The structure of nonlinear control systems possessing symmetries

Grizzle, Jessy W.; Marcus, Solomon

doi:10.1109/tac.1985.1103927

Cited by 122 publications

(93 citation statements)

References 17 publications

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“…This is also under study and will be addressed elsewhere. For example, integrability of the (2,3,5,8) problem of sub-Riemannian geometry, a problem with eight state variables, is currently an open question. This problem is out of the scope of the present Maple package: eleven hours of computing time were not enough for us to determine the symmetries of the problem.…”

Section: Discussionmentioning

confidence: 99%

Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Gouveia

Torres

2005

Comput. Methods Appl. Math.

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We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.

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Section: Discussionmentioning

confidence: 99%

Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Gouveia

Torres

2005

Comput. Methods Appl. Math.

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“…The following theorems assert that the extremals of problem (P ) are related to those extremals of problem (12) for which this constant is zero.…”

Section: Proposition 24 If the Homogeneity Conditionmentioning

confidence: 99%

“…The proof of Theorem 4.3 is based on the reduction of problem (P ) to problem (12) with Υ(t, z, v) = 1 L(t,z,v) ; on the subsequent Gamkrelidze's compactification of the space of admissible controls [10, §8.5]; on the abnormalCarathéodory-equivalence given by Theorems 2.5 and 2.6; and utilization of the classic Pontryagin maximum principle and the time-invariance property of problem (12). We remark that conditions (23) are not of the type of TonelliMorrey, and even for the basic problem of the calculus of variations one can cover new situations [19,20].…”

Section: Tonelli Full-regularitymentioning

confidence: 99%

Carathéodory Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

Torres¹

2004

Journal of Mathematical Sciences

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We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Carathéodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the theorems of Noether; iii) regularity conditions for the minimizers and the work pioneered by Tonelli.

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“…Lie algebraic and analytical methods were utilized; of particular interest was the estimation problem for linear systems with infrequently jumping parameters. In [5] and [6), the structure of nonlinear control systems with symmetries was studied, and the results obtained were used to derive reduced-order algorithms for certain classes of optimal control problems.…”

Section: Other Researchmentioning

confidence: 99%

Research in Adaptive and Decentralized Stochastic Control.

Marcus¹

1986

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The structure of nonlinear control systems possessing symmetries

Cited by 122 publications

References 17 publications

Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control

Carathéodory Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

Research in Adaptive and Decentralized Stochastic Control.

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