View source for Stability, absolute

Global stability of the trivial solution of a non-linear system of ordinary differential equations (or equations of other type), uniform for all systems of a certain class. The term  "absolute stability"  assumes given a class of systems and an indication of the sense in which stability and uniformity are to be understood. Besides ordinary differential equations one also considers finite-difference equations, integral equations, ordinary differential equations with delay argument, and partial differential equations.

Consider the system described by the differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

and by a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869402.png" /> of pairs of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869403.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869404.png" /> are constant complex matrices of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869406.png" />, respectively; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869408.png" /> are vectors of complex-valued functions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s0869409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694010.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694011.png" /> is locally summable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694012.png" /> is absolutely continuous. In applications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694016.png" /> are usually real, equation (1) describes the linear part of a system, while the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694017.png" /> is determined by the properties of the non-linear blocks of the system. In simple cases there is one non-linear block, which is described by an equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694020.png" /> are scalar functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694021.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694022.png" />-dimensional matrix; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694025.png" /> are real). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694026.png" /> is the set of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694027.png" /> for which (2) holds.

Numerous studies of particular non-linear systems have led to the understanding that in the first place one should take in consideration a certain quadratic relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694029.png" />. For example, suppose that about the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694030.png" /> in (2) it is known only that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694032.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694033.png" /></td> </tr></table>

In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694034.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694036.png" /> for which almost-everywhere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694038.png" />, or, otherwise,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Below, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694041.png" /> is a Hermitian form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694042.png" />. In the general case one considers the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694043.png" /> of all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694044.png" /> of functions satisfying almost-everywhere the local constraint

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694045.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

as well as the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694046.png" /> of pairs of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694047.png" /> satisfying the integral constraint

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694048.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

(the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694049.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694050.png" />). A variety of practically important non-linear blocks ( "air vents" , hysteresis non-linearity, impulse modulators of different types) satisfy a constraint (5), with a suitably chosen form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694051.png" />.

Below it is assumed that equation (1) is controllable (cf. [[#References|[1]]]), i.e. that the rank of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694052.png" />-dimensional matrix

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694053.png" /></td> </tr></table>

equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694054.png" />, and also that the following condition of minimal stability is fulfilled: There exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694055.png" />-dimensional matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694056.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694057.png" /> is a Hurwitz matrix (i.e. is stable) and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694058.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694059.png" /> is the form in (4) or (5). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694061.png" /> be arbitrary matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694063.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694064.png" />, and form the  "output"  of the system (1):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

One distinguishes between the real case, when all quantities in (1), (6) and the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694066.png" /> are real, and the complex case, when they are generally complex. The set of all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694067.png" /> satisfying (4) (or (5)) is denoted below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694068.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694069.png" />). Put

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694070.png" /></td> </tr></table>

The system (1) is called absolutely stable with respect to the output (6) in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694071.png" /> if there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694072.png" /> such that (1), (6) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694073.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694074.png" /> is finite and satisfies the estimate

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

Quadratic criteria for absolute stability. For the absolute stability of the system (1) with respect to output (6) in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694076.png" /> (in the real case in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694077.png" />) it is necessary and sufficient that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694081.png" />, and real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694082.png" /> connected by the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

If (8), (9) hold, then one can take in (7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694084.png" />, where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694086.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694087.png" /> in (5). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694088.png" /> and (4) is satisfied as well as (8) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694089.png" />), then one has global exponential stability:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694092.png" />.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694093.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694094.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694095.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694096.png" />-dimensional unit matrix). For the absolute stability of the system (1) with respect to the output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694097.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694098.png" /> it is necessary and sufficient that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s08694099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940100.png" />, and any complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940101.png" />, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

holds. For the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940103.png" /> an analogous assertion is true only relative to sufficiency. Necessary and sufficient conditions for the absolute stability in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940104.png" /> are known only for special forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940105.png" />, and an effectively verifiable condition only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940106.png" /> (cf. [[#References|[3]]], [[#References|[7]]]).

From relation (9) it follows that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940107.png" /></td> </tr></table>

the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940108.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940109.png" /> is called the frequency characteristic from input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940110.png" /> to output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940111.png" />. Criteria establishing certain properties of the system expressible by the frequency characteristics are called frequency stability criteria. The merit of frequency criteria lies in their usefulness in practical applications and in their invariance under a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940112.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940114.png" />) of the system (1).

In the real case with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940115.png" />, for the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940116.png" /> defined by the relation (3) condition (11) reduces to the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940117.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940118.png" /> is the frequency characteristic from input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940119.png" /> to output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940120.png" />. The frequency criterion (12) (circle criterion) means that the frequency characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940122.png" />, is non-intersecting with the circle with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940123.png" /> and passing through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940125.png" />. The condition of minimal stability in this case means asymptotic stability of the linear system (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940127.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940128.png" />. The criterion (12) is the natural extension of the Mikhailov–Nyquist criterion for non-linear systems (cf. [[Mikhailov criterion|Mikhailov criterion]]; [[Nyquist criterion|Nyquist criterion]]).

