On the Algebraic Classification of Fredholm Integral Operators
Using the well-known and specific connections between Fredholm integral equations, two-point boundary-value problems, and linear dynamics—quadratic cost control processes, we present a complete, independent set of algebraic invariants suitable for classifying a wide range of Fredholm integral operators with respect to a certain group of transformations. The group, termed the Riccati group, is naturally suggested by the control theoretic setting, but seems nonintuitive from a purely integral-equations point of view. Computational considerations resulting from this classification are discussed.
KeywordsRiccati Equation Fredholm Integral Equation Algebraic Riccati Equation Algebraic Invariant Matrix Riccati Equation
Unable to display preview. Download preview PDF.
- 1.Brockett, R., and Mayne, D., Proceedings of the NATO Advanced Study Institute on Geometric and Algebraic Methods for Nonlinear Systems, D. Reidel Publishing Company, New York, New York, 1973.Google Scholar
- 2.IEEE Proceedings, Special Issue on Recent Trends in System Theory, Vol. 64, No. 1, 1976.Google Scholar
- 3.Kalman, R., System Theoretic Aspects of Invariant Theory, University of Florida, Center for Mathematical Systems Theory, Gainesville, Florida, Private Communication, 1974.Google Scholar
- 4.Kalman, R., and Hazewinkel, M., On Invariants, Canonical Forms, and Moduli for Linear, Constant, Finite-Dimensional Dynamical Systems, Proceedings of the CISM Conference on Algebraic System Theory, Edited by A. Marzollo and G. Marchessini, New York, New York, 1976.Google Scholar
- 5.Hazewinkel, M., Representations of Quivers and Moduli of Linear Dynamical Systems, University of Rotterdam, Rotterdam, Holland, Economic Institute of Erasmus, 1976.Google Scholar
- 6.Byrnes, C., and Hurt, N., On the Moduli of Linear Dynamical Systems (to appear).Google Scholar
- 8.Casti, J., Invariant Theory, the Riccati Group and Linear Control Problems,IEEE Transactions on Automatic Control (to appear).Google Scholar
- 9.Casti, J., Generalized X—Y Functions, the Linear Matrix Inequality and Triangular Factorization for Linear Control Problems, IIASA, Laxenburg, Austria, Research Memorandum No. RM-76–10, 1976.Google Scholar
- 10.Casti, J., Dynamical Systems and Their Applications: Linear Theory, Academic Press, New York, New York, 1977.Google Scholar