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Global methods to improve control and optimal control of resonance interaction of light and matter

  • Stochastic And Quantum Systems
  • Conference paper
  • First Online:
Modeling and Control of Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 121))

Abstract

This paper presents computational methods for optimization of control, connected with development and application of an approach in which sufficient conditions for global minima of functionals in variational calculus and optimal control theory are used. The first results were given at the beginning of the sixties (1–5), see also (6,7). The main element which is looked for in this approach is a so called solving function depending on the state and the argument (time) of the process under consideration. Having properly chosen this function an optimal solution is found through maximization of some scalar function of state, control, and time with respect to first two variables.

Making use of these methods the problem of optimal control for resonance interaction of radiation with a quantum system is being investigated. An iteration method for solving this problem, applicable to a large dimensional system, is proposed. The author presents the numerical solutions of the following problems:

  1. 1)

    the obtaining of the maximal inverse population of a three level system excited by three fields with relaxation (dimension of a phase vector is equal to n=9);

  2. 2)

    the obtaining of the maximal population of the first oscilating zone of a molecule of a spherical top type, excited by one external field (dimensional of a phase vector equal to n=15202).

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References

  1. Krotov V.F., Methods of solution of variational problems on the basis of sufficient conditions for absolute minimum. I. Avtomat. i Telemeh., 23 (1962) 12, 1571–1583.

    Google Scholar 

  2. Krotov V.F., Methods of solving variational problems. II. Sliding regimes. Avtomat. i Telemeh., 24 (1963) 5, 581–598.

    Google Scholar 

  3. Krotov V.F., Methods for variational problem solution based on absolute minimum sufficient conditions. III. Avtomat. i Telemeh., 25 (1964) 7, 1037–1046.

    Google Scholar 

  4. Krotov V.F., Approximate synthesis of optimal control. Avtomat. i Telemeh., 25 (1964) 11, 1521–1527.

    Google Scholar 

  5. Krotov V.F., Sufficient optimality conditions for discrete control systems. Dokl. AN SSSR, 172 (1967) 1, 18–21.

    Google Scholar 

  6. Krotov V.F., Bukreev V.Z., Gurman V.I., New Variational Methods in Flight Dynamics. Mashinostroyenie, Moskva, 1969 (English transl.: NASA, Transl. TTF-657, 1971).

    Google Scholar 

  7. Krotov V.F., Gurman V.I., Methods and Problems of Optimal Control. Nauka, Moskva, 1973 (in Russian).

    Google Scholar 

  8. Bellman R., Dynamic Programming. Princeton University Press, New Jersey, 1957.

    Google Scholar 

  9. Ioffe A.D., Tihomirov V.M., Theory of Optimal Problems. Nauka, Moskva, 1974 (English transl.: North Holland, Amsterdam, 1973).

    Google Scholar 

  10. Pontryagin L.S., Boltianski V.G., Gamkrelidze P.V., Mishchenko E.F., Mathematical Theory of Optimal Processes. Fizmagtiz, Moskva, 1961 (English transl.: Interscience Publishers Inc. New York, 1962).

    Google Scholar 

  11. Mereau P.M., Powers W.F., A direct sufficient condition for free final time optimal control problems. SIAM J. Control Optim., 14 (1976) 4, 613–622.

    Google Scholar 

  12. Rozenberg G.S., On the necessity and the sufficient conditions for minimum in variational problems. Differencialnye Uravnenija, (1968) 2 (transl. to English as Differential Equations).

    Google Scholar 

  13. Ziedan V., 1st and 2nd order sufficient conditions for optimal control and the calculus of variations. Appl. Math. Optim., 11 (1984) 3, 209–226.

    Google Scholar 

  14. Girsanov I.V., On a relation between Krotov and Bellman functions in the dynamic programming method. Vestnik Moskov. Univ., Ser. I, Mat. Meh., (1968), 56–59 (in Russian).

    Google Scholar 

  15. Khrustalev M.M., On sufficient conditions of global maximum. Dok. AN SSSR, Ser. Mat., 174 (1967) 5 (in Russian).

    Google Scholar 

  16. Ioffe A.D., Convex functions occurring in variational problems and the absolute minimum problem. Mat. Sb., 88 (1972), 194–210.

