Dual approach to Lyapunov stability (Q387140)

The paper is concerned with stability at the origin, for a system of time-varying ordinary differential equations NEWLINE\[NEWLINE\dot x = f(t,x) \leqno(1)NEWLINE\]NEWLINE under Carathéodory assumptions. First, the author studies stability for the so-called dual equation NEWLINE\[NEWLINE\left( {{\partial W}\over{\partial y}} (t,y) \right)' = f \left(t, -{{\partial W}\over{\partial y}}(t,y) \right), \leqno(2)NEWLINE\]NEWLINE an implicit differential equation in the unknown \(y(t)\), where \(W(t,y)\) is some smooth function satisfying a suitable Hamilton-Jacobi inequality.NEWLINENEWLINESecond, the problem of relating the stability of (1) to the stability of (2) is addressed, and two possible approaches are described. Two illustrative examples are concluded in the paper.