Sums of squares II: Matrix functions

This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are defined, stabilization of the system is straight-forward. The sum of squares (SOS) decomposition and semidefinite programming has also provided an efficient methodology for analysis of nonlinear systems. The computational method used in this paper relies on the SOS decomposition of multivariate polynomials. By using SOS, we construct a (some) Lyapunov function(s) for the hybrid system. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be practically unsolvable. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Some examples are given to demonstrate the capabilities of the proposed approach.

Although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems, several fundamental questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a Lyapunov function which is a sum of squares of polynomials. In particular, the main result states that if a system is exponentially stable on a bounded nonempty set, then there exists a sum of squares Lyapunov function which is exponentially decreasing on that bounded set. Furthermore, we derive a bound on the degree of this converse Lyapunov function as a function of the continuity and stability properties of the vector field. The proof is constructive and uses the Picard iteration. Our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity.

Generalised absolute stability and sum of squares

We present a procedure that gives us an SOS (sum of squares) decomposition of a given real polynomial in variables, if there exists such decomposition. For the case of real polynomials in non-commutative variables we extend this procedure to obtain a sum of hermitian squares SOHS) decomposition whenever there exists any. This extended procedure is the main scientific contribution of the paper.

Certain second-order partial differential operators, which are expressed as sums of squares of real-analytic vector fields in R 3 {\mathbb {R}^3} and which are well known to be C ∞ {C^\infty } hypoelliptic, fail to be analytic hypoelliptic.

In this talk we give a report on a paper where we consider a model sum of squares of planar complex vector fields, being close to Kohn's operator but with a point singularity. The characteristic variety of the operator is the same symplectic real analytic manifold as Kohn's. We show that this operator is hypoelliptic and Gevrey hypoelliptic provided certain conditions are satisfied. We show that in the Gevrey spaces below a certain index the operator is not hypoelliptic. Moreover there are cases in which the operator is not even hypoelliptic. This fact leads to some general negative statement on the hypoellipticity properties of sums of squares of complex vector fields, even when the complex H\ormander condition is satisfied.

For a triangular n-by-n matrix function with a fixed ordered set of the indices of its diagonal entries, an explicit description is obtained for all possible n-tuples of the partial indices.

A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.

In this paper, a perceptually weighted quadratic optimization problem is formulated for the extraction of fine details from a vector field. To achieve better edge-preserving performance, a weighted sum square error (SSE) is adopted in the proposed optimization problem. The weighted SSE is used as data term in the proposed quadratic optimization problem to refine the extraction strength according to local intensity variance. Experimental results show that the perceptually weighted optimization problem can minimize sharp edge information in the extracted details.

Stability analysis is one of the most important problems in analysis of hybrid dynamical systems. In this paper, a computational method of Lyapunov functions is proposed for stability analysis of hybrid automata that have set-valued vector fields. For this purpose, a formulation of matrix-valued sums of squares is provided and applied to derive an LMI/LME problem whose solution yields a Lyapunov function.

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.

We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hormander condition.

We adapt Levi's parametrix method to construct a local fundamental solutions for operators of the form sum_i=1^m X_i^2, \nwhere X_1,ldots, X_m are H"ormander \nvector fields of step 2 having non-smooth coefficients. We\nalso provide estimates of the fundamental solution and of its derivatives.

Finite-time stability (FTS) requires that the state of a system does not exceed a certain bound during a specified time interval for given bound on the initial state. The concept of FTS introduced exogenous inputs is called finite time boundedness (FTB). This paper gives necessary and sufficient conditions for FTB of linear time-varying continuous-time systems. The conditions are extensions of existing necessary and sufficient for FTS. The sufficient conditions for FTB are also given in terms of differential linear matrix inequalities, which can be relaxed by matrix sum of squares, univariate polynomials. Numerical examples show the solvability of the sufficient conditions.

In this note we present a notion of harmonic oscillator on the Heisenberg group H n which forms the natural analogue of the harmonic oscillator on ℝ n under a few reasonable assumptions: the harmonic oscillator on H n should be a negative sum of squares of operators related to the sub-Laplacian on H n , essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L 2 (H n ). This approach leads to a differential operator on H n which is determined by the (stratified) Dynin–Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues.