Nonlinear Hypoellipticity of Infinite Type

0. Introduction. 1. General Calculus of Pseudodifferential Operators. 2. Model Classes of Degenerate Elliptic Differential Operators. 3. General Classes of Degenerate Elliptic Differential Operators. 4. Degenerate Elliptic Operators in Non--Power--Like Degeneration Case. 5. Lp-Theory for Degenerate Elliptic Operators. 6. Coerciveness of Degenerate Quadratic Forms. 7. Some Classes of Hypoelliptic Pseudodifferential Operators on a Closed Manifold. 8. Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary. 9. General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations. 10. Spectral Asymptotics of Degenerate Elliptic Operators. 11. Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics. A Brief Review of the Bibliography. Bibliography. Index of Notations. Index.

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $\sigma_2$-Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform $C^{0, 1}$ regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of $\sigma_2$-Yamabe problem is unique. A key ingredient is the convexity of Chang-Yang's $\cF$-functional along the (smooth) geodesic, in view of Gursky-Streets metric and a weighted Poincare inequality of B. Andrews on manifolds with positive Ricci curvature. In this paper we establish uniform $C^{1, 1}$ regularity of the Gursky-Streets' equation. As an application, we can establish strictly the geometric structure in terms of Gursky-Streets' metric, in particular the convexity of $\cF$-functional along $C^{1, 1}$ geodesic. This in particular gives a straightforward proof of the uniqueness of solutions of $\sigma_2$-Yamabe problem.

Asymptotics for Solution Branches Of Semilinear Elliptic Problems At Resonance

In this paper, a novel robust controller for a Static Synchronous Compensator (STATCOM) is presented to enhance the fault ride-through (FRT) capability of fixed speed induction generators (FSIGs), the most common type of generators that can be found in wind farms. The effects of STATCOM rating and wind farm integration on FRT capability of FSIGs are studied analytically using the power-voltage and torque-slip relationships as well as through simulations. The wind generator is a highly nonlinear system, which is modelled in this work as a linear part plus a nonlinear part, the nonlinear term being the Cauchy remainder term in the Taylor series expansion and of the equations used to model the wind farm. Bounds derived for this Cauchy remainder term are used to define an uncertain linear model for which a robust control design is performed. The controller resulting from this robust design provides an acceptable performance over a wide range of conditions needed to operate the wind farm during severe faults. The performance of the designed controller is demonstrated by large disturbance simulations on a test system.

This article examines the welfare impact of foreign direct investment (FDI) in a panel of 20 African countries over the period 2000–2013. We explore the multifactor and nonmonetary measures of welfare and the nonlinear effect of FDI on welfare. We used the Driscoll and Kraay standard errors and augmented mean group (AMG) estimator by Eberhardt and Teal (2010) to account for cross‐sectional dependency, endogeneity, and heterogeneity within panel units. The results indicate that although FDI is welfare enhancing, the nonlinear terms report mixed findings. When a multifactor indicator is employed, the increase in the nonlinear term is lower than the linear part. However, there is strong evidence that FDI is ultimately welfare enhancing when a nonmonetary indicator is employed. From an international business perspective, the findings have unlocked the welfare effects of international business on African host economies. International businesses through FDI can enhance welfare in Africa countries. However, the optimal efficacy of FDI‐welfare impact differs across the various dimensions of welfare. © 2017 Wiley Periodicals, Inc.

Variable step implementation of ETD methods for semilinear problems

In this paper, we suggest a proof method for the exact controllability of semilinear systems. Previously, it was applied in the context of the approximate controllability. In comparison to the traditional proof methods, the suggested method is easy because it does not require heavy estimations. Also, it is effective because it does not need some unnatural conditions, which are required for application of fixed‐point theorems. Using these advantages of the suggested method, we prove that a semilinear system in a Hilbert space, which satisfies general conditions for the existence of its solution, steers any initial state to any final state from the domain of the system operator if its linear part is exactly controllable at every noninitial time moment, the nonlinear term is bounded, and the norm of the inverse of the Gramian increases as time goes to zero not faster than the reciprocal function. The obtained result is demonstrated on examples.

SynopsisThe existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.

A methodology is proposed to design a robust Gain-Scheduled Model Predictive Control (MPC) strategy and to quantify the relative advantages of this controller versus a Linear MPC strategy. For the purpose of analysis and controller design, the process is represented by a nonlinear state-affine model identified from input−output data. This model can be split in linear and nonlinear terms where the linear part is used for controller design and the nonlinear part is accounted for as model uncertainty. Then, robust stability and robust performance tests are formulated based on linear matrix inequalities where the manipulated variables weight of the controllers is tuned to maximize performance. The uncertainty bounds used for the robustness tests are obtained in an iterative fashion by using the frequency response of the manipulated variable with respect to the feedback error. The control strategy performance is quantified by the ratio between the error norm and the disturbance norm. Finally, a case study involving a multiple-input−multiple-output bioreactor is presented. The study is able to predict for which range of operation the Gain-Scheduling MPC surpasses the performance of the Linear MPC.

Local existence and stability for some partial functional differential equations with infinite delay

In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.

Continuity properties of solutions to some degenerate elliptic equations

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space . To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.

Homogeneous turbulence in rotating shear flows is studied by means of pseudospectral direct numerical simulation and analytical spectral linear theory (SLT). The ratio of the Coriolis parameter to shear rate is varied over a wide range by changing the rotation strength, while a constant moderate shear rate is used to enable significant contributions to the nonlinear interscale energy transfer and to the nonlinear intercomponental redistribution terms. In the destabilized and neutral cases, in the sense of kinetic energy evolution, nonlinearity cannot saturate the growth of the largest scales. It permits the smallest scale to stabilize by a scale-by-scale quasibalance between the nonlinear energy transfer and the dissipation spectrum. In the stabilized cases, the role of rotation is mainly nonlinear, and interacting inertial waves can affect almost all scales as in purely rotating flows. In order to isolate the nonlinear effect of rotation, the two-dimensional manifold with vanishing spanwise wave number is revisited and both two-component spectra and single-point two-dimensional energy components exhibit an important effect of rotation, whereas the SLT as well as the purely two-dimensional nonlinear analysis are unaffected by rotation as stated by the Proudman theorem. The other two-dimensional manifold with vanishing streamwise wave number is analyzed with similar tools because it is essential for any shear flow. Finally, the spectral approach is used to disentangle, in an analytical way, the linear and nonlinear terms in the dynamical equations.