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Subsections

12 Numerical tests on stability and positiveness

At present there are no effective algorithms of computing the eigenvalues of linear systems with delays

$\displaystyle \dot{x}$ = A x + A$\scriptstyle \tau$ y(- $\displaystyle \tau$) + $\displaystyle \int\limits_{-\tau}^{0}$G(s) y(s) ds (12.1)

in order to check its stability.

In this section we discuss some methods for practical verification of stability of system (12.1) using computer simulation.

12.1 Stability via the fundamental matrix

The fundamental matrix

F[t] = $\displaystyle \left[\vphantom{ \begin{array}{cccc} f_{11}(t)&f_{12}(t)&\ldots&f... ...ldots&. [2mm] f_{n1}(t)&f_{n2}(t)&\ldots&f_{nn}(t)\\ \end{array} }\right.$$\displaystyle \begin{array}{cccc} f_{11}(t)&f_{12}(t)&\ldots&f_{1n}(t) [2mm]... ...m] . &. &\ldots&. [2mm] f_{n1}(t)&f_{n2}(t)&\ldots&f_{nn}(t)\\ \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cccc} f_{11}(t)&f_{12}(t)&\ldots&f... ...ldots&. [2mm] f_{n1}(t)&f_{n2}(t)&\ldots&f_{nn}(t)\\ \end{array} }\right]$ (12.2)

is the solution of the matrix delay differential equation

$\displaystyle {\frac{d F[t]}{d t}}$ = A F[t] + A$\scriptstyle \tau$ F[t - $\displaystyle \tau$] + $\displaystyle \int\limits_{-\tau}^{0}$G(s) F[t + s] ds ,  t > 0 , (12.3)

under the condition

$\displaystyle \left\{\vphantom{ \begin{array}{l} \vspace{2mm} \displaystyle F... ...laystyle F[t] = 0 \mbox{for} \qquad t < 0 .\end{array}}\right.$$\displaystyle \begin{array}{l} \vspace{2mm} \displaystyle F[0] = I ,... ...m} \displaystyle F[t] = 0 \mbox{for} \qquad t < 0 .\end{array}$ (12.4)

One can fomulate the following stability criteria in terms of fundamental matrix (12.2).

Theorem 12.1. System (12.1) is

1)
stable if and only if there exists a constant k > 0 such that

$\displaystyle \Bigl\Vert$ F[t] $\displaystyle \Bigr\Vert _{n \times n}^{}$$\displaystyle \le$k ,  t$\displaystyle \ge$0 ; (12.5)

2)
asymptotically stable if and only if there exist constants k > 0 and $ \alpha$ > 0 such that

$\displaystyle \Bigl\Vert$ F[t] $\displaystyle \Bigr\Vert _{n \times n}^{}$$\displaystyle \le$k e- $\scriptstyle \alpha$ t ,  t$\displaystyle \ge$0 . (12.6)

$ \Box$

The fundamental matrix can be found numerically using fundmatr (as the solution of system (12.3), (12.4)).

Remark 12.1. Thus, one can easily check stability (or instability) of system (12.1) solving numerically system (12.3), (12.4) and verifying the corresponding properties (12.5) or (12.6) of the matrix F[t]. Note, if at least one of the coefficients of the matrix F[t] is not uniformly bounded then system (12.1) is unstable. $ \Box$

12.2 Stability with respect to a class of functions

First of all it is necessary to note that, as it was emphasized by many authors, complete correct statement of a stability problem for a concrete system with delays should include description of a class of admissible initial functions (initial disturbances).

In this case it is sufficient to consider stability of solution of specific time-delay system only with respect to admissible initial disturbances.

Remark 12.2. In [32] one can find an example of time-delay system which is unstable with respect to the class of all continuous disturbances, but is stable with respect to more narrow class of admissible initial functions. $ \Box$

Of course, classes of admissible initial disturbances are different in different problems, so in general stability theory usually the class of all continuous or piece-wise continuous initial functions (disturbances) is considered. Nevertheless, in some problems such class of initial functions can be superfluous.

Let $ \cal {L}$ be a subset (a system of functions) of the space H.

