Differential Equations

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Routh-Hurwitz Criterion

Solved Problems

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Example 1

Investigate the stability of the zero solution of the equation

\[x^{\prime\prime\prime} + 6x^{\prime\prime} + 3x' + 2x = 0.\]

Example 2

Investigate the stability of the zero solution of the differential equation

\[{x^{IV}} + 2x^{\prime\prime\prime} + 4x^{\prime\prime} + 7x' + 3x = 0.\]

Example 3

For what values of the parameters\(\alpha\) and \(\beta\) the zero solution of the equation

\[{x^{IV}} + x^{\prime\prime\prime} + \alpha x^{\prime\prime} + \beta x' + x = 0.\]

Example 4

Investigate for which values of the parameters \(\alpha\) and \(\beta\) the zero solution of the system is asymptotically stable:

\[ \frac{{dx}}{{dt}} = \alpha x + \beta y,\;\; \frac{{dy}}{{dt}} = x - z,\;\; \frac{{dz}}{{dt}} = - x + y.\]

Example 1.

Investigate the stability of the zero solution of the equation

\[x^{\prime\prime\prime} + 6x^{\prime\prime} + 3x' + 2x = 0.\]

Solution.

We write the auxiliary equation:

\[{\lambda ^3} + 6{\lambda ^2} + 3\lambda + 2 = 0.\]

The coefficients \({a_i}\) are

\[ {a_0} = 1,\;\; {a_1} = 6,\;\; {a_2} = 3,\;\; {a_3} = 2.\]

Form the Hurwitz matrix

\[\left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_0}}&0\\ {{a_3}}&{{a_2}}&{{a_1}}\\ 0&{{a_4}}&{{a_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 6&1&0\\ 2&3&6\\ 0&0&2 \end{array}} \right]\]

and find all its principal minors:

\[ {\Delta _1} = 6 \gt 0,\;\; {\Delta _2} = \left| {\begin{array}{*{20}{c}} 6&1\\ 2&3 \end{array}} \right| = 18 - 2 = 16 \gt 0,\;\; {\Delta _3} = {a_3}{\Delta _2} = 2 \cdot 16 = 32 \gt 0.\]

As it can be seen, all minors are positive. Therefore, the zero solution of the equation is asymptotically stable.

Example 2.

Investigate the stability of the zero solution of the differential equation

\[{x^{IV}} + 2x^{\prime\prime\prime} + 4x^{\prime\prime} + 7x' + 3x = 0.\]

Solution.

The auxiliary equation is given by

\[{\lambda ^4} + 2{\lambda ^3} + 4{\lambda ^2} + 7\lambda + 3 = 0.\]

The coefficients \({a_i}\) in this equation are

\[ {a_0} = 1,\;\; {a_1} = 2,\;\; {a_2} = 4,\;\; {a_3} = 7,\;\; {a_4} = 3.\]

We write the Hurwitz matrix and find its principal diagonal minors:

\[\left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_0}}&0&0\\ {{a_3}}&{{a_2}}&{{a_1}}&{{a_0}}\\ 0&{{a_4}}&{{a_3}}&{{a_2}}\\ 0&0&0&{{a_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&1&0&0\\ 7&7&2&1\\ 0&3&7&4\\ 0&0&0&3 \end{array}} \right];\]
\[ {\Delta _1} = 2 \gt 0,\;\; {\Delta _2} = \left| {\begin{array}{*{20}{c}} 2&1\\ 7&4 \end{array}} \right| = 8 - 7 = 1 \gt 0,\;\; {\Delta _3} = \left| {\begin{array}{*{20}{c}} 2&1&0\\ 7&4&2\\ 0&3&7 \end{array}} \right| = - 2 \cdot \left| {\begin{array}{*{20}{c}} 2&1\\ 0&3 \end{array}} \right| + 7 \cdot \left| {\begin{array}{*{20}{c}} 2&1\\ 7&4 \end{array}} \right| = - 12 + 7 = - 5 \lt 0.\]

In the course of calculations, we've got a negative minor \({\Delta _3} \lt 0.\) This means that the zero solution is unstable.

Example 3.

For what values of the parameters\(\alpha\) and \(\beta\) the zero solution of the equation

\[{x^{IV}} + x^{\prime\prime\prime} + \alpha x^{\prime\prime} + \beta x' + x = 0.\]

Solution.

