A stability result for solutions of certain fourth order differential equations

Summary

A stability result is obtained for the solutions of the differential equation

$$x^{\left( 4 \right)} + a\mathop x\limits^{...} + b\mathop x\limits^{..} + g\left( {x, \dot x} \right) + h\left( x \right) = p\left( t \right)$$

by using Liapunov functions. Constraints are imposed on the functions$g\left( {x, \dot x} \right), g_x \left( {x, \dot x} \right), g_{\dot x} \left( {x, \dot x} \right)$ andh(x) relative to a set of Routh-Hurwitz numbersa, b, c, d. The result obtained is

$$x^2 \left( t \right) + \mathop {x^2 }\limits^ \cdot \left( t \right) + \mathop {x^2 }\limits^{ \cdot \cdot } \left( t \right) + \mathop {x^2 }\limits^{ \cdot \cdot \cdot } \left( t \right) \leqslant \left\{ {e^{ - \mu t} \left[ {D_1 + D_2 \int_{t_0 }^t {\left| {p\left( s \right)} \right|^{2\left( {1 - \mu } \right)} } e^{ - \mu t} ds} \right]} \right\}^{1/1 - \lambda } .$$