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Biswas A.,Center for Applicable Mathematics
Systems and Control Letters | Year: 2011

Risk sensitive control problem under near monotonicity condition is considered. We prove existence of a solution to the corresponding HamiltonJacobiBellman equation by an eigenvalue approach. Existence of an optimal control has also been proved. © 2010 Elsevier B.V. Source

Chandrashekar P.,Center for Applicable Mathematics
Communications in Computational Physics | Year: 2013

Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form {equation presented} are any consistent approximations to the pressure and the mass flux. This scheme thus leaves most terms in the numerical flux unspecified and various authors have used simple averaging. Here we enforce approximate or exact entropy consistency which leads to a unique choice of all the terms in the numerical fluxes. As a consequence novel entropy conservative flux that also preserves kinetic energy for the semi-discrete finite volume scheme has been proposed. These fluxes are centered and some dissipation has to be added if shocks are present or if the mesh is coarse. We construct scalar artificial dissipation terms which are kinetic energy stable and satisfy approximate/exact entropy condition. Secondly, we use entropy-variable based matrix dissipation flux which leads to kinetic energy and entropy stable schemes. These schemes are shown to be free of entropy violating solutions unlike the original Roe scheme. For hypersonic flows a blended scheme is proposed which gives carbuncle free solutions for blunt body flows. Numerical results for Euler and Navier-Stokes equations are presented to demonstrate the performance of the different schemes. © 2013 Global-Science Press. Source

Maity D.,Center for Applicable Mathematics
ESAIM - Control, Optimisation and Calculus of Variations | Year: 2015

In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around (ρ, 0, θ), with ρ > 0, θ > 0 is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state (ρ, υ, θ), with ρ > 0, υ > 0, θ > 0 is not null controllable by localized interior control or by boundary control for small time T. Next we consider two-dimensional compressible Navier-Stokes system for barotropic fluid linearized around a constant steady state (ρ, 0). We prove that this system is also not null controllable by localized interior control. © EDP Sciences, SMAI 2015. Source

Apte A.,Center for Applicable Mathematics | Jones C.K.R.T.,University of North Carolina at Chapel Hill
Nonlinear Processes in Geophysics | Year: 2013

The focus of this paper is on how two main manifestations of nonlinearity in low-dimensional systems-shear around a center fixed point (nonlinear center) and the differential divergence of trajectories passing by a saddle (nonlinear saddle)-strongly affect data assimilation. The impact is felt through their leading to non-Gaussian distribution functions. The major factors that control the strength of these effects is time between observations, and covariance of the prior relative to covariance of the observational noise. Both these factors-less frequent observations and larger prior covariance-allow the nonlinearity to take hold. To expose these nonlinear effects, we use the comparison between exact posterior distributions conditioned on observations and the ensemble Kalman filter (EnKF) approximation of these posteriors. We discuss the serious limitations of the EnKF in handling these effects. © 2013 Author(s). Source

Apte A.,Center for Applicable Mathematics
Physica D: Nonlinear Phenomena | Year: 2011

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE. © 2010 Elsevier B.V. All rights reserved. Source

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