the iterative method for solving nonlinear matrix equation xs a x ta q abstract nonlinear matrix equation x s a x t a q where q is an hermitian positive definite matrix a is the conjugate transpose of the matrix a and both of s and t are positive integers has widely applied background in this paper necessary conditions and sufficient conditions for the existence of hermitian positive definite solutions of this equation are derived an iterative method for obtaining hermitian positive definite solutions is proposed moreover a perturbation bound for the hermitian positive definite solutions is given

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

on the positive definite solutions of nonlinear matrix equation x a x δa q abstract in this paper we consider the positive definite solutions of nonlinear matrix equation x a x a q where 0 1 which appears for the first time in s m el sayed a c m ran on an iteration methods for solving a class of nonlinear matrix equations siam j matrix anal appl 23 2001 632 645 the necessary and sufficient conditions for the existence of a solution are derived an iterative algorithm for obtaining the positive definite solutions of the equation is discussed the error estimations are found

on the positive definite solutions of the matrix equations xs atx ta in abstract the two matrix equations x s a t x t a i n and x s a t x t a i n are studied based on the fixed point theory the existence of the symmetric positive definite solutions are proved sensitivity analysis of the maximal solution is presented some elegant estimates of the positive definite solutions are obtained three iterative methods for computing the positive solutions are proposed

tracking for fully actuated mechanical systems a geometric framework we present a general framework for the control of lagrangian systems with as many inputs as degrees of freedom relying on the geometry of mechanical systems on manifolds we propose a design algorithm for the tracking problem the notions of error function and transport map lead to a proper definition of configuration and velocity error these are the crucial ingredients in designing a proportional derivative feedback and feedforward controller the proposed approach includes as special cases a variety of results on control of manipulators pointing devices and autonomous vehicles our design provides particular insight into both aerospace and underwater applications where the configuration manifold is a lie group

on positive definite solutions of the nonlinear matrix equation x a x n a i in this paper some properties of a positive definite solution of the nonlinear matrix equation x a x na i are discussed effective iterative methods to obtain the smallest positive definite solution are established moreover the necessary and sufficient conditions for existence the smallest positive definite solution are obtained finally some numerical results are given

some properties of the nonlinear matrix equation xs a x ta q abstract in this paper the nonlinear matrix equation x s a x t a q is investigated where q is a hermitian positive definite matrix we consider three cases of this equation the case s t 0 the case s 1 0 t 1 and the case 0 s 1 t 1 in the case s t 0 we derive necessary conditions and sufficient condition for the existence of hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions as compared to earlier works on these topics the results we present here are more general and the analysis here is much simpler in the cases s 1 0 t 1 and 0 s 1 t 1 necessary conditions for the existence of a hermitian positive definite solution is given which is sharper than that of cai and chen 2010 9

some inequalities for the nonlinear matrix equation x s a x t a q trace determinant and eigenvalue the nonlinear matrix equation x s a x ta q is investigated where q is a hermitian positive definite matrix we present the upper bound of det aa 1 n when the nonlinear matrix equation x s a x ta q has hermitian positive definite solution the bound of det aa 1 n improves the corresponding result in duan and liao 2008 6 theorem 3 3 and yin et al 2009 17 theorem 2 1 we also get the bounds of detx 1 n and trx for the existence of a hermitian positive definite solution which generalize and improve the corresponding conclusions of zhao 2013 22 we obtain some bounds for the eigenvalues of the hermitian positive definite solution as compared with earlier works on these topics the results we present here are more general and the analysis here is much simpler finally we derive tight bounds about partial sum and partial product about the eigenvalues of the solution x for the nonlinear matrix equation x s a x ta q these present improvements and completions for the existing bounds

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

stabilization of a 3d axially symmetric pendulum stabilizing controllers are developed for a 3d pendulum assuming that the pendulum has a single axis of symmetry and that the center of mass lies on the axis of symmetry this assumption allows development of a reduced model that forms the basis for controller design and global closed loop analysis this reduced model is parameterized by the constant angular velocity component of the 3d pendulum about its axis of symmetry several different controllers are proposed controllers based on angular velocity feedback only asymptotically stabilize the hanging equilibrium then controllers are introduced based on angular velocity and reduced attitude feedback that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium these problems can be viewed as stabilization of a lagrange top finally if the angular velocity about the axis of symmetry is assumed to be zero controllers are introduced based on angular velocity and reduced attitude feedback that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium this problem can be viewed as stabilization of a spherical pendulum

some investigation on hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the hermitian positive definite solutions of the matrix equation x s a x t a q are considered where q is an hermitian positive definite matrix s and t are positive integers necessary and sufficient conditions for the existence of an hermitian positive definite solution are derived a sufficient condition for the equation to have only two different hermitian positive definite solutions and the formulas for these solutions are obtained in particular the equation with the case aq 1 2 q 1 2 a is discussed a necessary condition for the existence of an hermitian positive definite solution and some new properties of the hermitian positive definite solutions are given which generalize the existing related results

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

on the hermitian positive definite solutions of nonlinear matrix equation xs a x ta q abstract nonlinear matrix equation xs a x ta q where a q are n n complex matrices with q hermitian positive definite has widely applied background in this paper we consider the hermitian positive definite solutions of this matrix equation with two cases s 1 0

global stabilization of spherical orientation by synergistic hybrid feedback with application to reduced attitude tracking for rigid bodies in this paper we propose a hybrid feedback based on a synergistic potential function that achieves global asymptotic stabilization of a desired orientation on the n sphere with a nominal robustness to measurement disturbances a task that is not possible by classical feedback be it smooth nonsmooth periodic or any combination thereof due to the topological structure of the sphere we extend this basic result to a tracking controller for the reduced attitude or pointing direction of a rigid body and provide a method to remove jumps in the controlled torque by backstepping the proposed hybrid feedback is compared with a similar smooth feedback in simulation where it is illustrated that the hybrid feedback overcomes performance limitations inherent to the smooth feedback we provide two examples of a synergistic potential function one defined on a general sphere and the other on the unit circle

on the existence of hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the existence of hermitian positive definite solutions of the general nonlinear matrix equation x s a x t a q is studied systematically and deeply a new estimate of hermitian positive definite solutions is derived based on a fixed point theorem some new sufficient conditions and new necessary conditions for the existence of hermitian positive definite solutions are obtained in the end a necessary and sufficient condition for the existence of a hermitian positive definite solution is proved

lifts of automorphisms of vertex operator algebras in simple current extensions in this article we study isomorphisms between simple current extensions of a simple voa for example we classify the isomorphism classes of simple current extensions of the voas v sqrt2e 8 and v lambda 16 where 16 is the barnes wall lattice of rank 16 moreover we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension in particular we regard the moonshine module v natural as simple current extensions of five subvoas v l for 2 elementary totally even lattices l and describe corresponding five normalizers of elementary abelian 2 group in the automorphism group of v natural in terms of v l by using this description we show that three of them form a monster amalgam

on hermitian positive definite solutions of matrix equation x a x 2a i abstract the hermitian positive definite solutions of the matrix equation x a x 2 a i are studied a necessary and sufficient condition for existence of solutions is given in case a is normal the basic fixed point iterations for the equation in case a is nonnormal with a 2 3 3 are discussed in some detail some of ivanov s hasanov s and minchev s results in linear algebra appl 326 2001 27 are improved