The auxiliary equation can be written as

\[{\lambda ^4} + {\lambda ^3} + \alpha {\lambda ^2} + \beta \lambda + 1 = 0.\]

The coefficients have the following values:

\[ {a_0} = 1,\;\; {a_1} = 1,\;\; {a_2} = \alpha,\;\; {a_3} = \beta,\;\; {a_4} = 1.\]

Form the Hurwitz matrix

\[\left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_0}}&0&0\\ {{a_3}}&{{a_2}}&{{a_1}}&{{a_0}}\\ 0&{{a_4}}&{{a_3}}&{{a_2}}\\ 0&0&0&{{a_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&1&0&0\\ \beta&\alpha&1&1\\ 0&1&\beta&\alpha\\ 0&0&0&1 \end{array}} \right]\]

and write the Routh-Hurwitz stability condition:

\[{\Delta _1} = 1 \gt 0,\]
\[{\Delta _2} = \left| {\begin{array}{*{20}{c}} 1&1\\ \beta &\alpha \end{array}} \right| = \alpha - \beta \gt 0,\]
\[ {\Delta _3} = \left| {\begin{array}{*{20}{c}} 1&1&0\\ \beta &\alpha &1\\ 0&1&\beta \end{array}} \right| = \left| {\begin{array}{*{20}{c}} \alpha &1\\ 1&\beta \end{array}} \right| - \beta \left| {\begin{array}{*{20}{c}} 1&0\\ 0&\beta \end{array}} \right| = \alpha \beta - 1 - {\beta ^2} = \beta \left( {\alpha - \beta } \right) - 1 \gt 0,\]
\[{\Delta _4} = {a_4}{\Delta _3} \gt 0,\;\; \Rightarrow {a_4} = 1 \gt 0.\]

Thus, the system of inequalities that describes the stability region is as follows:

\[\left\{ \begin{array}{l} {\Delta _2} \gt 0\\ {\Delta _3} \gt 0 \end{array} \right.,\;\; \Rightarrow \left\{ {\begin{array}{*{20}{l}} {\alpha - \beta \gt 0}\\ {\beta \left( {\alpha - \beta } \right) \gt 1} \end{array}} \right..\]

Let us first consider the second inequality. We solve it with respect to \(\alpha.\) Here, there are two cases:

  1. If \(\beta \gt 0,\) then dividing both sides of the inequality by \(\beta,\) we obtain:
    \[\alpha - \beta \gt \frac{1}{\beta },\;\; \Rightarrow \alpha \gt \beta + \frac{1}{\beta };\]
  2. If \(\beta \lt 0,\) then we have:
    \[\alpha - \beta \lt \frac{1}{\beta },\;\; \Rightarrow \alpha \lt \beta + \frac{1}{\beta }.\]

Obviously, the case \(\beta = 0\) does not satisfy this inequality.

So, the solution of the inequality are two infinite regions, which are symmetric about the origin and bounded by the curve \(\alpha = \beta + {\frac{1}{\beta }}\) (Figure \(1\)).

The region (shaded in green) where the system in Example 3 is asymptotically stable
Figure 1.

The first inequality \(\alpha \gt \beta\) is satisfied by the set of points above the line \(\alpha = \beta.\) As a result, the solution of the whole system is the only region \(\alpha \gt \beta + {\frac{1}{\beta }}\) in the upper right quadrant (with \(\beta \gt 0,\) \(\alpha \gt 0\)). In Figure \(1,\) this area is shown in green. This region of the plane \(\left( {\beta ,\alpha } \right)\) is a region of asymptotic stability of the original equation.

Example 4.

Investigate for which values of the parameters \(\alpha\) and \(\beta\) the zero solution of the system is asymptotically stable:

\[ \frac{{dx}}{{dt}} = \alpha x + \beta y,\;\; \frac{{dy}}{{dt}} = x - z,\;\; \frac{{dz}}{{dt}} = - x + y.\]

Solution.

The auxiliary equation is given by

\[ \left| {\begin{array}{*{20}{c}} {\alpha - \lambda }&\beta &0\\ 1&{ - \lambda }&{ - 1}\\ { - 1}&1&{ - \lambda } \end{array}} \right| = 0,\;\; \Rightarrow \left( {\alpha - \lambda } \right)\left| {\begin{array}{*{20}{r}} { - \lambda }&{ - 1}\\ 1&{ - \lambda } \end{array}} \right| - \beta \left| {\begin{array}{*{20}{r}} 1&{ - 1}\\ { - 1}&{ - \lambda } \end{array}} \right| = 0,\;\; \Rightarrow \left( {\alpha - \lambda } \right)\left( {{\lambda ^2} + 1} \right) - \beta \left( { - \lambda - 1} \right) = 0,\;\; \Rightarrow \alpha {\lambda ^2} - {\lambda ^3} + \alpha - \lambda + \beta \lambda + \beta = 0,\;\; \Rightarrow {\lambda ^3} - \alpha {\lambda ^2} + \left( {1 - \beta } \right)\lambda - \alpha - \beta = 0.\]

