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In this study, we propose new global stabilization approaches for a class of polynomial systems in both model-based and data-driven settings. The existing model-based approach guarantees global asymptotic stability of the closed-loop system…
S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a…
This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…
This paper studies robust solutions and semidefinite linear programming (SDP) relaxations of a class of convex polynomial programs in the face of data uncertainty. The class of convex programs, called robust SOS-convex programs, includes…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations, or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length,…
The stability number of a graph $G$, denoted as $\alpha(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lov\'asz's theta number and expanded through…
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…
We show that if a system of degree-$k$ polynomial constraints on~$n$ Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most~$s$ many monomials, then it also has one whose degree is of the order of the square…
High-order tensor methods that employ Taylor-based local models (of degree $p\ge 3$) within adaptive regularization frameworks have been recently proposed for both convex and nonconvex optimization problems. They have been shown to have…
We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of…
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled…
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…
We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar-Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if $f$ is a polynomial of degree at most $2 d_f$ nonnegative on…
The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…
This paper studies distributionally robust optimization (DRO) with polynomial robust constraints. We give a Moment-SOS relaxation approach to solve the DRO. This reduces to solving linear conic optimization with semidefinite constraints.…
Given a closed, convex cone $K\subseteq \mathbb{R}^n$, a multivariate polynomial $f\in\mathbb{C}[\mathbf{z}]$ is called $K$-stable if the imaginary parts of its roots are not contained in the relative interior of $K$. If $K$ is the…
We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete…
We study polynomial stability to the one-dimensional system in the linear isothermal theory of swelling porous elastic soils with an internal fractional damping. We establish an optimal decay result by frequency domain method
Hybrid systems exhibit phenomena which do not occur in systems with continuous vector fields. One such phenomenon - Zeno executions - is characterized by an infinite number of discrete events or transitions occurring over a finite interval…