Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Firefly algorithm for polynomial Bézier surface parameterization

A classical issue in many applied fields is to obtain an approximating surface to a given set of data points. This problem arises in Computer-Aided Design and Manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy

A New Z=0 Metagalactic Ultraviolet Background Limit

We present new integral-field spectroscopy in the outskirts of two nearby, edge-on, late-type galaxies to search for the H alpha emission that is expected from the exposure of their hydrogen gas to the metagalactic ultraviolet background (UVB). Despite the sensitivity of the VIRUS-P spectrograph on the McDonald 2.7 m telescope to low surface brightness emission and the large field of view, we do not detect H alpha to 5 sigma upper limits of 6.4 x 10(-19) erg s(-1) cm(-2) arcsec(-2) in UGC 7321 and of 25 x 10(-19) erg s(-1) cm(-2) arcsec(-2) in UGC 1281 in each of the hundreds of independent spatial elements (fibers). We fit gas distribution models from overlapping 21 cm data of HI, extrapolate one scale length beyond the HI data, and estimate predicted H alpha surface brightness maps. We analyze three types of limits from the data with stacks formed from increasingly large spatial regions and compare to the model predictions: (1) single fibers, (2) convolution of the fiber grid with a Gaussian, circular kernel (10('') full width at half-maximum), and (3) the co-added spectra from a few hundred fibers over the brightest model regions. None of these methods produce a significant detection (>5 sigma) with the most stringent constraints on the Hi photoionization rate of Gamma(z = 0) < 1.7 x 10(-14) s(-1) in UGC 7321 and Gamma(z = 0) < 14 x 10(-14) s(-1) in UGC 1281. The UGC 7321 limit is below previous measurement limits and also below current theoretical models. Restricting the analysis to the fibers bound by the HI data leads to a comparable limit; the limit is Gamma(z = 0) < 2.3 x 10(-14) s(-1) in UGC 7321. We discuss how a low Lyman limit escape fraction in z similar to 0 redshift star-forming galaxies might explain this lower than predicted UVB strength and the prospects of deeper data to make a direct detection.U.S. Government NAG W-2166National Science FoundationUT David BrutonTexas Norman Hackerman Advanced Research Program 003658-0295-2007Cynthia and George Mitchell FoundationMcDonald Observator

Firefly algorithm for explicit B-spline curve fitting to data points

ABSTRACT. This paper introduces a new method to compute the approximating explicit B-spline curve to a given set of noisy data points.The proposed method computes all parameters of the B-spline fitting curve of a given order.This requires to solve a difficult continuous, multimodal, and multivariate nonlinear least-squares optimization problem. In our approach, this optimization problem is solved by applying the firefly algorithm, a powerful metaheuristic nature-inspired algorithm well suited for optimization. The method has been applied to three illustrative real-world engineering examples from different fields. Our experimental results show that the presented method performs very well, being able to fit the data points with a high degree of accuracy. Furthermore, our scheme outperforms some popular previous approaches in terms of different fitting error criteria