    Google Scholar 

  17. Vinter R.B., Weakest conditions for existence of Lipchitz continuous Krotov functions in optimal control theory. SIAM J. Control Optim., 21 (1983) 2, 215–234.

    Google Scholar 

  18. Vinter R.B., New global optimality conditions in optimal control theory. SIAM J. Control Optim., 21 (1983) 2, 235–245.

    Google Scholar 

  19. Vinter R.B., Lewis R.M., A verification theorem which provides a necessary and sufficient condition for optimality. IEEE Trans. Autom. Control, AC-25 (1980) 1, 84–89.

    Google Scholar 

  20. Klotzler R., Starke Qualitat in der Steuerungstheorie. Math. Nachr., 95 (1980), 253–263.

    Google Scholar 

  21. Khrustalev M.M., On sufficient optimality conditions in the problem with phase coordinates constraints. Avtomat. i Telemeh., (1967) 4, 1829 (in Russian).

    Google Scholar 

  22. Kelley H.J., Gradient theory of optimal flight path. ARS J., 30 (1960) 10.

    Google Scholar 

  23. Eneev T.M., On application of gradient method in optimal control theory problems. Kosmicheskiye Issledovania, 4 (1966) 5 (in Russian).

    Google Scholar 

  24. Krylov I.A., Chernousko F.L., On a method of consecutive approximations for solving optimal control problems. Z. Vycisl. Mat. i Fiz., 2 (1962) 6, 1132–1139 (in Russian).

    Google Scholar 

  25. Bryson A.E., Ho Y.Ch., Applied Optimal Control. Hemisphere Publ. Corp., Washington, 1975.

    Google Scholar 

  26. Krotov V.F., Feldman I.N., An iterative method for solving problems of optimal control. Izv. AN SSSR, Technic. Kibern. (1983) 2, 162–167 (in Russian).

    Google Scholar 

  27. Propoi A.I., Elements of the Theory of Optimal Discrete Systems. Nauka, Moskva, 1981 (in Russian).

    Google Scholar 

  28. Kazakov V.A., Krotov V.F., Optimal control of resonance interaction of the light and a substance. Avtomat. i Telemeh., (1987) 4, 9–15.

    Google Scholar 

  29. Emelicev V.A., Komlik V.I., A Method for Constructing a Sequence of Plans for Solving Discrete Optimal Problems. Nauka, Moskva, 1981 (in Russian).

    Google Scholar 

  30. Wagner G., Principles of Operations Research with Application to Managerial Decisions. Prentice Hall, Englewood Cliffs, 1969.

    Google Scholar 

  31. Gurman V.I., Baturin V.A., Danilina E.V., New Methods of Improving Controlled Process. Nauka, Novosibirsk, 1987 (in Russian).

    Google Scholar 

  32. Krotov V.F., Computing algorithms for solution and optimization of a controlled system of equations. I. Izv. AN SSSR, Technic. Kibern., (1975) 5, 3–15 (in Russian).

    Google Scholar 

  33. Krotov V.F., Computing algorithms for solution and optimization of a controlled system of equations. II. Izv. AN SSSR, Technic. Kibern., (1975) 6, 3–13 (in Russian).

    Google Scholar 

  34. Krotov V.F., Sergeev S.I., Computing algorithms for solution of some problems in linear and linear-integer programming. Avtomat. i Telemeh., Part I (1980) 12, 86–96, Part II (1981) 1, 86–96, Part III (1981) 3, 83–94, Part IV (1981) 4, 103–112.

    Google Scholar 

  35. Egorov V.A., Gusev L.I., The Dynamics of the Flight between the Earth and the Moon. Nauka, Moskva, 1980 (in Russian).

    Google Scholar 

  36. Gukasian M.H., On computing algorithms for optimization in linear controlled systems with distributed parameters. in: Investigations in Mechanics of a Solid Deforming Body. Erewan, 1981, 93–97.

    Google Scholar 

  37. Bukreev V.Z., Concerning certain method of approximate synthesis of optimal control. Avtomat. i Telemeh., (1968) 11, 5–13.

