## Automatic Computation of Conservation Laws in the Calculus of Variations and

Venue: | Optimal Control, Comput. Methods Appl. Math |

Citations: | 8 - 6 self |

### BibTeX

@ARTICLE{Gouveia_automaticcomputation,

author = {Paulo D. F. Gouveia and Delfim F. M. Torres},

title = {Automatic Computation of Conservation Laws in the Calculus of Variations and},

journal = {Optimal Control, Comput. Methods Appl. Math},

year = {},

pages = {387--409}

}

### OpenURL

### Abstract

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.

### Citations

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Citation Context ... Such conservation laws can be used to simplify the problem [8, 9]. The question is then the following: how to determine these conservation laws? It turns out that the classic results of Emmy Noether =-=[14, 15]-=- of the calculus of variations, relating the existence of conservation laws with the existence of symmetries, can be generalized to the wider context of optimal control [4, 6, 23], reducing the proble... |

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Citation Context ...pt/delfim/maple.htm 1Automatic Computation of Conservation Laws in Optimal Control 1 Introduction Optimal control problems are usually solved with the help of the famous Pontryagin maximum principle =-=[17]-=-, which is a generalization of the classic Euler-Lagrange and Weierstrass necessary optimality conditions of the calculus of variations. The method of finding optimal solutions via Pontryagin’s maximu... |

23 | On the Noether Theorem for Optimal Control
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Citation Context ...esults of Emmy Noether [14, 15] of the calculus of variations, relating the existence of conservation laws with the existence of symmetries, can be generalized to the wider context of optimal control =-=[4, 6, 23]-=-, reducing the problem to the one of discovering the invariance-symmetries. The difficulty resides precisely in the determination of the variational symmetries. While in Physics and Economics the ques... |

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Citation Context ...er Algebra Systems, Maple as an example, facilitate the interplay of conventional mathematics with computers. They are, in some sense, changing the way we learn, teach, and do research in mathematics =-=[1]-=-. They can perform a myriad of symbolic mathematical operations, like analytic differentiation, integration of algebraic formulae, factoring polynomials, computing the complex roots of analytic functi... |

18 |
The Calculus of Variations
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Citation Context ...following control system: ⎧ ⎪⎨ In Maple we have: ⎪⎩ ˙x1(t) = 1 + x2(t), ˙x2(t) = x3(t), ˙x3(t) = u(t), ˙x4(t) = x3(t) 2 − x2(t) 2 . 12 }Paulo D. F. Gouveia, Delfim F. M. Torres > L:=1; phi:=[1+x[2],x=-=[3]-=-,u,x[3]^2-x[2]^2]; L := 1 ϕ := [1 + x2,x3,u,x3 2 − x2 2 ] > Symmetry(L, phi, t, [x[1],x[2],x[3],x[4]], u); { ( T = C5, X1 = − 1 2 C2 − 1 2 C1 ) t + 1 2 C2x1 + C4, X2 = − 1 2 C1 + 1 2 C2x2, X3 = 1 2 C2... |

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Citation Context ... existence of conservation laws [14]. This relation constitutes a universal principle that can be formulated, as a theorem, in several different contexts, under several different hypotheses (see e.g. =-=[4, 6, 9, 10, 15, 26, 28]-=-). Contributions in the literature go, however, further than extending Noether’s theorem to different contexts, and weakening its assumptions. Since the pioneering work by Noether [14], several defini... |

14 |
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Citation Context ...ent contexts, and weakening its assumptions. Since the pioneering work by Noether [14], several definitions of invariance have been introduced for the problems of the calculus of variations (see e.g. =-=[10, 11, 20, 26]-=-); and for the problems of optimal control (see e.g. [4, 6, 23, 27]). All these definitions are given with respect to a one-parameter group of transformations (9). Although written in a different way ... |

10 | Symmetries of flat rank two distributions and sub-Riemannian structures
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Citation Context ...r Maple package to three important problems of geodesics in sub-Riemannian geometry. The reader, interested in the study of symmetries of flat distributions of sub-Riemannian geometry, is referred to =-=[19]-=-. Example 16 (Martinet – (2,2,3) problem) Given the problem (n = 3, m = 2) ∫b ( u1(t) 2 + u2(t) 2) dt −→ min, a ⎧ ⎪⎨ ⎪⎩ ˙x1(t) = u1(t), u2(t) ˙x2(t) = 1 + αx1(t) , ˙x3(t) = x2(t) 2u1(t), α ∈ R, we con... |

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Citation Context ...k by Noether [14], several definitions of invariance have been introduced for the problems of the calculus of variations (see e.g. [10, 11, 20, 26]); and for the problems of optimal control (see e.g. =-=[4, 6, 23, 27]-=-). All these definitions are given with respect to a one-parameter group of transformations (9). Although written in a different way (some of these invariance/symmetry notions involve the integral fun... |

9 | Proper extensions of Noether’s symmetry theorem for nonsmooth extremals of the calculus of variations
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(Show Context)
Citation Context ... existence of conservation laws [14]. This relation constitutes a universal principle that can be formulated, as a theorem, in several different contexts, under several different hypotheses (see e.g. =-=[4, 6, 9, 10, 15, 26, 28]-=-). Contributions in the literature go, however, further than extending Noether’s theorem to different contexts, and weakening its assumptions. Since the pioneering work by Noether [14], several defini... |

8 |
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Citation Context ... [21]. One way to address the problem is to find conservation laws, i.e., quantities which are preserved along the extremals of the problem. Such conservation laws can be used to simplify the problem =-=[8, 9]-=-. The question is then the following: how to determine these conservation laws? It turns out that the classic results of Emmy Noether [14, 15] of the calculus of variations, relating the existence of ... |

