27 research outputs found
Differential inequalities and the matrix Riccati equation
The matrix Riccati equation has attracted attention recently because of its occurrence in a number of different situations. Its solutions determine solutions of the optimal linear regulator problem; the existence of solutions on an interval is related to disconjugacy of a linear Hamiltonian system on an interval has demonstrated an equivalence between solutions of matrix Riccati equations and Fredholm resolvents. Most recently, Fair has written about continued fraction solutions of a Riccati equation in a Banach algebra
A New Representation for Volterra Factors and the Fredholm Resolvent
Generalizations of the Chandrasekhar-Ambartsumian X-Y functions of radiative transfer are used to give a new representation of the Bellman-Krein formula for the Fredholm resolvent, as well as to represent the Volterra factors of the Gohberg-Krein factorization theory for Fredholm integral operators. It is shown that the new formulas have direct connections to matrix Riccati equations and may be used to advantage in various imbedding procedures used to calculate the Fredholm resolvent
First-order asymptotic perturbation theory for extensions of symmetric operators
This work offers a new prospective on asymptotic perturbation theory for
varying self-adjoint extensions of symmetric operators. Employing symplectic
formulation of self-adjointness we obtain a new version of Krein formula for
resolvent difference which facilitates asymptotic analysis of resolvent
operators via first order expansion for the family of Lagrangian planes
associated with perturbed operators. Specifically, we derive a Riccati-type
differential equation and the first order asymptotic expansion for resolvents
of self-adjoint extensions determined by smooth one-parameter families of
Lagrangian planes. This asymptotic perturbation theory yields a symplectic
version of the abstract Kato selection theorem and Hadamard-Rellich-type
variational formula for slopes of multiple eigenvalue curves bifurcating from
an eigenvalue of the unperturbed operator. The latter, in turn, gives a general
infinitesimal version of the celebrated formula equating the spectral flow of a
path of self-adjoint extensions and the Maslov index of the corresponding path
of Lagrangian planes. Applications are given to quantum graphs, periodic
Kronig-Penney model, elliptic second order partial differential operators with
Robin boundary conditions, and physically relevant heat equations with thermal
conductivity
Algebraic aspects of spectral theory
We describe some aspects of spectral theory that involve algebraic
considerations but need no analysis. Some of the important applications of the
results are to the algebra of matrices with entries that are
polynomials or more general analytic functions
Recursive mass matrix factorization and inversion: An operator approach to open- and closed-chain multibody dynamics
This report advances a linear operator approach for analyzing the dynamics of systems of joint-connected rigid bodies.It is established that the mass matrix M for such a system can be factored as M=(I+H phi L)D(I+H phi L) sup T. This yields an immediate inversion M sup -1=(I-H psi L) sup T D sup -1 (I-H psi L), where H and phi are given by known link geometric parameters, and L, psi and D are obtained recursively by a spatial discrete-step Kalman filter and by the corresponding Riccati equation associated with this filter. The factors (I+H phi L) and (I-H psi L) are lower triangular matrices which are inverses of each other, and D is a diagonal matrix. This factorization and inversion of the mass matrix leads to recursive algortihms for forward dynamics based on spatially recursive filtering and smoothing. The primary motivation for advancing the operator approach is to provide a better means to formulate, analyze and understand spatial recursions in multibody dynamics. This is achieved because the linear operator notation allows manipulation of the equations of motion using a very high-level analytical framework (a spatial operator algebra) that is easy to understand and use. Detailed lower-level recursive algorithms can readily be obtained for inspection from the expressions involving spatial operators. The report consists of two main sections. In Part 1, the problem of serial chain manipulators is analyzed and solved. Extensions to a closed-chain system formed by multiple manipulators moving a common task object are contained in Part 2. To retain ease of exposition in the report, only these two types of multibody systems are considered. However, the same methods can be easily applied to arbitrary multibody systems formed by a collection of joint-connected regid bodies