C. van der Mee

Optimal control of switched autonomous linear systems

The paper deals with the optimal control of switched piecewise linear autonomous systems, where the objective is that of minimizing a quadratic performance index over an infinite time horizon. We assume that the switching sequence and the corresponding jump matrix sequence is known, while the unknown switching times are the optimization parameters. The optimal control for this class of systems, assuming a switching sequence of finite length, takes the form of a homogeneous state feedback, i.e., it is possible to identify a homogeneous region of the state space such that an optimal switch should occur if and only if the present state belongs to this region. We show how such a region can be computed with a numerical procedure. As the number of allowed switches goes to infinity, we study the stability of the system and discuss some preliminary results related to the convergence of the state feedback law.

An eigenvalue criterion for matrices transforming Stokes parameters

Simple eigenvalue tests are given to ascertain that a given real 4×4 matrix transforms the four‐vector of Stokes parameters of a beam of light into the four‐vector of Stokes parameters of another beam of light, and to determine whether a given 4×4 matrix is a weighted sum of pure Mueller matrices. The latter result is derived for matrices satisfying a certain symmetry condition. To derive these results indefinite inner products are applied.

Testing scattering matrices: A compendium of recipes

Abstract Scattering matrices describe the transformation of the Stokes parameters of a beam of radiation upon scattering of that beam. The problems of testing scattering matrices for scattering by one particle and for single scattering by an assembly of particles are addressed. The treatment concerns arbitrary particles, orientations and scattering geometries. A synopsis of tests that appear to be the most useful ones from a practical point of view is presented. Special attention is given to matrices with uncertainties due, e.g., to experimental errors. In particular, it is shown how a matrix Emod can be constructed which is closest (in the sense of the Frobenius norm) to a given real 4 × 4 matrix E such that Emod is a proper scattering matrix of one particle or of an assembly of particles, respectively. Criteria for the rejection of E are also discussed. To illustrate the theoretical treatment a practical example is treated. Finally, it is shown that all results given for scattering matrices of one particle are applicable for all pure Mueller matrices, while all results for scattering matrices of assemblies of particles hold for sums of pure Mueller matrices.

Structure of matrices transforming Stokes parameters

The structure of matrices that represent a linear transformation of the Stokes parameters of a beam of light into the Stokes parameters of another beam of light is investigated by means of the so‐called Stokes criterion. This holds that the degree of polarization of a beam of light can never be changed into a number larger than unity. Several general properties are derived for matrices satisfying the Stokes criterion. These are used to establish conditions for the elements of such matrices. Conditions that are either necessary, or sufficient or both are presented. General 4×4 matrices are treated and a number of special cases is worked out analytically. Several applications are pointed out.

Optimal control of autonomous linear systems switched with a pre-assigned finite sequence

The paper deals with the optimal control of switched piecewise linear autonomous systems, where the objective is to minimize the performance index over an infinite time horizon. We assume that the switching sequence has a finite length and is pre-assigned, while the unknown switching times are the optimization parameters. We also assume that at each switch a jump in the state space may occur and that a cost may be associated to each switch. The optimal control for this class of systems takes the form of a state feedback, i.e., it is possible to identify a region of the state space such that an optimal switch should occur if and only if the present state belongs to this region. We show how such a region can be computed with a numerical procedure and show that, in the particular case in which the switching costs is null, the region is homogeneous.

Parameterized scattering matrices for small particles in planetary atmospheres.

Abstract Parameterized matrices are discussed that may be used as (single) scattering matrices for interpretations of the brightness and polarization of planetary atmospheres containing randomly oriented small particles. A number of guidelines are developed for the construction of such matrices. These guidelines are based on (i) physical conditions for the elements of a natural scattering matrix, some holding for arbitrary scattering angles and some for the exact forward and backward scattering directions only, as well as (ii) theorems for the asymptotic behavior of coefficients in expansions of the matrix elements in generalized spherical functions of the scattering angle. A set of parameterized matrices is introduced and assessed according to these guidelines. These particular parameterizations are especially useful for scattering by particles that are not large compared to the wavelength, particles in the Rayleigh–Gans domain and for a variety of irregularly shaped particles in the visible part of the spectrum. The use of parameterized matrices as scattering matrices is illustrated by deriving their elements as functions of the scattering angle from simulated measurements of the brightness and polarization of light reflected by plane–parallel atmospheres containing aggregated or spheroidal particles. In both cases, the scattering angle dependences of the original elements are retrieved in fair approximation.

The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

The inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schrodinger (NLS) equation with non-zero boundary values ql/r(t)≡Al/re−2iAl/r2t+iθl/r as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with Al ≠ Ar and θl ≠ θr. The direct problem is shown to be well-defined for NLS solutions q(x, t) such that q(x,t)−ql/r(t)∈L1,1(R∓) with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λl/r=k2+Al/r2, where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have ...

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

In this article various existence results for theLDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by theLDU-factorizations of finite sections are discussed. The generalization of the results to theLDU-factorization of multi-index Toeplitz matrices is outlined.

Fast computation of two-level circulant preconditioners

In this paper we present an algorithm for the construction of the superoptimal circulant preconditioner for a two-level Toeplitz linear system. The algorithm is fast, in the sense that it operates in FFT time. Numerical results are given to assess its performance when applied to the solution of two-level Toeplitz systems by the conjugate gradient method, compared with the Strang and optimal circulant preconditioners.

Well‐posedness of stationary and time‐dependent Spencer–Lewis equations modeling electron slowing down

The unique solvability of the time‐dependent and stationary Spencer–Lewis equations is established under natural assumptions on the solution and the data of the problem. The strategy used is the method of characteristics followed by perturbation and monotone approximation arguments. The evolution operator in the time‐dependent Spencer–Lewis equation is proved to generate a strongly continuous contraction semigroup.