7 | A remarkable property of the dynamic optimization extremals - Torres |

6 |
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Citation Context ...t from the family of conservation laws just obtained, three independent conservation laws. We just need to fix one constant to a non-zero value, and choose all the other constants to be zero: > subs(C=-=[4]-=-=1,seq(C[i]=0,i=1..5), CL); > subs(C[1]=1,seq(C[i]=0,i=1..5), CL); > simplify(subs(C[5]=-1,seq(C[i]=0,i=1..5), CL)); ψ3(t) = const x2(t)ψ3(t) + ψ1(t) = const 1 2 ψ1(t) 2 + 1 2 ψ2(t) 2 + ψ2(t)x1(t)ψ3(t... |

5 |
Applied Mathematics: A Contemporary Approach
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Citation Context ...ent contexts, and weakening its assumptions. Since the pioneering work by Noether [14], several definitions of invariance have been introduced for the problems of the calculus of variations (see e.g. =-=[10, 11, 20, 26]-=-); and for the problems of optimal control (see e.g. [4, 6, 23, 27]). All these definitions are given with respect to a one-parameter group of transformations (9). Although written in a different way ... |

4 |
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Citation Context ...controls (m = 2): ∫b ( u1(t) 2 + u2(t) 2) dt −→ min, a ⎧ ⎪⎨ ⎪⎩ ˙x1(t) = x3(t), ˙x2(t) = x4(t), ˙x3(t) = −x1(t) (x1(t) 2 + x2(t) 2 ) + u1(t), ˙x4(t) = −x2(t) (x1(t) 2 + x2(t) 2 ) + u2(t), > L:=u[1]^2+u=-=[2]-=-^2; phi:=[x[3],x[4],-x[1]*(x[1]^2+x[2]^2)+u[1], -x[2]*(x[1]^2+x[2]^2)+u[2]]; L := u1 2 + u2 2 ϕ := [ x3,x4, −x1 ( x1 2 + x2 2) ( + u1, −x2 x1 2 + x2 2) ] + u2 > Symmetry(L, phi, t, [x[1],x[2],x[3],x[4... |

4 | 2005a) Computação Algébrica no Cálculo das Variações: Determinação de Simetrias e Leis de Conservação, (in Portuguese - Gouveia, Torres |

2 |
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Citation Context ...esults of Emmy Noether [14, 15] of the calculus of variations, relating the existence of conservation laws with the existence of symmetries, can be generalized to the wider context of optimal control =-=[4, 6, 23]-=-, reducing the problem to the one of discovering the invariance-symmetries. The difficulty resides precisely in the determination of the variational symmetries. While in Physics and Economics the ques... |

2 |
Reduction in optimal control theory, Rep
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Citation Context ...conservation laws, one can even integrate and solve the problems completely. Like Noether’s theorem, also the classical reduction theory can be extended to the more general setting of optimal control =-=[5, 13]-=-. However, the reduction theory in optimal control is an area not yet completed. More theoretical results are needed in order to be possible to automatize the whole process, from the computation of sy... |

2 | Integrable geodesics flows of non-holonomic metrics
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Citation Context ...ws: 1 2 ⎧ ⎨ ⎩ ∫b a ( u1(t) 2 + u2(t) 2) dt −→ min, ˙x1(t) = u1(t) , ˙x2(t) = u2(t) , ˙x3(t) = u2(t)x1(t) . The problem was proved to be completely integrable using three independent conservation laws =-=[22]-=-. Such conservation laws can now be easily obtained with our Maple functions. > L:=1/2*(u[1]^2+u[2]^2); > phi:=[u[1], u[2], u[2]*x[1]]; L := 1 2 u1 2 + 1 2 u2 2 ϕ := [u1,u2,u2x1] > Symmetry(L, phi, t,... |

2 | Weak conservation laws for minimizers which are not Pontryagin extremals
- Torres
(Show Context)
Citation Context ...n. We propose a computational method that permits to obtain conservation laws for a given optimal control problem. Our method is based in the version of Noether’s theorem established in [4] (see also =-=[28]-=-). To describe a systematic method to compute conservation laws, first we need to recall the standard definitions of extremal, and conservation law. The central result in optimal control theory is the... |

1 |
Symmetries and reduction in optimal control theory
- Echeverría-Enríquez, Marín-Solano, et al.
(Show Context)
Citation Context ...2 C4 − 1 4 K3 2 C4 − K5 K1 C4 = const conduces to a true proposition (constant equal constant). Finally, substituting only the Pontryagin multipliers, > subs({psi[1](t)=K[6], psi[3](t)=0, psi[4](t)=-K=-=[5]-=-*t+K[3], psi[2](t)=K[5]}, CL); ( (C3x1(t) + C5)K6 + (C1t + 3C3x2(t) + C6)K5 + C1 + C3 ( ( d − − dt x1(t) ) 2 ( d2 − dt2x2(t) ) 2 + K6 ) + (−K5 t + K3) d2 dt 2x2(t) d dt x2(t) ) (−K5 t + K3) d dt x1(t)... |

1 |
Counterexamples in optimal control theory
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(Show Context)
Citation Context ... the solutions of this system of ordinary differential equations are sought. Steps (ii) and (iii) are, generally speaking, nontrivial, and very difficult (or even impossible) to implement in practice =-=[21]-=-. One way to address the problem is to find conservation laws, i.e., quantities which are preserved along the extremals of the problem. Such conservation laws can be used to simplify the problem [8, 9... |