Stability by Liapunov's Direct Method: With Applications

assume asymptotic stability bounded set boundedness characteristic equation characteristic roots cofactors complete stability conditions of Theorem conjugate continuous first partial critical point defined denote determinant differential equations E₁ equilibrium EXAMPLE finite escape Hence implies inequality invariant set Lagrange stable Lemma Let V(x Liapunov function V(x limit-cycle matrix n-vector negative definite negative real nonlinear nonsingular nonsingular matrix origin is asymptotically origin is unstable P₁ parameters partial derivatives path periodic solution positive definite function positive solution power series practical stability quadratic form RICHARD BELLMAN satisfied scalar function Section set Q SOLOMON LEFSCHETZ solution is bounded solution starting solution x(t spherical region stability theorem sufficiently large Suppose symmetric matrix system F t₁ theorems of Liapunov trajectory transformation of coordinates ultimately bounded V₁ variable vector x₁ Y₁ zero