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While abstract interpretation is not theoretically restricted to specific kinds of properties, it is, in practice, mainly developed to compute linear over-approximations of reachable sets, aka. the collecting semantics of the program. The…
In this paper, we present a novel approach to determine the stability of switched linear and nonlinear systems using Sum of Squares optimisation. Particularly, we use Sum of Squares optimisation to search for a Lyapunov function that…
This study covers an analytical approach to calculate positively invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an…
Sum of Squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems but several questions remain unanswered. In this paper, we show that exponential stability of a polynomial vector…
This article presents a novel numerically tractable technique for synthesizing Lyapunov functions for equilibria of nonlinear vector fields. In broad strokes, corresponding to an isolated equilibrium point of a given vector field, a…
Stochastic gradient methods with momentum are widely used in applications and at the core of optimization subroutines in many popular machine learning libraries. However, their sample complexities have not been obtained for problems beyond…
We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not…
The quasipotential function allows for comprehension and prediction of the escape mechanisms from metastable states in nonlinear dynamical systems. This function acts as a natural extension of the potential function for non-gradient systems…
We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which…
This paper presents a method to approximately solve stochastic optimal control problems in which the cost function and the system dynamics are polynomial. For stochastic systems with polynomial dynamics, the moments of the state can be…
We describe a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our…
For the class of nonlinear input-affine systems with polynomial dynamics, we consider the problem of designing an input-to-state stabilizing controller with respect to typical exogenous signals in a feedback control system, such as actuator…
Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics,…
In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the…
We develop a landscape-flux framework to investigate observed frequency distributions of vegetation and the stability of these ecological systems under fluctuations. The frequency distributions can characterize the population-potential…
The kinetic Langevin dynamics finds diverse applications in various disciplines such as molecular dynamics and Hamiltonian Monte Carlo sampling. In this paper, a novel splitting scalar auxiliary variable (SSAV) scheme is proposed for the…
The push-sum algorithm is probably the most important distributed averaging approach over directed graphs, which has been applied to various problems including distributed optimization. This paper establishes the explicit absolute…
When a system has more than one stable state, how can the stability of these states be compared? This deceptively simple question has important consequences for ecosystems, because systems with alternative stable states can undergo dramatic…
This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. First, we present a method which exactly certifies sufficient conditions for invariance…
Policy gradient methods are widely used in reinforcement learning. Yet, the nonconvexity of policy optimization poses significant challenges in understanding the global convergence of policy gradient methods. For a class of finite-horizon…