×

On the optimal control of implicit systems. (English) Zbl 0898.93009

From the abstract: “We define the geometric framework of a \(q\)-\(\pi\)-submanifold in the tangent bundle of a surrounding manifold…With this geometric framework, we define a class of well-posed implicit differential equations for which we obtain locally a controlled vector field on a submanifold \(W\) of the surrounding manifold \(X\dots\)” – Now, at the very beginning of chapter 3.1 \(q\)-\(\pi\)-SUBMANIFOLD, we find Proposition 3.3 consisting of an ‘if and only if’ statement; the proof cannot be understood, but apparently ‘only if’ is wrong, ‘if’ being trivial. The name ‘\(q\)-\(\pi\)-submanifold’ is not well chosen, because a \(q\)-\(\pi\)-submanifold may also be a \(q'\)-\(\pi\)-submanifold with \(q'\neq q\). – Local trivializations of certain \(q\)-\(\pi\)-submanifolds are considered in the following Theorem 3.10; by deleting the word “unique” the statement concerning existence would become trivial but correct, while the statement relating different local trivializations by diffeomorphisms is nonsense. – The ‘geometric framework’ seems to need thorough revision!
Reviewer: P.Kraut (Egmating)

MSC:

93B29 Differential-geometric methods in systems theory (MSC2000)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
49N99 Miscellaneous topics in calculus of variations and optimal control
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] V. Alexeev, V.E. Galeev, V. Tikhomirov: Recueil de problèmes d’optimisation, Mir, Moscou, 1987.
[2] V. Alexeev, V. Tikhomirov, S. Fomin: Optimal control, Plenum Publishing Corporation, New York, 1987. Zbl0689.49001 · Zbl 0689.49001
[3] G.A. Bliss: The problem of Lagrange in the calculus of variations, American Journal of Mathematics 52 ( 1930), 673-744. MR1506783 JFM56.0435.01 · JFM 56.0435.01
[4] G.A. Bliss: Lectures on the calculus of variations, University of Chicago press, Chicago, 1946. Zbl0063.00459 MR17881 · Zbl 0063.00459
[5] C. Carathéodory: Calculus of variations and partial differential equations of the first order, Series in Mathematical Physics, Holden-Day, San francisco, Cambridge, London, Amsterdam, 1967. Zbl0134.31004 MR232264 · Zbl 0134.31004
[6] E. Cartan: Sur l’intégration de certains systèmes indéterminés d’équations différentielles, vol. 2, CNRS, Paris, 1984.
[7] L. Cesari: Optimization-theory and applications, Applications of Mathematics, Springer-Verlag, New York Heidelberg Berlin, 1983. Zbl0506.49001 MR688142 · Zbl 0506.49001
[8] F. Clarke: Optimization and nonsmooth analysis, Les publications du Centre de Recherche Mathématiques de l’Université de Montréal, 1989. Zbl0727.90045 MR1019086 · Zbl 0727.90045
[9] J. Dieudonné: Eléments d’analyse, vol. 3, Gauthier-Villars, Paris, 1970. Zbl0208.31802 MR270377 · Zbl 0208.31802
[10] P.A. Griffiths: Exterior differential systems and the calculus of variations, Birkäuser, Boston, 1983. Zbl0512.49003 MR684663 · Zbl 0512.49003
[11] J. Hadamard: Leçons sur le calcul des variations, Herman, Paris, 1940. JFM41.0432.02 · JFM 41.0432.02
[12] V. Jurdjevic: Geometric control theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1997. Zbl0940.93005 MR1425878 · Zbl 0940.93005
[13] Ph. Loewen: Optimal control via nonsmooth analysis, CRM Proceedings and Lecture Notes, vol. 2, American Mathematical Society, Providence, Rhode Island USA, 1993. Zbl0874.49002 MR1232864 · Zbl 0874.49002
[14] L. Pontryagin et al.: Théorie mathématique de processus optimaux, Mir, Moscou, 1974. MR358482
[15] P.-J. Rabier, W.C. Rheinboldt: A general existence and uniqueness theory for implicit differential-algebraic equations, Differential Integral Equations 4 ( 1991), 563-582. Zbl0722.34004 MR1097919 · Zbl 0722.34004
[16] P.-J. Rabier, W.C. Rheinboldt: A geometric treatment of implicit differential-algebraic equations, Journal of Differential Equations 109 ( 1994), 110-146. Zbl0804.34004 MR1272402 · Zbl 0804.34004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.