We propose a technique for synthesizing switching guards for hybrid systems to satisfy a given state-based safety constraint. Using techniques from sum of squares (SOS) optimization, we design guards defined by semialgebraic sets that trigger mode switches, and we guarantee that the synthesized switching policy does not allow Zeno executions. We demonstrate our approach on an example of switche...

This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that 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 KKT ideal if f ≥ 0 on K.

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear nonlinear systems theory. However, the scale of problems that can be addressed reliably efficiently is still limited. In this paper, we introduce a new notion block factor-width-two matrices build hierarchy inner outer approximations cone positive semidefinite (PSD) matrices. T...

In 1888, Hilbert described how to find real polynomials which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert’s construction and present many such polynomials. 1. History and Overview A real polynomial f(x1, . . . , xn) is psd or positive if f...

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for polynomial optimization problems, where sparsity includes correlative and term sparsity. We compare strengths moment-HSOS with real moment-SOS on either randomly generated problems or AC optimal power flow problem. The results numerical experiments show that provides a trade-off betw...

We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing real-time certificates of collision ...

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite di...

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. propose data-driven approach to stabilize systems when only sample trajectories dynamics accessible. Our method is built on density-function-based stability certificate that dual Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lea...

We present a general formulation for estimation of the region of attraction (ROA) for nonlinear systems with parametric uncertainties using a combination of the polynomial chaos expansion (PCE) theorem and the sum of squares (SOS) method. The uncertain parameters in the nonlinear system are treated as random variables with a probability distribution. First, the decomposition of the uncertain no...

For an n-variate order-d tensor A, define Amax := sup‖x‖2=1〈A, x ⊗d〉 to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d. ±1 entries, Amax . √ n · d · log d w.h.p. We study the problem of efficiently certifying upper bounds on Amax via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: When A is a rand...