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Heterogeneous formation shape control with interagent bearing and distance constraints involves the design of a distributed control law that ensures the formation moves such that these interagent constraints are achieved and maintained. This paper looks at the design of a distributed control scheme to solve different formation shape control problems in an ambient two-dimensional space with bearing, distance and mixed bearing and distance constraints. The proposed control law allows the agents in the formation to move in any direction on a half-plane and guarantees that despite this freedom, the proposed shape control algorithm ensures convergence to a formation shape meeting the prescribed constraints. This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only constraints are typically maintained and where each agent's motion is typically restricted to follow a very particular path. A stability analysis is sketched, and a number of illustrative examples are also given.

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The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instance of the Bayes filter for continuous state-space models. Consequently, this filter has been studied extensively in the literature since the seminal 1961 paper of Kalman and Bucy. The purpose of this work is to review, re-prove and refine existing results concerning the dynamical properties of the KalmanBucy filter so far as they pertain to filter stability and convergence. The associated differential matrix Riccati equation is a focal point of this study with a number of bounds, convergence, and eigenvalue inequalities rigorously proven. New results are also given in the form of exponential and comparison inequalities for both the filter and the Riccati flow.

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This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.

The stability properties of matrix-valued Riccati diffusions are investigated. The matrixvalued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. In addition, this class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. The analysis developed here applies to filtering problems with unstable signals. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation.1/2This special case (κ = 0) defines, in some sense, a minimal prototype of a forward-in-time matrixvalued Riccati diffusion in the space of symmetric positive (semi-)definite matrices. We let φ ǫ t (Q) := Q t be the stochastic flow of the matrix diffusion equation (1.2). Whenever it exists, the inverse stochastic flow of (1.2) is denoted by φ −ǫ t (Q) := Q −1 t . For any 0 ≤ s ≤ t, we let E ǫ s,t (Q) be the transition semigroup associated with the flow of random matrices [A − φ ǫ t (Q)S], i.e. the solution of the forward and backward equationsfor any 0 ≤ s ≤ t, with E ǫ t,t (Q) = I. When s = 0 we write E ǫ t (Q) instead of E ǫ 0,t (Q). We write φ t (Q) and E s,t (Q) instead of φ 0 t (Q), and E 0 s,t (Q), to denote the flow of the deterministic matrix Riccati differential equation when ǫ = 0, and the exponential semigroup defined via φ t (Q). 2