Class Of Linear Evolution Equations Research Articles

On Analytical in a Sector Resolving Families of Operators for Strongly Degenerate Evolution Equations of Higher and Fractional Orders

In this paper, we study a class of linear evolution equations of fractional order that are degenerate on the kernel of the operator under the sign of the derivative and on its relatively generalized eigenvectors. We prove that in the case considered, in contrast to the case of first-order degenerate equations and equations of fractional order with weak degeneration (i.e., degeneration only on the kernel of the operator under the sign of the derivative), the family of analytical in a sector operators does not vanish on relative generalized eigenspaces of the operator under the sign of the derivative, has a singularity at zero, and hence does not determine any solution of a strongly degenerate equation of fractional order. For the case of a strongly degenerate equation of integer order this fact does not hold, but the behavior of the family of resolving operators at zero cannot be examined by ordinary method.

Controllability of block diagonal systems and applications

In this paper we present necessary a sufficient conditions for the exact and approximate controllability of Block-Diagonal Control Systems representing a broad class of linear evolution equations in Hilbert spaces. We apply these results to the controlled nD wave equation, heat equation and thermoelastic plate equation.

A time reversal based algorithm for solving initial data inverse problems

We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schrödinger and the Maxwell equations in a bounded domain $\Omega$. We assume that the only available information is a distributed observation (i.e. partial observation of the solution on a sub-domain $\omega$ during a finite time interval $(0,\tau)$). Under some quite natural assumptions (essentially : the exact observability of the system for some time $\tau_{obs}>0$, $\tau\ge \tau_{obs}$ and the existence of a time-reversal operator for the problem), an iterative algorithm based on a Neumann series expansion is proposed. Numerical examples are presented to show the efficiency of the method.

Initial-boundary value problems for linear PDEs with variable coefficients

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

Canonical formalism and quantization for a class of classical fields with application to radiative atomic decay in dielectrics

For the description of spectral and radiative decay properties of atoms or molecules, placed in a photonic material, the electromagnetic field in the material must be quantized and active research is taking place in this area. Here a unified account is given of such quantization procedures. Led by the Maxwell example, we consider the canonical formalism and its quantization for a class of linear evolution equations ${\ensuremath{\partial}}_{t}F=NF\ensuremath{-}G$, obeying a conservation law for $G=0$. If $N$ has a nonempty null space (zero is an eigenvalue with associated nonpropagating solutions), an abstract form of the gauge concept makes its appearance and generalizations of the familiar Coulomb and Lorentz gauges are obtained. A canonical formalism is set up and quantized. The application to spatially inhomogeneous nonconducting electrodynamical systems is immediate, including the interaction with matter. Next atomic decay in a medium is considered, in particular in the presence of band gaps. For a simple two-level model with transition frequency in the gap, single-photon decay is inhibited and also a different stable eigenvalue of the combined system is found. An open problem in connection with random dielectrics, showing Anderson localization, is discussed. Finally, a mass renormalization, by means of a Kramers transformation, is presented. In general, the renormalized mass is no longer a scalar quantity.

Kolmogorov equation for solutions of Cauchy problems for a class of linear evolution equations

$S Here W(t) is a multidimensional Wiener process defined on .{fL St p} with mutually independent components w(t), wk(t), Wkj(t); D1(t), D2(t) are second-order differential operators with nonrandom coefficients a(t), ak(t), akj(t ) and ~(t), ~k(t), ~kj(t) belonging to the spaces Ll([s, T]) and L2([s, t]), respectively, -. OxhOxj n 0 n Oz O.~(t)dW(t) = o~(t)dw(t)+~ ~. (t) -~ dw.(t)+ + ~, =~j(t) dw.j (t). k-~'--I k-~l j~-k

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.

Abstract singular evolution equations of “hyperbolic” type

This paper is concerned with the study of a large class of linear abstract evolution equations with singular coefficients under Kato's hyperbolicity assumptions. Existence and uniqueness results for strong and strict solutions are obtained in suitable weighted spaces.

Boundary Control Problems with Convex Cost Criterion

A class of boundary-distributed linear control systems in Banach spaces is studied. A maximum principle for a convex control problem associated with such systems is obtained.

On a class of linear evolution equations of ``hyperbolic'' type in reflexive Banach spaces

On a class of linear evolution equations of ``hyperbolic'' type in reflexive Banach spaces