We propose a novel discontinuous mixed finite element formulation for the solution of second-order elliptic problems. Fully discontinuous piecewise polynomial finite element spaces are used for the trial and test functions. The discontinuous nature of the test functions at the element interfaces allows to introduce new boundary unknowns that, on the one hand enforce the weak continuity of the trial functions, and on the other avoid the need to define a priori algorithmic fluxes as in standard discontinuous Galerkin methods. Static condensation is performed at the element level, leading to a solution procedure based on the sole interface unknowns. The resulting family of discontinuous dual-primal mixed finite element methods is presented in the one and two-dimensional cases. In the one-dimensional case, we show the equivalence of the method with implicit Runge-Kutta schemes of the collocation type exhibiting optimal behavior. Numerical experiments in one and two dimensions demonstrate the order accuracy of the new method, confirming the results of the analysis.
Let us assume somebody in a truck manufacturing company has the idea to improve the driver’s comfort by damping the oscillations of the driver’s cabin. For this end he plans to introduce a new type of dampers governed by a sophisticated control law and a complicated mechanical coupling device. The best argument for convincing the decision makers in his company would be a demonstration of his idea. The classical “demonstration” of such an idea would require the construction of the new device and a lot of field tests. And this costs money and takes time and might be too expensive just for checking an idea.
In the last chapter we discussed the numerical treatment of explicit ordinary differential equations. Here, we will consider the more general case, implicit ordinary differential equations.
The equations of motion of multibody systems often contain discontinuities in the right hand side, in the solution or in its higher derivatives. In the sequel we summarize discontinuities in the function itself as well as in its higher derivatives under the expression “discontinuity”.
In this chapter we are seeking an equilibrium position of a multibody system. This is a position where the system does not move. The velocity and acceleration are zero.
In the preceding chapters we started from a mathematical description of a mechanical system and used it to predict its behavior.
Most ordinary differential equations (ODEs) cannot be solved analytically. Only for a restricted class of ODEs computing the analytical solution can be taken into consideration. But even in these cases it is often more efficient to apply numerical techniques to approximate the solution.
Until recently stochastic realization theory has been primarily addressed to modeling of random processes in the absence of exogenous signals (inputs). However stochastic realization of models with exogenous inputs is also of interest. In particular it is of interest for the new class of " subspace " type identification algorithms. These algorithms can be formulated as stochastic realization algorithms in an appropriate data Hilbert space [23, 24] (as it is well-known subspace methods have important advantages over the traditional parametric optimization approach to identification). We discuss here procedures for constructing minimal state-space models in presence of inputs, based on a generalization of the idea of Markovian splitting subspace which is central in stochastic realization theory for random processes. In particular we discuss a geometric procedure for constructing the minimal state-space (the predictor space) of a process with inputs which leads to an interesting identification algorithm.
In this paper we extend the theory of supervisory control of nondeterministic discrete-event systems, subject to nondeterministic specification, developed in [9]. We focus our attention on nonblocking and liveness considerations and develop algorithms for nonblocking-supervisor synthesis.
We discuss here some open issues concerning the problem of modeling signals with low multiplicity within the context of systems theory. In particular we focus on the issue of approximating a stochastic process with a low multiplicity one, and we propose some applications to the theory of interest rates models.
Pairs of linear transformations on a finite dimensional vector space are of great relevance in the analysis of two-dimensional (2D) systems evolutions. In this paper, special properties of matrix pairs, such as finite memory, separability, property L and property P, as well as their dynamical interpretations, are investigated. Practical criteria for testing property L and property P in a finite number of steps are also presented. The nonnegativity requirement on a matrix pair allows for much stronger characterizations of finite memory and separability, which in fact prove to be structural properties. Finally, the irreducibility and primitivity notions of positive matrix pairs are discussed, and connected with the dynamical behavior of the associated 2D systems.
In this paper, we discuss the use of differential invariants, and curvature driven flows for active vision. We concentrate on three problem areas: invariant flows, active contours, and L 1-based methods for optical flow and stereo. The solutions to these key problems will all be based on curvature based evolutions which are obtained in a completely natural manner from geometric and physical principles.
The problem of factoring a polynomial matrix B = B* into B(ξ) = H(ξ)*H(ξ), with H Hurwitz, is called the Hurwitz spectral factorization problem. We show that the Newton iteration, applied to the computation of H, converges.
A dynamical system may be called discontinuous if it is subject to abrupt changes in its dynamic characteristics. Such changes may be induced either externally (“time events”) or internally (“state events”). Here we discuss so-called linear complementarity systems. These are piecewise linear systems in which state events occur due to transitions from one branch of an ideal-diode-type characteristic to the other. We present a precise definition of the dynamics of such systems and give sufficient conditions for well-posedness.
In this paper we extend Kalman’s concept of partial realization and define generalized partial realizations of finite matrix sequences by descriptor type systems. The aim is to prove a counterpart of Kalman’s main theorem of realization theory for generalized partial realizations. The paper ends with some results concerning topological aspects of generalized partial realizations.
Completely J-positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J-positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J-symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J-symmetric realizations are discussed.
We show how a certain class of Hamiltonian systems give rise to differential equations on spaces of matrices whose elements are rational functions. In particular, we reinterpret the results of Kac and van Moerbeke on the periodic Toda lattice in terms of such differential equations and relate the action-angle coordinates found by them to the evolution of a certain two by two symmetric matrix of rational functions flowing on a space of fixed McMillian degree and fixed Cauchy index. Realization theory is used to pass from a description of the flow in terms of rational matrices to a description in terms of the original coordinates.
