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admissible control arbitrary Assume asymptotically stable attractor boundary of G brings the system change sign column vector condition for optimal continuous first partial control theory define dynamic programming eigenvalues eigenvectors equation of dynamic equilibrium point error in control Exercise exponential function on G Hence integral Liapunov function LIAPUNOV STABILITY THEORY limit set linear approximation linear system linear time optimal linearly independent mathematical matrix exponential n x n matrix n-vector necessary and sufficient necessary condition negative definite negative real neighborhood nonsingular obtain optimal control law optimal control problem optimal problem origin in finite origin is asymptotically performance criterion polynomial Pontryagin's maximum principle positive definite matrix proper control systems quadrant quadratic form reach the origin region of attraction remain in G satisfying Section shown in Fig solution of 3.8 solution starting solution x(t starting in G sufficient condition system x Theorem 7.1 unstable