Nonlinear Volterra Operators Research Articles

Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators

This paper deals with direct and inverse source problems for parabolic or byperbolic PDEs containing nonlinear Volterra operators (including the variable order time-fractional derivatives). We show the well posedness (existence and uniqueness of a solution) of various settings. The proof-technique is based on the contraction mapping theorem and on a bootstrap argument, which ensures the existence of a global in time solution.

Transformational invariance in compact process modeling

Background: Modern one-digit technological nodes demand strict reproduction of the optical proximity corrections for repeatable congruent patterns. To ensure this property, the optical and process simulations must be invariant to the geometrical transformations of the translation, rotation, and reflection. Simulators must support invariance both in theory, mathematically, and in practice, numerically. The invariance of compact modeling operators has never been scrutinized before. Aim: We aim to examine manner and conditions under which optical simulations preserve or violate intrinsic invariances of exact imaging. We analyze invariances of Volterra operators, which are widely used in compact process modeling. Our goal is to determine necessary and sufficient conditions under which such operators become fully invariant Approach: We use theoretical analysis to deduce full invariance conditions and numerical simulations to illustrate the results. Results: The linear fully invariant operators are convolutions with rotationally symmetrical kernels. The fully invariant quadratic operators have special functional form with two radial and one polar argument and are not necessarily rotationally symmetrical. We deduced invariance conditions for the kernels of high-order Volterra operators. Conclusions: We suggest to use fully invariant nonlinear Volterra operators as atomic construction blocks in machine learning and neural networks for compact process modeling.

Глобальная экспоненциальная устойчивость для дифференциальных систем с нелинейными задержками

Березанский Леонид, отделение математики, Университет имени Бен-Гуриона, Беэр-Шева, Израиль; brznsky@cs.bgu.ac.il. Leonid Berezansky, brznsky@cs.bgu.ac.il Ben-Gurion University of the Negev, Beer-Sheva, Israel

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time-dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time-discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time-discrete approximation scheme in space. Finally, full-discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015

Existence of heteroclinic orbits in simplified longitudinal flight dynamics

The existence of an infinite chain of heteroclinic orbits connecting saddle-nodes is proved, in the case of a simplified system of differential equations governing the longitudinal flight of an aircraft, using fixed point theory for nonlinear Volterra operators with convolution kernels. Numerical simulations are given and the importance of the obtained results is discussed from the point of view of the appearance of oscillatory behavior in longitudinal flight.

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T n of increasing dimensions n + 1 and with a domain shape in the form of a “hyper-pyramid”, 0 ≤ ξ n ≤ ξ n − 1 ⋯ ≤ ξ 1 ≤ x ≤ 1 . We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L 2 and H 1 local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.

Optical implementation of second-order nonlinear Volterra operators with use of triple correlation.

Following the Volterra theorem, every nonlinear operator can be implemented with a sum of integrals applied over the input. The second-order Volterra operator that describes many useful systems can be related to a single integral that is a projection of an operation called the triple correlation. This operation may be easily implemented optically and thus be incorporated into fast real-time nonlinear control systems. We present a theoretical investigation of the relation existing between the Volterra operators and the triple correlation as well as an experimental demonstration that validates the theory.

Complete Continuity of Some Nonlinear Volterra Operator in Banach Spaces

We consider continuity and compactness of the particular Volterra operator Hx(t) = \int ^t_{t_0} K(r)x(r)dr , where K(t) is a nonlinear continuous and compact or a-Lipschitz operator in some Banach space.

Sensitivity and robust stability of nonlinear input-output systems consisting of finitely many blocks

We construct a general model of a nonlinear input-output system consisting of finitely many blocks whose variables are in extended spaces. This model encompasses all configurations used in control which have several feedback and feedforward loops. We show that such a system is essentially the traditional MI-MO feedback system. Moreover, assuming that one block of the system is a plant affected by perturbations, we derive conditions guaranteeing linear insensitivity and/or robust stability of the whole system. In particular, we consider systems whose variables are continuous, vector-valued functions on [0, ?), and whose blocks are described by nonlinear Volterra operators. The applications of the results are illustrated by several examples of feedback-feedforward systems.

A note on nonlinear Volterra operators

In this paper we discuss a class of nonlinear Volterra operators and show that they are globally invertible.

Solutions of multidimensional nonlinear volterra operator equations

Solutions of multidimensional nonlinear volterra operator equations

On the theory of solutions of nonlinear Volterra operator equations on a half-axis

On the theory of solutions of nonlinear Volterra operator equations on a half-axis