Historically, the first frequency criterion of absolute stability for non-linear systems was Popov's criterion for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940129.png" /> and the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940130.png" /> of stationary non-linearities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940131.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940132.png" /> (cf. [[#References|[2]]]). It has the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940133.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940134.png" /></td> </tr></table>

The condition of minimal stability in this case is equivalent to requiring the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940135.png" /> in (1) to be a Hurwitz matrix. This criterion can be simply verified in a geometrical manner.

There exists a definite connection between the frequency criteria (6), (12), (13), and others and the existence of a global [[Lyapunov function|Lyapunov function]]. The frequency criteria of absolute stability usually cover all criteria which can be obtained by means of a Lyapunov function in certain multi-parameter classes of functions. For example, the criterion (12) is a necessary and sufficient condition for the existence of a function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940136.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940137.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940138.png" />-dimensional matrix, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940139.png" /> is the sign for Hermitian conjugation) such that its derivative along the trajectories of the systems (1), (2) with an arbitrary non-linearity (2) (for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940140.png" />) satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940141.png" /></td> </tr></table>

Similarly, Popov's frequency condition (13) includes all criteria which can be established using Lyapunov functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940142.png" /></td> </tr></table>

Many other frequency criteria for absolute stability are known for different classes of non-linearities (cf. [[#References|[3]]]–[[#References|[6]]]). In particular, they cover many important cases in applications, such as non-unique equilibrium positions (cf. [[#References|[1]]]). Frequency criteria of absolute stability allow one to distinguish classes of non-linear systems of a general form for which the fact of global stability is rather simple to establish. E.g., the system (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940146.png" /> (i.e. an arbitrary system of order at most 3 with a single non-linearity), is globally asymptotically stable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940147.png" /> and if any linear system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940148.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940149.png" />, is asymptotically stable. For systems of order 4 (or higher) an analogous assertion is incorrect. Moreover, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940150.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940151.png" /> there exists a system (1) and a non-linearity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940153.png" />, such that the matrix of any linearized system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940154.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940155.png" />, is a Hurwitz matrix, while the non-linear system has a periodic solution.

After replacing the condition of minimal stability by the analogous condition of minimal instability, the inequalities (8), (11), (12), (13) become criteria of absolute instability (with a corresponding meaning for the last term). For example, consider the real case with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940156.png" />, let the matrix of coefficients of the system (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940157.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940158.png" /> (i.e. the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940159.png" />) for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940161.png" />, have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940162.png" /> eigenvalues in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940163.png" />, and let the frequency condition (12) be satisfied. Then the system (1), (2) with function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940164.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940165.png" /> (as well as the system (1), (3)) possesses a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940166.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940167.png" /></td> </tr></table>

where the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940169.png" /> are the same for all systems of the class considered. The corresponding vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940170.png" /> fill a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940171.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940172.png" /> is a matrix having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940173.png" /> negative eigenvalues.

Similarly, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940174.png" /> the condition (13) is a frequency criterion for the absolute instability of the system (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940175.png" /> in the class of stationary non-linearities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940176.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940177.png" />, if in (1) the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940178.png" /> has an eigenvalue in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086940/s086940179.png" />.

In the theory of absolute stability there are similar frequency criteria for dissipation, convergence, existence of periodic motions (self-oscillations and forced regimes), and others (cf. e.g. [[#References|[3]]], [[#References|[5]]] and the references in [[#References|[1]]], [[#References|[3]]], [[#References|[5]]]; see also [[#References|[8]]]–[[#References|[10]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.Kh. Gelig,   "Stability of non-linear systems with non-unique equilibrium positions" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Aizerman,   F.R. Gantmakher,   "Absolute stability of non-linear control sytems" , Moscow  (1963)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Yakubovich,  , ''Methods for studing non-linear systems of automatic control'' , Moscow  (1975)  pp. 74–180  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Popov,   "Hyperstability of control systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Voronov,   "Stability, controllability, observability" , Moscow  (1979)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V. Rezvan,   "Absolute stability of automatic systems with delay" , Moscow  (1983)  (In Rumanian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.S. Pyatnitskii,   "New research on the absolute stability of automatic control systems"  ''Automat. Remote Control'' :  6  (1968)  pp. 885–881  ''Avtomatika i Telemekhanika'' :  6  (1968)  pp. 5–36</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  D.D. Ŝiljak,   "Nonlinear systems. Parameter analysis and design" , Wiley  (1969)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  K.S. Narendra,   I.H. Taylor,   "Frequency domain criteria for absolute stability" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.L. Willems,   "Stability theory of dynamical systems" , Nelson  (1970)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. Willems,   "Least squares stationary optimal control and the algebraic Riccati equation"  ''IEEE Trans. Aut. Control'' , '''AC-16'''  (1971)  pp. 621–634</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. La Salle,   S. Lefschetz,   "Stability by Lyapunov's direct method with applications" , Acad. Press  (1961)</TD></TR></table>