Definition 12.1. System (12.1) is stable with respect to a class of functions $ \cal {L}$ if for any h $ \in$ $ \cal {L}$ the corresponding solution is bounded. $ \Box$

Definition 12.2. System (12.1) is asymptotically stable with respect to a class of functions $ \cal {L}$ if

$\displaystyle \lim_{t \to \infty}^{}$| x(t;h)| = 0 (12.7)

for any h $ \in$ $ \cal {L}$. $ \Box$

Note, from linearity of system (12.1) it follows that if the system is (asymptotically) stable with respect to a class $ \cal {L}$ then the system will be also (asymptotically) stable with respect to the space L* = span{$ \cal {L}$} spanned on $ \cal {L}$, and, moreover, the system will be (asymptotically) stable with respect to the class

$\displaystyle \bar{L}$ = span$\displaystyle \left\{\vphantom{ \bigcup_{h \in {\cal L}} \bigcup_{t \ge 0} x_t(h) }\right.$$\displaystyle \bigcup_{h \in {\cal L}}^{}$$\displaystyle \bigcup_{t \ge 0}^{}$xt(h)$\displaystyle \left.\vphantom{ \bigcup_{h \in {\cal L}} \bigcup_{t \ge 0} x_t(h) }\right\}$ .

As we already mentioned, in many cases it is difficult to prove stability of a system with respect to the class all continuous initial functions. In this case one can check stability of the system with respect to a class of test initial functions $ \cal {L}$ by computer simulation.

The corresponding class of functions can be chosen, for example, in the following way.

It is well known, that there exist orthogonal systems of continuous on the interval [- $ \tau$, 0] functions {$ \phi_{i}^{}$( . )}i = 0$\scriptstyle \infty$ such that every function $ \psi$( . ) $ \in$ C[- $ \tau$, 0] can be expanded in series 6

$\displaystyle \psi$(s) = $\displaystyle \sum_{i=0}^{\infty}$$\displaystyle \gamma_{i}^{}$ $\displaystyle \phi_{i}^{}$(s) ,    - $\displaystyle \tau$$\displaystyle \le$s$\displaystyle \le$0 , (12.8)

with some coefficients {$ \gamma_{i}^{}$}i = 0$\scriptstyle \infty$ $ \subset$ $ \bf R$.

One can consider first k functions

$\displaystyle \phi_{1}^{}$( . ),$\displaystyle \phi_{2}^{}$( . ),...,$\displaystyle \phi_{k}^{}$( . ) $\displaystyle \in$ C[- $\displaystyle \tau$, 0] (12.9)

as basic (test) functions, and investigate stability of system (12.1) with respect to this finite class of functions. In this case the system will be stable with respect to subspace of functions, which are linear combinations of "basic" functions (12.9).

From the linearity of system (12.1) it follows that if for every basic function $ \phi_{1}^{}$,...,$ \phi_{k}^{}$ the corresponding solution x(t,$ \phi_{i}^{}$) tends to zero as t$ \to$$ \infty$, then for arbitrary constants $ \gamma_{1}^{}$,...,$ \gamma_{k}^{}$ the solution x(t,$ \phi$), corresponding to an initial function (12.11), also tends to zero. So, it is sufficient to check convergence to zero only for functions $ \phi_{1}^{}$,...,$ \phi_{k}^{}$.

Definition 12.3. System (12.1) is asymptotic stable with respect to a class of functions (12.9) if

$\displaystyle \lim_{t \to \infty}^{}$| x(t;$\displaystyle \phi_{i}^{}$)| = 0 (12.10)

i = 1,..., k. $ \Box$

Also it is necessary to note, though the series (12.8) contains infinite number of terms, nevertheless taking into account presence of some uncertainties at every specific (applied) problem one can consider a class of admissible initial disturbances as a finite sum

$\displaystyle \psi$(s) = $\displaystyle \sum_{i=0}^{k}$$\displaystyle \gamma_{i}^{}$ $\displaystyle \phi_{i}^{}$(s) ,    - $\displaystyle \tau$$\displaystyle \le$s$\displaystyle \le$0 , (12.11)

$ \gamma_{1}^{}$,..., $ \gamma_{k}^{}$ $ \in$ $ \bf R$, assuming that remainder part $\displaystyle \sum_{i=k+1}^{\infty}$$\displaystyle \gamma_{i}^{}$ $\displaystyle \phi_{i}^{}$( . ) of the series (12.8) corresponds to uncertainties.

