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We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
We prove the existence, uniqueness, regularity and smooth dependence
of the weak solution on the initial data for a certain class of semilinear first order
dissipative hyperbolic systems with spacially discontinuous coefficients. Such
kind of hyperbolic problems have succesfully been used to describe the dynamics
of distributed feedback multisection semiconductor lasers in recent years. We
show that in a suitable function space of continuous functions the weak solutions
generate a smooth semiflow.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.
For a system of globally pulse-coupled phase-oscillators,
we derive conditions for stability of the completely synchronous
state and all possible two-cluster states and explain
how the different states are naturally connected via
bifurcations. The coupling is modeled using the phaseresponse-
curve (PRC), which measures the sensitivity of
each oscillator’s phase to perturbations. For large systems
with a PRC, which turns to zero at the spiking threshold,
we are able to find the parameter regions where multiple
stable two-cluster states coexist and illustrate this by an
example. In addition, we explain how a locally unstable
one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon
of intermittent, asymptotic synchronization with abating
beats away from the perfect synchrony.
We use the traveling wave model for simulating and analyzing
nonlinear dynamics of complex semiconductor ring laser devices.
This modeling allows to consider temporal-spatial distributions
of the coun\-ter-pro\-pa\-ga\-ting slowly varying optical fields
and the carriers, what can be important when studying
non-homogeneous ring cavities, propagation of short pulses or fast switching.
By performing numerical integration of the model
equations we observe several dynamic regimes as well as transitions
between them. The computation of ring cavity modes explains some
peculiarities of these regimes.
A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions.
Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some
problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are
sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schroedinger and the drift-diffusion equation
but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.
Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting group-velocity horizon can be exploited for adiabatic compression of the soliton down into the few-cycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum- well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of $p(x)$-Laplacian-type with discontinuous exponent $p$, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori $L^1$ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two diffculties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
We show that the spectrum of linear delay differential equations with
large delay splits into two different parts. One part, called the
strong spectrum, converges to isolated points when the delay parameter
tends to infinity. The other part, called the pseudocontinuous spectrum,
accumulates near criticality and converges after rescaling to a set
of spectral curves, called the asymptotic continuous spectrum. We
show that the spectral curves and strong spectral points provide a
complete description of the spectrum for sufficiently large delay
and can be comparatively easily calculated by approximating expressions.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The perturbation and ADAE index of a degenerated hyperbolic system modelling a heat exchanger
(2007)
The heat exchanger in a heat pump can be modelled by the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a~priori $L^1$ term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the $L^1$ right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.
The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations. We derive such a deflation procedure and apply this new technique to symmetric, Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils. We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples.
Classical stability properties of solutions
that are well-known for ordinary differential
equations (ODEs) are generalized to differential-algebraic equations (DAEs).
A new test equation is derived for the analysis of numerical methods applied
to DAEs with respect to the stability of the numerical approximations.
Morevover, a stabilization technique is developed to improve the stability of classical DAE integration methods. The stability regions for these stabilized discretization methods are determined and it is shown that they much better reproduce the stability properties known for the ODE case
than in the unstabilized form.
Movies that depict the stability regions for several methods are included for interactive use.
The Lang-Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues grows to infinity. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.
In this paper we discuss the stability and model order reduction of coupled linear
time-invariant systems. Sufficient conditions for a closed-loop system to be asymptotically stable are
given. We present a model reduction approach for coupled systems based on reducing the order of the
subsystems and coupling the reduced-order subsystems by the same interconnection matrices as for
the original model. Such an approach allows to obtain error bounds for the reduced-order closed-loop
system in terms of the errors in the reduced-order subsystems. Model reduction of coupled systems
with unstable subsystems is also considered. Numerical examples are given.
Chimera states are particular trajectories
in systems of phase oscillators with non-local coupling
that display a spatio-temporal pattern of coherent and incoherent motion.
We present here a detailed analysis
of the spectral properties for such trajectories.
First, we study numerically their Lyapunov spectrum
and its behavior for an increasing number of oscillators.
The spectra demonstrate the hyperchaotic nature of the chimera states
and show a correspondence of the Lyapunov dimension
with the number of incoherent oscillators.
Then, we pass to the thermodynamic limit equation
and present an analytic approach
to the spectrum of a corresponding linearized evolution operator.
We show that in this setting, the chimera state is neutrally stable
and that the continuous spectrum coincides with the limit
of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.
Three properties of matrices: the spark, the mutual incoherence and the restricted isometry property have recently been introduced in the context of compressed sensing. We study these properties for matrices that are Kronecker products and show how these properties relate to those of the factors. For the mutual incoherence we also
discuss results for sums of Kronecker products.
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely
representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to nonzero expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.
The purpose of this paper is the analysis of dynamic iteration methods for
the numerical integration of coupled systems of ODEs and DAEs.
We will investigate convergence of these methods and put special emphasis
on the {\sc Jacobi}- and {\sc Gauss-Seidel} methods. Furthermore, the
fundamental difference in the convergence behaviour of coupled ODEs and DAEs
is pointed out. This difference is used to explain why certain relaxation methods
for coupled DAEs may fail. Finally, a remedy to this undesirable
effect is proposed that makes use of a so-called {\em preconditioned dynamic
iteration} strategy. This regularization also allows significant reduction of
dynamic iteration steps.
Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field.
In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.
In the case of the equidistant discretization of the Airy differential equation (\discrete
Airy equation") the exact solution can be found explicitly. This fact is used
to derive a discrete transparent boundary condition (TBC) for a Schroedinger
equation with linear varying potential, which can be used in \parabolic equation"
simulations in (underwater) acoustics and for radar propagation in the troposphere.
We propose different strategies for the discrete TBC and show an efficient implementation.
Finally a stability proof for the resulting scheme is given. A numerical
example in the application to underwater acoustics shows the superiority of the new
discrete TBC.
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers
(parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
The classical singular value decomposition for a matrix $A\in\Cmn$ is a
canonical form for $A$ that also displays the eigenvalues
of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop
a corresponding decomposition for $A$ that provides the Jordan canonical forms
for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider
the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$
are invertible and either complex symmetric and complex skew-symmetric, and we
provide a canonical form under transformations of the form
$(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.
Im Zentrum der Arbeiten soll die Chaos- und Kohärenzkontrolle von Halbleiterlasern mit gegenseitiger optischer Kopplung stehen. Diese Fragestellung ist von erheblicher praktischer Relevanz, da Rauschen und chaotisches Verhalten generelle Probleme in der optischen Hochgeschwindigkeitskommunikation sind. Die Kontrolle von optischen Systemen mit komplexer Selbstorganisation stellt aber auch aus grundsätzlicher Sicht Neuland dar. Hierfür geeignete Konzepte sind bisher weder überzeugend theoretisch beschrieben noch experimentell umgesetzt.
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.
We study the dynamics of a ring of unidirectionally coupled autonomous
Duffing oscillators. Starting from a situation where the individual
oscillator without coupling has only trivial equilibrium dynamics,
the coupling induces complicated transitions to periodic, quasiperiodic,
chaotic, and hyperchaotic behavior. We study these transitions in
detail for small and large numbers of oscillators. Particular attention
is paid to the role of unstable periodic solutions for the appearance
of chaotic rotating waves, spatiotemporal structures and the Eckhaus
effect for a large number of oscillators. Our analytical and numerical
results are confirmed by a simple experiment based on the electronic
implementation of coupled Duffing oscillators.
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite element method (FEM) can be avoided by using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses $d$, where the accuracy is best possible independent of the conductivity or the frequency being small or large -- this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs which are accurate up to $O(d^2)$.
Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
In this paper we consider the rational interpolation problem consisting in finding a rational matrix-valued function that
interpolates a given set of parameters. We briefly describe two different numerical methods for solving this problem. These are
the vector fitting and the frequency domain subspace identification method. Several numerical examples are given that compare
the properties of these methods. Furthermore, we discuss the computation of a (minimal) state space realization of a rational
function. Model order reduction methods such as modal approximation and balanced truncation are also presented. These
methods can be used to compute a reduced-order approximation of the realized dynamical system.
In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for
linear differential-algebraic equations.
Furthermore, a perturbation and error analysis for these methods is presented. We
investigate how errors in the data and in the numerical integration affect the
accuracy of the approximate spectral intervals. Although we need to integrate
numerically some differential-algebraic systems on usually very long
time-intervals, under certain assumptions, it is shown that the error of the
computed spectral intervals can be controlled by the local error of numerical
integration and the error in solving the algebraic constraint.
Some numerical examples are presented to illustrate the theoretical results.
We propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Positivity and stability are preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
We propose a model reduction method for positive systems that ensures the positivity of the reduced model. For both, continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Stability is preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations
(2007)
We discuss the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations. We
show that there is a substantial difference in the behavior under these perturbations. We also discuss the perturbation of real eigenvalues of real
skew-Hamiltonian matrices under structured perturbations and use these results to analyze the properties of the URV method of computing the
eigenvalues of Hamiltonian matrices.
Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case
(2012)
We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
(2013)
We consider boundary value problems for semilinear hyperbolic systems of the type
$$
\partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n
$$
with smooth coefficient functions $a_j$
and $b_j$
such that
$b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$.
We state conditions for Hopf bifurcation, i.e.,
for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence
on $\la$
of time-periodic solutions bifurcating from the zero stationary solution. Furthermore,
we derive a formula which determines the bifurcation direction.
The proof is done by means of a Liapunov-Schmidt reduction procedure.
For this purpose, Fredholm properties of the linearized
system and implicit function
theorem techniques are used.
There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs:
First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters
is much more delicate. And second,
a sufficient amount of dissipativity is needed in the system, and a priori
it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
(2013)
This paper concerns linear first-order hyperbolic systems in one space dimension of the type
$$
\partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space. We state a kind of dissipativity condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem, i.e., either there is a nontrivial solution to the homogeneous problem (in this case the space of such solutions has finite dimension) or the nonhomogeneous problem is uniquely solvable for any right-hand side (in this case the solution depends continuously on the right-hand side). In particular, under those conditions no small denominator effects occur.
Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
Finally, in the case that all coefficients $a_j$ are $t$-independent, we show that the solutions are $C^\infty$-smooth if the data are $C^\infty$-smooth.