Sufficient Conditions for Stability of General Systems

Abstract

In this section, we give sufficient conditions for stability of the solution x = 0 of (3.1) and illustrate the results with examples. If V: R+ × CH→ R is continuous we let V˙(t,ϕ)=lim⁡‾h→0+1h[V(t+h,xt+h(t,ϕ))−V(t,ϕ)]{\rm{\dot V}}\left( {{\rm{t}},{\rm{\phi }}} \right) = \mathop {\overline {\lim } }\limits_{{\rm{h}} \to 0^ + } \frac{1}{{\rm{h}}}\left[ {{\rm{V}}\left( {{\rm{t + h,x}}_{{\rm{t}} + {\rm{h}}} \left( {{\rm{t}},{\rm{\phi }}} \right)} \right) - {\rm{V}}\left( {{\rm{t}},{\rm{\phi }}} \right)} \right] where xt+h(t,ϕ) is the solution of (3.1) through (t,ϕ). V(t,ϕ) is the upper right hand derivate of V(t,ϕ) along the solutions of (3.1).