When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all t≥t0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n).

Interpolation on the special orthogonal group with high-dimensional Kuramoto model

The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Approximating Correlation Matrices Using Stochastic Lie Group Methods

Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.

The distinction problems for Sp4 and SO3,3

This paper studies the Prasad conjecture for the special orthogonal group \mathrm{SO}_{3,3}. Then we use the local theta correspondence between \mathrm{Sp}_{4} and \mathrm{O}(V) to study the \mathrm{Sp}_{4}-distinction problems over a quadratic field extension E/F and \dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of \mathrm{Sp}_{4}(E).

A Novel Coarse Alignment Method for SINS Using Special Orthogonal Group Optimal Estimation

Aimed at the alignment problem of strapdown inertial navigation system (SINS) on the swing base, a novel coarse alignment method using special orthogonal group optimal estimation is proposed. There are two main contributions in this paper. First, based on the Lie group differential equation, the rotation matrix is updated directly by using error Lie algebra, which avoids the non-convexity of traditional methods and the need for non-collinear vector observation. Second is that a novel optimal estimation method is developed by using the exact error Lie algebra, which is calculated based on the physical definition of Lie algebra, as the innovation term to compensate the initial special orthogonal group in the estimation process. The asymptotic convergence of the proposed optimal estimation method is proved by Lyapunov's second law. The simulation and experimental results demonstrate that the proposed method exhibits better performance than existing methods in alignment accuracy and time, which can achieve the self-alignment of SINS on the swing base.