Constructive Multidimensional Systems Theory

We propose to continue the work in our recent submitted papers Stability and Stabilization of Multidimensional Input/Output (IO-) Systems (Stabilization paper) and Canonical State Representations and Hilbert Functions of Multidimensional Systems (State paper) on multidimensional behaviors which are governed by linear systems of partial differential (continuous case) or difference (discrete case) equations with constant coefficients. The present research project intends to both advance the theory and to develop and implement in MAPLE various algorithms which are contained in the two papers. The novelty of our stabilization approach, even for one-dimensional systems, lies in the treatment of continuous or discrete input/output behaviors whereas most papers on multidimensional stabilization treat the stabilization of discrete IO-maps or transfer matrices. Usually discrete transfer matrices are also assumed proper, an assumption which was not made in the stabilization paper, but is basic for one-dimensional and many discrete multidimensional systems. Therefore we propose to investigate, in the spirit of the stabilization paper, the proper stabilization of IO-systems where the stable feedback IO-system has a proper transfer matrix. As far as possible the algorithms of the stabilization paper and their proper stable counter-parts should be implemented in MAPLE. The main result of the state paper is the construction of the canonical and normal state representation of a multidimensional system which, as a by-product, furnishes a new algorithm for the computation of the Hilbert function. A substantial task for the new project is the implementation in MAPLE of all algorithms of the paper for all dimensions including the constructive solution of the analytic Cauchy problem for general multidimensional systems. Further theoretical work will concern the characterization of systems whose normal state space representations are of the first order and the (difficult) solution of the non-analytic Cauchy problem for the general state space systems of the state paper. Some parts of the research described above will enter into M. Scheicher`s doctoral thesis. The theorems and algorithms of the two papers are new even for one-dimensional systems. One or two master`s theses should be devoted to the implementation of the algorithms in MAPLE and to work out the proper stabilization theory for one-dimensional IO-systems and further state space representations. This is justified in addition to the multidimensional work above since no fundamental theoretical or algorithmic obstacles and more complete solutions are expected in this important classical case. These results will enter into lecture notes of the applicant on one-dimensional linear systems theory for mathematicians which exist in hand-written form already.

This project was a continuation of the project leader`s articles "Stability and Stabilization of Multidimensional Input/Output Systems" and "Canonical State Representations and Hilbert Functions of Multidimensional Systems" on multidimensional systems described by linear systems of partial differential equations (continuous case) or partial difference equations (discrete case). Such systems appear amongst others in physics (Maxwell`s equation in electrodynamics and the Navier Stokes equation in fluid dynamics), in digital image processing (e.g. filter for sharpening images) and engineering (analysis and synthesis of electrical networks, design of controllers for machines such as cruise control in cars). The main result with respect to the first mentioned article is the refinement of the method developed therein for stabilising input/output systems to the case where the transfer matrices of the feedback system and of the controller are proper. Furthermore, the thus obtained new algorithms as well as the ones described in the article about state representations have been implemented in the computer algebra system SINGULAR. Apart from that, this project led to significant results regarding the derivation of an impulse response of multidimensional systems in the continuous case. Picturing a multidimensional system as a machine accepting an input and producing some output which is initially at rest, one can compute the output from the input using the impulse response and thus predict the performance of the machine. Using these results we were able to solve Jury`s conjecture on BIBO (bounded input/bounded output) stability of multidimensional continuous systems which was open for more than 20 years, i.e., we answered the following question: Which properties has a system to have, such that any bounded input leads to a bounded output? For applications, BIBO stability is fundamental, because if some of the system`s components can take arbitrarily large values, this results in damage or even destruction of the described device. This project was focused on mathematical aspects of systems theory, a field with a wide range of application as indicated in the first paragraph. Thus it will need time for our results and algorithms to diffuse to the areas of applications. However, with our algorithms one can compute what until now could not be computed and we expect that in the long term they will applied regularly.