_{i,j=1}^{n} B_{ij}(t)x_ix_j,\quad B_{ij}=B_{ji}. \tag{5} \]
The coefficients of the forms (4) and (5) are connected with one another by the system of differential equations, independent of one another,
\[ \frac{dA_i}{dt}+2A_ip_{ii}=-B_{ii} \tag{6} \]
and by the relations
\[ A_ip_{ij}+A_jp_{ji}=-B_{ij}\quad (i\ne j). \tag{7} \]
Solving equations (6) with respect to \(A_i(t)\), we find
\[ A_i(t)=\left(a_i+\int_{0}^{t} B_{ii}(t)e^{F_i(t)}\,dt\right)e^{-F_i(t)}, \tag{8} \]
where
\[ F_i(t)=2\int_{0}^{t}p_{ii}(t)\,dt, \]
and \(a_i\) is an arbitrary constant of integration.
Choose \(B_{ii}(t)\) \((i=1,2,\ldots,n)\), and let
\[ B_{ii}(t)=-b_ie^{-F_{2i}}F'_{1i}, \tag{9} \]
where \(b_i\) is an arbitrary positive constant.
Substituting the chosen functions into formulas (8), we obtain
\[ A_i(t)=\left[a_i-b_ie^{F_{1i}(0)}+b_ie^{F_{1i}(t)}\right]e^{-F_i(t)}. \]
Let the arbitrary constants \(a_i\) be chosen so that
\[ a_i-b_i e^{F_{1i}(0)}=0. \]
Then
\[ A_i(t)=b_i e^{-F_{2i}(t)}. \]
In view of condition (*) the \(A_i=b_i e^{-F_{2i}(t)}\) are bounded functions, and therefore the positive-definite function
\[ V(t,x)=\sum_{i=1}^{n} b_i e^{-F_{2i}(t)}x_i^2 \tag{4¹} \]
admits an infinitesimal upper limit. Formula (7) for the constructed \(A_i(t)\) has the form
\[ -B_{ij}=b_i e^{-F_{2i}(t)}p_{ij}(t)+b_j e^{-F_{2j}(t)}p_{ji}(t)\quad (i\ne j). \tag{7¹} \]
According to Lemma 1 and condition (*), the fulfillment of the conditions
\[ B_{11}>0,\quad \begin{vmatrix} B_{11} & B_{12}\\ B_{21} & B_{22} \end{vmatrix}>0,\ \ldots,\quad \begin{vmatrix} B_{11} & \cdot & \cdot & \cdot & B_{1,n-1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ B_{n-1,1} & \cdot & \cdot & \cdot & B_{n-1,n-1} \end{vmatrix}>0, \]
\[ \inf_{t\ge 0} \begin{vmatrix} B_{11} & \cdot & \cdot & B_{1n}\\ B_{21} & \cdot & \cdot & B_{2n}\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ B_{n1} & \cdot & \cdot & B_{nn} \end{vmatrix}>0, \tag{10} \]
where \(B_{ij}(t)\) are computed by formulas (7¹) and (9), ensures for the form (5) sign-definiteness in the sense of Lyapunov, the signs of the forms (4) and (5) being opposite.
On the basis of A. M. Lyapunov’s theorem on asymptotic stability, when condition (10) is satisfied the trivial solution \(x_i=0\) of system (1) will be asymptotically stable.
The inequalities (10) contain \(n\) arbitrary positive constants \(b_i\). The necessary and sufficient conditions for the compatibility of the inequalities (10) with respect to \(b_i>0\), which can be obtained by induction on \(n\), will be necessary and sufficient for the form (4¹) to satisfy Lyapunov’s theorem on asymptotic stability.
Remark 1. The conditions (*), formulated at the beginning of the note, are necessarily satisfied if for system 1 there exists a positive Lyapunov function of the form \(V(t,x)=\sum_{i=1}^{n} A_i(t)x_i^2\) with bounded coefficients \(A_i(t)\).
Indeed, denoting by
\[ F_{2i}(t)=-\ln\frac{A_i(t)}{b_i}, \]
we find
\[ F_{1i}=F_i-F_{2i}=2\int_{0}^{t}p_{ii}(t)\,dt+\ln\frac{A_i(t)}{b_i}, \]
whence
\[ F'_{1i}=2p_{ii}(t)+\frac{\dfrac{dA_i(t)}{dt}}{A_i(t)} =-\frac{B_{ii}(t)}{A_i(t)}<0. \]
Remark 2. If in conditions \((*)\), instead of \(F'_{1i}<0\), one requires \(F'_{1i}>0\), then the compatibility conditions for the inequalities \((10^1)\)
\[ -B_{11}>0,\quad \left|\begin{matrix} B_{11}&B_{12}\\ B_{21}&B_{22} \end{matrix}\right|>0,\ \ldots,\ (-1)^i \left|\begin{matrix} B_{11}&\cdots&B_{1i}\\ \cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\\ B_{i1}&\cdots&B_{ii} \end{matrix}\right|>0, \]
\[ >0,\ \ldots,\ \inf_t(-1)^n \left|\begin{matrix} B_{11}&\cdots&B_{1n}\\ \cdot&\cdot&\cdot\\ \cdot&\cdot&\cdot\\ B_{n1}&\cdots&B_{nn} \end{matrix}\right|>0 \tag{10^1} \]
will be sufficient for the form \((4^1)\) to satisfy the instability theorem.