Here, the coefficients \({a_i}\) have the following values:

\[ {a_0} = 1,\;\; {a_1} = - \alpha ,\;\; {a_2} = 1 - \beta ,\;\; {a_3} = - \alpha - \beta .\]

First we determine at which values of \(\alpha\) and \(\beta\) the necessary condition of the stability holds:

\[{a_i} \gt 0,\;\;i = 0,1,2,3.\]

We obtain the following system of inequalities:

\[\left\{ \begin{array}{l} - \alpha \gt 0\\ 1 - \beta \gt 0\\ - \alpha - \beta \gt 0 \end{array} \right.,\;\; \Rightarrow \left\{ \begin{array}{l} \alpha \lt 0\\ \beta \lt 1\\ \beta \lt - \alpha \end{array} \right..\]

It is convenient to draw the solution to this system in the plane \(\left( {\alpha ,\beta } \right).\) Figure \(2\) shows the straight lines \(\alpha = 0,\) \(\beta = 1,\) \(\beta = -\alpha\) and the regions corresponding to the solution of each inequality. The intersection of these areas is marked in yellow.

\[\left\{ \begin{array}{l} - \alpha \gt 0\\ 1 - \beta \gt 0\\ - \alpha - \beta \gt 0 \end{array} \right.,\;\; \Rightarrow \left\{ \begin{array}{l} \alpha \lt 0\\ \beta \lt 1\\ \beta \lt - \alpha \end{array} \right..\]
The region (shaded in green) where the system in Example 4 is asymptotically stable
Figure 2.

Now consider the sufficient conditions of the stability. Write the Hurwitz matrix:

\[\left[ {\begin{array}{*{20}{c}} {{a_1}}&{{a_0}}&0\\ {{a_3}}&{{a_2}}&{{a_1}}\\ 0&0&{{a_3}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - \alpha }&1&0\\ { - \alpha - \beta }&{1 - \beta }&{ - \alpha }\\ 0&0&{ - \alpha - \beta } \end{array}} \right].\]

The principal diagonal minors \({\Delta _i}\) are given by

\[{\Delta _1} = - \alpha ,\]
\[{\Delta _2} = \left| {\begin{array}{*{20}{c}} { - \alpha }&1\\ { - \alpha - \beta }&{1 - \beta } \end{array}} \right| = \alpha \left( {\beta - 1} \right) + \alpha + \beta = \alpha \beta - \cancel{\alpha} + \cancel{\alpha} + \beta = \left( {\alpha + 1} \right)\beta ,\]
\[{\Delta _3} = {a_3}{\Delta _2} = \left( { - \alpha - \beta } \right)\left( {\alpha + 1} \right)\beta .\]

According to the Routh-Hurwitz stability criterion, the following inequalities must be satisfied:

\[{\Delta _1} = - \alpha \gt 0,\;\;\; {\Delta _2} = \left( {\alpha + 1} \right)\beta \gt 0,\;\;\; {a_3} \gt 0.\]

From these inequalities, the additional one (compared with the necessary conditions) is the only inequality

\[{\Delta _2} \gt 0\;\;\;\text{or}\;\;\; \left( {\alpha + 1} \right)\beta \gt 0.\]

Its solution includes two cases:

  1. \(\beta \gt 0,\;\; \) \(\Rightarrow \alpha + 1 \gt 0\;\;\) \(\text{or}\;\;\alpha \gt - 1;\)
  2. \(\beta \lt 0,\;\; \) \(\Rightarrow \alpha + 1 \lt 0\;\;\) \(\text{or}\;\;\alpha \lt - 1.\)

These two cases are represented by two quadrants bounded by the lines \(\alpha = -1,\) \(\beta = 0\) (They are depicted in purple in Figure \(2\)).

The region of stability of the system is bounded by the values of the parameters \(\alpha, \beta,\) at which both necessary and sufficient conditions of the asymptotic stability are satisfied. This region consists of the left lower quadrant bounded by the lines \(\alpha = -1,\) \(\beta = 0,\) and the right triangle with vertices \(\left( {0,0} \right),\) \(\left( {-1,0} \right),\) \(\left( {-1,1} \right).\) The resulting stability region is shown in the figure in green.

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