| Title | ABSOLUTE EXTREMA OF INVARIANT OPTIMAL CONTROL PROBLEMS |
| Publication Type | Journal Article |
| Year of Publication | 2006 |
| Authors | Silva, CJ, Torres, DFM |
| Secondary Title | Communications in Applied Analysis |
| Volume | 10 |
| Issue | 4 |
| Start Page | 517 |
| Pagination | 529 |
| Date Published | 12/2006 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 49J15, 49M30 |
| Abstract | Optimal control problems are usually addressed with the help of the famous Pontryagin Maximum Principle (PMP) which gives a generalization of the classical Euler-Lagrange and Weierstrass necessary optimality conditions of the calculus of variations. Success in applying the PMP permits to obtain candidates for a local minimum. In 1967 a direct method, which permits to obtain global minimizers directly, without using necessary conditions, was introduced by Leitmann. Leitmann’s approach is connected, as showed by Carlson in 2002, with “Carath ́eodory’s royal road of the calculus of variations”. Here we propose a related but different direct approach to problems of the calculus of variations and optimal control, which permit to obtain global minima directly, without recourse to needle variations and necessary conditions. Our method is inspired by the classical Noether’s Theorem and its recent extensions to optimal control. We make use of the variational symmetries of the problem, considering parameter-invariance transformations and substituting the original problem by a parameter-family of optimal control problems. Parameters are then fixed in order to make the problem trivial, in some sense. Finally, by applying the inverse of the chosen invariance-transformation, we get the global minimizer for the original problem. The proposed method is illustrated, by solving concrete problems, and compared with Leitmann’s approach. |
| URL | http://www.acadsol.eu/en/articles/10/4/6.pdf |
| Short Title | Absolute Extrema |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] C. Caratheodory, Calculus of Variations and Partial Differential Equations, Chelsea Publishing Company, New York, 1982.
[2] D.A. Carlson, An observation on two methods of obtaining solutions to variational problems, J. Optim. Theory Appl., 114 (2002), no. 2, 345-361. [3] D.A. Carlson and G. Leitmann, Coordinate transformation method for the extremization of multiple integrals, J. Optim. Theory Appl., 127 (2005), no. 3, 523-533. [4] P.D.F. Gouveia and D.F.M. Torres, Computacaoalgebrica no calculo das variacoes: determinacao de simetrias e leis de conservacao, TEMA Tend. Mat. Apl. Comput., 6 (2005), no. 1, 81-90. [5] P.D.F. Gouveia and D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Computational Methods in Applied Mathematics, 5 (2005), no. 4, 387-409. [6] P.D.F. Gouveia, D.F.M. Torres, and E.A.M. Rocha, Symbolic computation of variational symmetries in optimal control, Control and Cybernetics, To Appear. [7] G. Leitmann, A note on absolute extrema of certain integrals, International Journal of Nonlinear Mechanics, 2 (1967), 55-59. [8] G. Leitmann, The Calculus of Variations and Optimal Control – An Introduction, Plenum Press, New York, 1981. [9] G. Leitmann, On a class of direct optimization problems, J. Optim. Theory Appl., 108 (2001), no. 3, 467-481. [10] G. Leitmann, Some extensions to a direct optimization method, J. Optim. Theory Appl., 111 (2001), no. 1, 1-6. [11] J. W. Macki and A. Strauss, Introduction to Optimal Control Theory, Springer, New York, 1982. [12] E. Noether, Invariante variationsprobleme, Gott. Nachr. (1918), 235-257. [13] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley and Sons, Inc., New York-London, 1962. [14] D.F.M. Torres, On the Noether Theorem for optimal control, European Journal of Control, 8 (2002), no. 1, 56-63.
[15] D.F.M. Torres, Carath ́eodory equivalence, Noether Theorems and Tonelli full regularity in the calculus of variations and optimal control, Journal of Mathematical Sciences, 120 (2004), no. 1, 1032-1050. [16] D.F.M. Torres, Quasi-invariant optimal control problems, Port. Math., 61 (2004), no. 1, 97-114. |