In conclusion, let us obtain the compatibility condition for inequalities \((10)\) and \((10^1)\) for \(n=2\). It consists of the solvability condition for the inequality
\[ \inf_t \left| \begin{matrix} b_1e^{-F_{21}}F'_{11} & -\bigl(b_1e^{-F_{21}}p_{12}+b_2e^{-F_{22}}p_{21}\bigr) \\ -\bigl(b_1e^{-F_{21}}p_{12}+b_2e^{-F_{22}}p_{21}\bigr) & b_2e^{-F_{22}}F'_{12} \end{matrix} \right|>0 \tag{11} \]
with respect to the positive constants \(b_1,\ b_2\). Transform inequality \((11)\) to the form
\[ \sup_t\{\alpha(t)k_0^2+\beta(t)k_0+\gamma(t)\}<0, \tag{12} \]
where
\[ k_0=\frac{b_2}{b_1}, \]
\[ \alpha(t)=\frac{[p_{21}(t)]^2}{e^{2F_{22}(t)}}\geq 0,\quad \gamma(t)=\frac{[p_{12}(t)]^2}{e^{2F_{21}(t)}}\geq 0,\quad \beta(t)=\frac{2p_{12}p_{21}-F'_{11}F'_{12}}{e^{F_{21}+F_{22}}}. \]
Inequality \((12)\), for \(k_0>0\), is a criterion for asymptotic stability when \(F'_{1i}<0\), and for instability when \(F'_{1i}>0\), equivalent to inequalities \((10)\) and \((10^1)\) for \(n=2\).
Denote the roots of the trinomial
\[ \alpha(t)k^2+\beta(t)k+\gamma(t) \tag{12a} \]
by \(k_1(t)\) and \(k_2(t)\), and let \(k_1(t)>k_2(t)\). Carrying out an investigation of the quadratic trinomial \((12a)\), we establish that, in order that there exist at least one positive value \(k_0=\mathrm{const}\) satisfying inequality \((12)\), it is necessary and sufficient that the roots \(k_1(t)\) and \(k_2(t)\) be real and satisfy the relation
\[ \inf_t k_1(t)>\sup_t k_2(t)\geq 0. \tag{13} \]
In the former notation, inequality \((13)\) has the form
\[ \inf_t \frac{ F'_{11}F'_{12}-2p_{12}p_{21} +\sqrt{F'_{11}F'_{12}\bigl(F'_{11}F'_{12}-4p_{12}p_{21}\bigr)} }{ 2p_{21}^{2}e^{F_{21}-F_{22}} } > \]
\[ > \sup_t \frac{F'_{11}F'_{12}-2p_{12}p_{21} -\sqrt{F'_{11}F'_{12}\left(F'_{11}F'_{12}-4p_{12}p_{21}\right)}} {2p_{21}^{2}e^{-F_{21}-F_{22}}} \geqslant 0. \tag{13a} \]
Let us note that the function appearing on the right in (13a) has removable discontinuities at the zeros of the denominator and can be completed to a continuous function. In using inequality (13a), we shall assume this to have been done.
We summarize the foregoing in the form of a theorem.
In order that, for system (1) with \(n=2\), there exist a Lyapunov function of the form
\[ V(t,x_1,x_2)=+b_1e^{-F_{21}(t)}x_1^2+b_2e^{-F_{22}}x_2^2, \]
satisfying all the conditions of Lyapunov’s theorem on asymptotic stability or instability, it is necessary and sufficient that the coefficients of system (1) satisfy inequality (13a). When inequalities (13a) are fulfilled, the inequality \(F'_{1i}<0\) \((i=1,2)\) ensures asymptotic stability, while \(F'_{1i}>0\) \((i=1,2)\) ensures instability of the trivial solution \(x_i=0\).
Remark. If in inequality (13a) the sign \(>\) is replaced by the sign \(\geqslant\), then for \(F'_{1i}<0\) a sufficient stability criterion is obtained.
References
- Lyapunov A. M. The general problem of the stability of motion. Gostekhizdat, 1950.
- Ivanov E. G. Candidate dissertation, Moscow State University, 1961.
- Ivanov E. G. Scientific works of the Tula Mining Institute, coll. 3. Gosgortekhizdat, 1961.
Received by the editors
December 9, 1964
Novosibirsk Pedagogical Institute