Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets

We consider the problem of overbounding and underbounding both the backward and forward reachable set for a given polynomial vector field, nonlinear in both state and input, with a given semialgebriac set of initial conditions and with inputs constrained pointwise to lie in a semialgebraic set. Specifically, we represent the forward reachable set using the value function which gives the optimal cost to go of an optimal control problems and if smooth satisfies the Hamilton-Jacobi- Bellman PDE. We then show that there exist polynomial upper and lower bounds to this value function and furthermore, these polynomial sub-value and super-value functions provide provable upper and lower bounds to the forward reachable set. Finally, by minimizing the distance between these sub-value and super-value functions in the L1-norm, we are able to construct inner and outer bounds for the reachable set and show numerically on several examples that for relatively small degree, the Hausdorff distance between these bounds is negligible

Production of Millisecond Dips in Sco X-1 Count Rates by Dead Time Effects

Chang et al. (2006) reported millisecond duration dips in the X-ray intensity of Sco X-1 and attributed them to occultations of the source by small trans-Neptunian objects (TNOs). We have found multiple lines of evidence that these dips are not astronomical in origin, but rather the result of high-energy charged particle events in the RXTE PCA detectors. Our analysis of the RXTE data indicates that at most 10% of the observed dips in Sco X-1 could be due to occultations by TNOs, and, furthermore, we find no positive or supporting evidence for any of them being due to TNOs. We therefore believe that it is a mistake to conclude that any TNOs have been detected via occultation of Sco X-1.Comment: Submitted to ApJ; uses emulateapj.cls, 8 pages with 8 figure

Existence of Partially Quadratic Lyapunov Functions That Can Certify The Local Asymptotic Stability of Nonlinear Systems

This paper proposes a method for certifying the local asymptotic stability of a given nonlinear Ordinary Differential Equation (ODE) by using Sum-of-Squares (SOS) programming to search for a partially quadratic Lyapunov Function (LF). The proposed method is particularly well suited to the stability analysis of ODEs with high dimensional state spaces. This is due to the fact that partially quadratic LFs are parametrized by fewer decision variables when compared with general SOS LFs. The main contribution of this paper is using the Center Manifold Theorem to show that partially quadratic LFs that certify the local asymptotic stability of a given ODE exist under certain conditions

The Structural Genomics Consortium: successful organisational technology experiment or new institutional infrastructure for health research?

In a sector characterised by patenting, direct appropriations and returns from investment, the Structural Genomics Consortium (SGC) constitutes a radically different public-private and entirely open access approach to pre-competitive research. This paper discusses the significance of findings from the first independent review of the SGC. We argue that the SGC offers a shared knowledge resource for drug discovery which is distinctive from other types of knowledge production and, as such, provides a knowledge infrastructure for the wider scientific community. We distinguish three ways in which this infrastructure functions as a model for investing in, extracting value from, and generating knowledge for the field. Our analysis suggests there is a future for open science models such as the SGC in health research and innovation, but that such models raise a set of challenges over the role of different public and private institutional actors and the way in which value is extracted

Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems

This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems that yields a sequence of polynomials that converges to our proposed converse Lyapunov function uniformly from above in the L1 norm. We show that each member of the sequence of 1-sublevel sets of the polynomial solutions to our proposed sequence of SOS programming problems are certifiably contained inside the maximal region of attraction of the ODE, and moreover, we show that this sequence of sublevel sets converges to the maximal region of attraction of the ODE with respect to the volume metric. We provide numerical examples of estimations of the maximal region of attraction for the Van der Pol oscillator and a three dimensional servomechanism

Polynomial Approximation of Value Functions and Nonlinear Controller Design with Performance Bounds

For any suitable Optimal Control Problem (OCP) which satisfies the Principle of Optimality, there exists a value function, defined as the unique viscosity solution to a HJB PDE, and which can be used to design an optimal feedback controller for the given OCP. Solving the HJB analytically is rarely possible, and existing numerical approximation schemes largely rely on discretization - implying that the resulting approximate value functions may not have the useful property of being uniformly less than or equal to the true value function (ie be sub-value functions). Furthermore, controllers obtained from such schemes currently have no associated bound on performance. To address these issues, for a given OCP, we propose a sequence of Sum-Of-Squares (SOS) problems, each of which yields a polynomial sub-solution to the HJB PDE, and show that the resulting sequence of polynomial sub-solutions converges to the value function of the OCP in the $L^1$ norm. Furthermore, for each polynomial sub-solution in this sequence we define an associated sublevel set, and show that the resulting sequence of sublevel sets converges to the sub-level set of the value function of the OCP in the volume metric. Next, for any approximate value function, obtained from an SOS program or any other method (e.g. discretization), we construct an associated feedback controller, and show that sub-optimality of this controller as applied to the OCP is bounded by the distance between the approximate and true value function of the OCP in the $W^{1,\infty}$ (Sobolev) norm. This result implies approximation of value functions in the $W^{1,\infty}$ norm results in feedback controllers with performance that can be made arbitrarily close to optimality. Finally, we demonstrate numerically that by solving our proposed SOS problem we are able to accurately approximate value functions, design controllers and estimate reachable sets