    Google Scholar 

  38. Bukreev V.Z., Rozenblat G.M., Approximate design of optimal control for discrete-time systems. Avtomat. i Telemeh., (1976) 1, 90–93.

    Google Scholar 

  39. Bakhito R.U., Krapchetkov N.P., Krotov V.F., The synthesis of an approximately optimal control for one class of controlled systems. Avtomat. i Telemeh., (1972) 10, 33–43.

    Google Scholar 

  40. Trigub M.B., Approximately optimal stabilization of one class of nonlinear systems. Avtomat. i Telemeh., (1987) 1, 34–47.

    Google Scholar 

  41. Lekkerke P.I., Derleman T.W., Calculation method for dynamic optimization using quadratic approximations to minimum surface. Automatica, 7 (1971) 6, 713.

    Google Scholar 

  42. Butkovski A.G., Samoilenko Yu I., Control for Quantum-mechanical processes. Nauka. 1984.

    Google Scholar 

  43. Krasnov I.V., Shaparev N. Ya., Shkedov I.M., Optimal Control for photo-processes in gas. — Preprints of Academy of Sci. 1979, No10.

    Google Scholar 

  44. Khorozov O.A., Control of coherent quantum-mechanical processes. — Dissertation, Institute of Cybernetics, Ukrained SSR Academy of Sci., Kiev, 1984.

    Google Scholar 

  45. Bunkin F.V., Kirichenko N.A., Lukyonchuk B.S., Optimal modes for heating material by laser radiation. Preprint of 1979, No146.

    Google Scholar 

  46. Vagners J., Neal R., Vlasas G., Optimal laser plasma heating in a solenoidal magnetic field. Physics of fluids, 1975, v.18, 1314–1320.

    Google Scholar 

  47. Fain V.M., Photons and nonlinear media "Sov. Radio", 1972.

    Google Scholar 

  48. Butylkin V.S., Kaplan A.E., Tronopulo Yu. G., Yakubovich E.I., Light resonance interactions with substance, Publ. Nauka, 1977.

    Google Scholar 

  49. Artamonova N.D., Platonenko V.T., Khokhlov R.V., On control of chemical reactions by resonance photoeffects on molecules. Publ., 1970, vol.58, 2195–2201.

    Google Scholar 

  50. Keldysh L.V., Ionization in the field of a strong light wave, 1964, vol.47, 945–1957.

    Google Scholar 

  51. Voronov G.S., Probability of atoms ionization and its dependence on the intensity of photons flow. 1966, vol.51, 1496–1498.

    Google Scholar 

  52. Krotov V.F., Feldman I.N., Iteration method of solving the problems of optimal control. USSR Academy of Sci. Technical Cybernetics, No2, 1983, 160–169.

    Google Scholar 

  53. Krotov V.F., Bukreev V.Z., Gurman V.I., New technique of variational computation in flight dynamics. Publ. Mashinostroyenie, 1969.

    Google Scholar 

  54. Somoilenko Yu. I., Optimal Control of Quantum Statistical set. Automatics and Telemechanics.

    Google Scholar 

  55. Delone N.B., Krainov V.P., Atom in strong light field. Publ. Energoatomizdat, 1984.

    Google Scholar 

  56. Salomaa R., Stendoholm S., Field optimization aspects in two-photon excitations. Appl. Phys., 1978, vol.17, 309–316.

    Google Scholar 

  57. Gurman V.I., Rasina I.V., On practical application sufficient conditions for strong relative minimum. Automatics and Telemechanics, 1979, No10.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Control Sciences, Profsouznaya 65, Moscow, USSR

    V. F. Krotov

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  1. V. F. Krotov
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Austin Blaquiére

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© 1989 Springer-Verlag

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Krotov, V.F. (1989). Global methods to improve control and optimal control of resonance interaction of light and matter. In: Blaquiére, A. (eds) Modeling and Control of Systems. Lecture Notes in Control and Information Sciences, vol 121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0041198

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  • DOI: https://doi.org/10.1007/BFb0041198

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50790-1

  • Online ISBN: 978-3-540-46087-9

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