The solution to the matrix equationAV + BW = EVJ + R

AbstractThis note considers the solution to the generalized Sylvester matrix equation AV + BW = EVJ + R, where A, B, E, and R are given matrices of appropriate dimensions, J is an arbitrary given Jordan matrix, while V and W are matrices to be determined. A general parametric solution for this equation is proposed, based on the Smith form reduction of the matrix [A − sE B]. The solution possesses a very simple and neat form, and does not require the eigenvalues of matrix J to be known. An example is presented to illustrate the proposed solution

Solving the generalized Sylvester matrix equation AV+BW=EVF via a Kronecker map

AbstractThis note considers the solution to the generalized Sylvester matrix equation AV+BW=EVF with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, some properties of the Sylvester sum are first proposed. By applying the Sylvester sum as tools, an explicit parametric solution to this matrix equation is established. The proposed solution is expressed by the Sylvester sum, and allows the matrix F to be undetermined

Toward Solution of Matrix Equation X=Af(X)B+C

This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation $X=Af(X) B+C$ with $f(X) =X^{\mathrm{T}},$ $f(X) =\bar{X}$ and $f(X) =X^{\mathrm{H}},$ where $X$ is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of $W=\mathcal{A}W\mathcal{B}+\mathcal{C}$ where the dimensions of the coefficient matrices $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are the same as those of the original equation. Closed-form solutions of equation $X=Af(X) B+C$ can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation $X=Af(X) B+C$. Necessary and sufficient conditions are established to guarantee the convergence of the iterations

Coupling of light from an optical fiber taper into silver nanowires

We report the coupling of photons from an optical fiber taper to surface plasmon modes of silver nanowires. The launch of propagating plasmons can be realized not only at ends of the nanowires, but also at the midsection. The degree of the coupling can be controlled by adjusting the light polarization. In addition, we present the coupling of light into multiple nanowires from a single optical fiber taper simultaneously. Our demonstration offers a novel method for optimizing plasmon coupling into nanoscale metallic waveguides and promotes the realization of highly integrated plasmonic devices.Comment: 5 pages, 4 figure

Path-Following Control of Wheeled Planetary Exploration Robots Moving on Deformable Rough Terrain

The control of planetary rovers, which are high performance mobile robots that move on deformable rough terrain, is a challenging problem. Taking lateral skid into account, this paper presents a rough terrain model and nonholonomic kinematics model for planetary rovers. An approach is proposed in which the reference path is generated according to the planned path by combining look-ahead distance and path updating distance on the basis of the carrot following method. A path-following strategy for wheeled planetary exploration robots incorporating slip compensation is designed. Simulation results of a four-wheeled robot on deformable rough terrain verify that it can be controlled to follow a planned path with good precision, despite the fact that the wheels will obviously skid and slip

A neighboring extremal solution for an optimal switched impulsive control problem

This paper presents a neighboring extremal solution for a class of optimal switched impulsive control problems with perturbations in the initial state, terminal condition and system's parameters. The sequence of mode's switching is pre-specified, and the decision variables, i.e. the switching times and parameters of the system involved, have inequality constraints. It is assumed that the active status of these constraints is unchanged with the perturbations. We derive this solution by expanding the necessary conditions for optimality to first-order and then solving the resulting multiple-point boundary-value problem by the backward sweep technique. Numerical simulations are presented to illustrate this solution method