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Descriptor systems present a general
mathematical framework for the modelling, simulation and control of complex dynamical systems arising in many areas of mechanical,
electrical and chemical engineering. This
paper presents a survey of the current theory of descriptor systems,concerning
solvability, stability, model reduction, controllability, observability and optimal control.
We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index.
In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case.
We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls.
Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.
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.
We present a new extension of the well-known
Perron-Frobenius theorem to regular matrix pairs $(E,A)$.
The new extension is based on projector chains and is motivated from
the solution of positive differential-algebraic systems or descriptor
systems. We present several examples where the new condition holds, whereas conditions
in previous literature are not satisfied.
Being one of the key tools in conformation dynamics, the identification of
meta-stable states of Markov chains has been subject to extensive research in
recent years, especially when the Markov chains represent energy states of biomolecules. Some previous work on this topic involved the computation
of the eigenvalue cluster close to one, as well as the corresponding
eigenvectors and the stationary probability distribution of the associated stochastic
matrix. Later, since the eigenvalue cluster algorithm turned out to be non-robust, an optimisation approach was developed. As a possible less costly alternative, we present an SVD approach to identifying
meta-stable states of a stochastic matrix, where we only need
the second largest singular vector. We outline some theoretical background
and discuss the advantages of this strategy. Some simulated and real
numerical examples illustrate the effectiveness of the proposed algorithm.
Lagrangian invariant subspaces for symplectic matrices play an important role in the numerical solution of discrete time, robust and optimal control problems. The sensitivity (perturbation) analysis of these subspaces, however, is a difficult problem, in particular, when the eigenvalues are on or close to some critical regions in the complex plane, such as the unit circle.
We present a detailed perturbation analysis for several different cases of real and complex symplectic matrices. We analyze stability and conditional stability
as well as the index of stability for these subspaces.
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.
We consider hybrid systems of differential-algebraic equations and present
a general framework for general nonlinear over- and underdetermined hybrid
systems that allows the
analysis of existence and uniqueness and the application of index reduction
methods for hybrid differential-algebraic systems.
A particular difficulty in the numerical simulation of hybrid systems is
(numerical) chattering, i.e., fast oscillations between modes of operations.
A regularization technique using sliding modes allows to regularize the
system behavior in the case of chattering.
Further, we show how chattering behavior during the numerical solution can
be prevented using sliding mode simulation. The advantage of the sliding mode
simulation is illustrated by numerical examples.
Canonical forms for matrix triples $(A,G,\hat G)$, where
$A$ is arbitrary rectangular and $G$, $\hat G$ are either real symmetric
or skew symmetric, or complex Hermitian or skew Hermitian, are derived.
These forms generalize classical product Schur forms as well as
singular value decompositions.
An new proof for the complex case is given, where there is no need to
distinguish whether $G$ and $\hat G$ are Hermitian or skew Hermitian.
This proof is independent from the results in Bolschakov/Reichstein 1995, where
a similar canonical form has been obtained for the complex case,
and it allows generalization to the real case. Here,
the three cases, i.e., that
$G$ and $\hat G$ are both symmetric, both skew symmetric or one each,
are treated separately.