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1995 | 32 | 1 | 361-378
Tytuł artykułu

Minima in control problems with constraints

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This paper is devoted to describing second order conditions in the framework of extremal problems, that is, conditions obtained by reducing the optimal control problem to an abstract one in a suitable Banach (or Hilbert) space. The studied problem includes equality constraints both on the end-points and on the state-control trajectory. The second goal is to give a complete description of necessary and sufficient second order conditions for weak local optimality by describing first the associated linear-quadratic problem and then by giving a conjugate point theory for this linear quadratic problem with constraints.
Rocznik
Tom
32
Numer
1
Strony
361-378
Opis fizyczny
Daty
wydano
1995
Twórcy
  • Dipartimento di Matematica e Applicazioni, Via Mezzocannone 8, 80134 Napoli, Italy
  • Dipartimento di Matematica Defas, Via C. Lombroso 6/17, 50134 Firenze, Italy
Bibliografia
  • [1] A. A. Agrachev, Quadratic mappings in geometric control theory, Itogi Nauki, Problemy Geometrii, 20 (1988), 111-205 (in Russian); English transl. in: J. Soviet Math. 51 (1990).
  • [2] A. A. Agrachev and R. Gamkrelidze, Symplectic methods for optimization and control, in: B. Jakubczyk and W. Respondek (eds.), Geometry of Feedback and Optimal Control, Pure and Appl. Math., Marcel Dekker, New York, 1995.
  • [3] P. Bernhard, La théorie de la seconde variation et le problème linéaire quadratique, in: J. P. Aubin, A. Bensoussan and I. Ekeland (eds.), Advances in Hamiltonian Systems, Birkhäuser, Boston, 1983, 109-142.
  • [4] A. Dmitruk, The Euler-Jacobi equation in variational calculus, Mat. Zametki 20 (1976), 847-858 (in Russian); English transl.: Math. Notes 20 (1976), 1032-1038.
  • [5] A. L. Dontchev, W. W. Hager, A. B. Poore and B. Yang, Optimal stability and convergence in nonlinear control, preprint, 1992.
  • [6] W. H. Fleming and R. Rishel, Optimal Deterministic and Stochastic Control, Springer, Berlin, 1975.
  • [7] E. G. Gilbert and D. S. Bernstein, Second order necessary conditions in optimal control: accessory-problem results without normality conditions, J. Optim. Theory Appl. 41 (1983), 75-106.
  • [8] K. A. Grasse, Controllability and accessibility in nonlinear control systems, PhD thesis, University of Illinois at Urbana-Champaign, 1979.
  • [9] M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to calculus of variations, Pacific J. Math. 1 (1951), 525-581.
  • [10] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966.
  • [11] E. Levitin, A. Milyutin and N. Osmolovskiĭ, Conditions of high order for a local minimum in problems with constraints, Uspekhi Mat. Nauk, 33 (1978), 85-148 (in Russian); English transl.: Russian Math. Surveys 33 (1978), 97-168.
  • [12] K. Makowski and L. Neustadt, Optimal control problems with mixed control-phase variable equality and inequality constraints, SIAM J. Control 12 (1974), 184-228.
  • [13] H. Maurer, First and second order sufficient optimality conditions in mathematical programming and optimal control, Math. Programming Stud. 14 (1981), 163-177.
  • [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
  • [15] E. Mikami, Focal points in a control problem, Pacific J. Math. 35 (1970), 473-485.
  • [16] M. Morse, The Calculus of Variations in the Large, Amer. Math. Soc. Colloq. Publ. 18, New York, 1934.
  • [17] N. Osmolovskiĭ, Second order conditions for a weak local minimum in an optimal control problem (necessity, sufficiency), Dokl. Akad. Nauk SSSR 225 (1975) (in Russian); English transl.: Soviet Math. Dokl. 16 (1975), 1480-1483.
  • [18] G. Stefani and P. Zezza, A new type of sufficient optimality conditions for a nonlinear constrained optimal control problem, in: M. Fliess (ed.), Proceedings of the Nonlinear Control System Design Symposium, Bordeaux, 1992, 713-719.
  • [19] G. Stefani and P. Zezza, Optimal control problems with mixed state-control constraints: necessary conditions, J. Math. Syst. Est. Control 2 (1992), 155-189.
  • [20] G. Stefani and P. Zezza, The Jacobi condition for LQ-control problems with constraints, in: J. W. Nieuwenhuis, C. Praagman and H. L. Trentelman (eds.), Proceedings of the Second European Control Conference, 1993, 1003-1007.
  • [21] G. Stefani and P. Zezza, Optimality conditions for a constrained optimal control problem, preprint, University of Florence, DiMaDEFAS, 1993.
  • [22] G. Stefani and P. Zezza, Regular constrained LQ-control problems, preprint #12-94, University of Florence, DiMaDEFAS.
  • [23] J. Warga, Second order necessary conditions in optimization, SIAM J. Control Optim. 22 (1984), 524-528.
  • [24] V. Zeidan, Sufficiency conditions for variational problems with variable endpoints: coupled points, Appl. Math. Optim. 27 (1993), 191-209.
  • [25] V. Zeidan and P. Zezza, Necessary conditions for optimal control problems: conjugate points, SIAM J. Control Optim. 26 (1988), 592-608.
  • [26] V. Zeidan and P. Zezza, Coupled points in optimal control, IEEE Trans. Automat. Control 36 (1991), 1276-1281.
  • [27] P. Zezza, The Jacobi condition for elliptic forms in Hilbert spaces, J. Optim. Theory Appl. 76 (1993), 357-380.
Typ dokumentu
Bibliografia
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