Backstepping PDE Design: A Convex Optimization Approach

Abstract\u2014Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof- Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite delay

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned. Finally, we also prove the existence of a more general attractor for the problem known as D-pullback attractor.Conselho Nacional de Desenvolvimento Científico e TecnológicoMinisterio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de Andalucí

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system

Parametric amplification of optical phonons

Amplification of light through stimulated emission or nonlinear optical interactions has had a transformative impact on modern science and technology. The amplification of other bosonic excitations, like phonons in solids, is likely to open up new remarkable physical phenomena. Here, we report on an experimental demonstration of optical phonon amplification. A coherent mid-infrared optical field is used to drive large amplitude oscillations of the Si-C stretching mode in silicon carbide. Upon nonlinear phonon excitation, a second probe pulse experiences parametric optical gain at all wavelengths throughout the reststrahlen band, which reflects the amplification of optical-phonon fluctuations. Starting from first principle calculations, we show that the high-frequency dielectric permittivity and the phonon oscillator strength depend quadratically on the lattice coordinate. In the experimental conditions explored here, these oscillate then at twice the frequency of the optical field and provide a parametric drive for lattice fluctuations. Parametric gain in phononic four wave mixing is a generic mechanism that can be extended to all polar modes of solids, as a new means to control the kinetics of phase transitions, to amplify many body interactions or to control phonon-polariton waves