Stability by Liapunov's Direct Method: With Applications

assume asymptotic stability bounded for t≥0 bounded set boundedness characteristic equation characteristic roots cofactors complete stability conditions of Theorem continuous first partial critical point d₁ defined denote determinant diag differential equations E₁ equilibrium equivalent system EXAMPLE finite escape Hence implies inequality invariant set Lagrange stable Lemma Let V(x Liapunov function Liapunov function V(x limit-cycle matrix n-vector negative definite negative real nonlinear nonsingular nonsingular matrix origin is asymptotically origin is unstable parameters partial derivatives path periodic solution positive definite function positive solution power series practical stability quadratic form RICHARD BELLMAN satisfied scalar function Section SOLOMON LEFSCHETZ solution is bounded solution starting solution x(t spherical region stability theorem sufficiently large Suppose symmetric matrix system F trajectory transformation of coordinates ultimately bounded V₁ variable vector x₁ y₁ zero