Given an ideal I and a polynomial f the Ideal Membership Problem (IMP) is to test if ϵ . This problem fundamental algorithmic with important applications notoriously intractable. We study complexity of IMP for combinatorial ideals that arise from constrained problems over Boolean domain. As our main result, we identify borderline tractability. By using Gröbner bases techniques, extend Schaefer’...

This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using frequency-distributed fractional integrator model. There are two types frequency and temporal stabilization methods that global semi-global conditions attain sum-of-squares (SOS) method. Substantiation asymptotical complicated systems. According to recent studies on nonlinear systems, t...

Hybrid dynamic models describe hierarchical processes, which evolve according to different sets of lower level dynamic components depending on the upper level logical/discrete mode that characterizes the system, at any given point in time. In general hybrid systems consist of a finite set of modes of operation described in continuous time with differential equations. Continuous modes of operati...

This paper studies the representations of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its critical ideal. Under the assumption that the semi-algebraic set K is regular and f satisfies the boundary Hessian conditions (BHC) at each zero of f in K. We show that f can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if f ≥ 0 on K. P...

This paper provides an exact linear matrix inequality (LMI) condition for robust asymptotic stability of uncertain systems depending polynomially on a scalar parameter in both continuous-time and discrete-time cases. Specifically, this condition exploits sum of squares (SOS) techniques and is based on the construction of polynomials of known degree that detect the presence of eigenvalues on the...

—this paper presents a new forum an integral sliding mode manifold named as a general forum, and then proposes an algorithmic approach based on sum of squares (SOS) programming, to design a generalized integral sliding mode manifold and controller for nonlinear systems with both matched and unmatched uncertainty. The method gives sufficient conditions to obtain a solution for designing the cont...

In this paper, we propose a computational scheme of solving the output feedback H∞ control problem for a class of nonlinear systems with polynomial vector field. The output feedback control design problem will be decomposed into a state feedback and an output estimation problems. Resorting to higher order Lyapunov functions, two Hamilton-JacobianIsaacs (HJI) inequalities are first formulated as...

Our interest lies in solving large-scale unconstrained SOS (sum of squares) polynomial optimization problems. Because interior-point methods for solving these problems are severely limited by the large-scale, we are motivated to explore efficient implementations of an accelerated first-order method to solve this class of problems. By exploiting special structural properties of this problem clas...

This blog post is an introduction to the “Sum of Squares” (SOS) algorithm from my biased perspective. This post is rather long I apologize. If you’d rather ”see the movie”, I’ll be giving a TCS+ seminar about this topic on Wednesday, February 26th 1pm EST. If you prefer the live version, I’ll be giving a summer course in Stockholm from June 30 till July 4th. I’ve been told that there are worse ...