Depending on concrete problem one can choose his own system of (linear independent) test functions.

12.3 Test on positiveness of quadratic functionals (with respect to a class of functions)

At present there are no effective criteria of positiveness for general quadratic functionals.

In this subsection we describe an approximate method of verifying positiveness of a quadratic functional

Z[x, y( . )] = x'$\displaystyle \Phi_{0}^{}$ x + 2x'$\displaystyle \int\limits_{-\tau}^{0}$$\displaystyle \Phi_{1}^{}$(s) y(s) ds + $\displaystyle \int\limits_{-\tau}^{0}$$\displaystyle \int\limits_{-\tau}^{0}$y'(s) $\displaystyle \Phi_{2}^{}$(s,$\displaystyle \nu$) y($\displaystyle \nu$) ds d$\displaystyle \nu$ , (12.12)

here $ \Phi_{0}^{}$ is a constant n×n matrix, $ \Phi_{1}^{}$ is n×n matrix with piece-wise continuous elements on [- $ \tau$, 0], $ \Phi_{2}^{}$ is n×n matrix with piece-wise continuous elements on [- $ \tau$, 0]×[- $ \tau$, 0].

Let us consider a finite number of elements

h(i) = {x(i), y(i)( . )} $\displaystyle \in$ H ,     i = 1,..., m , (12.13)

and their linear combinations

$\displaystyle \bar{h}$ = $\displaystyle \lambda_{1}^{}$ h(1) +...+ $\displaystyle \lambda_{m}^{}$ h(m) (12.14)

$ \lambda_{1}^{}$,..., $ \lambda_{m}^{}$ $ \in$ $ \bf R$.

If we substitute (12.14) into quadratic functional (12.12) we obtain the quadratic function $ \omega$ of variables $ \lambda_{1}^{}$,...,$ \lambda_{m}^{}$

$\displaystyle \omega$($\displaystyle \lambda_{1}^{}$,...,$\displaystyle \lambda_{m}^{}$) = Z[$\displaystyle \lambda_{1}^{}$ h(1) +...+ $\displaystyle \lambda_{m}^{}$ h(m)] =

$\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ x(1) +...+ $\displaystyle \lambda_{m}^{}$ x(m)$\displaystyle \Bigr){^\prime}$$\displaystyle \Phi_{0}^{}$ $\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ x(1) +...+ $\displaystyle \lambda_{m}^{}$ x(m)$\displaystyle \Bigr)$ +

+2$\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ x(1) +...+ $\displaystyle \lambda_{m}^{}$ x(m)$\displaystyle \Bigr){^\prime}$$\displaystyle \int\limits_{-\tau}^{0}$$\displaystyle \Phi_{1}^{}$(s) $\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ y(1)(s) +...+ $\displaystyle \lambda_{m}^{}$ y(m)(s)$\displaystyle \Bigr)$ ds +

+ $\displaystyle \int\limits_{-\tau}^{0}$$\displaystyle \int\limits_{-\tau}^{0}$$\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ y(1)(s) +...+ $\displaystyle \lambda_{m}^{}$ y(m)(s)$\displaystyle \Bigr){^\prime}$ $\displaystyle \Phi_{2}^{}$(s,$\displaystyle \nu$) $\displaystyle \Bigl($$\displaystyle \lambda_{1}^{}$ y(1)($\displaystyle \nu$) +...+ $\displaystyle \lambda_{m}^{}$ y(m)($\displaystyle \nu$)$\displaystyle \Bigr)$ ds d$\displaystyle \nu$ . (12.15)


Definition 12.4. Functional (12.12) is called positive definite with respect to a class of functions (12.13) if the quadratic function $ \omega$($ \lambda_{1}^{}$,...,$ \lambda_{m}^{}$) is positive definite with respect to $ \lambda_{1}^{}$,...,$ \lambda_{m}^{}$. $ \Box$

Representation (12.15) can be used for approximate verification of positiveness of quadratic functionals.

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