Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Linear control problems with total differential equations without convexity


Author: M. B. Suryanarayana
Journal: Trans. Amer. Math. Soc. 200 (1974), 233-249
MSC: Primary 49A35
DOI: https://doi.org/10.1090/S0002-9947-1974-0355729-X
MathSciNet review: 0355729
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Neustadt type existence theorems are given for optimal control problems described by Dieudonné-Rashevsky type total differential equations which are linear in the state variable. The multipliers from the corresponding conjugate problem are used to obtain an integral representation for the functional which in turn is used in conjunction with a Lyapunov type theorem on convexity of range of integrals to derive the existence of a usual solution from that of a generalized solution, which thus needs no convexity. Existence of optimal solutions is also proved in certain cases using an implicit function theorem along with the sufficiency of the maximum principle for optimality in the case of linear systems. Bang bang type controls are shown to exist when the system is linear in the control variable also.


References [Enhancements On Off] (What's this?)

  • [1] D. L. Bernstein, Existence theorems in partial differential equations, Ann. of Math. Studies, no. 23, Princeton Univ. Press, Princeton, N. J., 1950. MR 12, 262. MR 0037440 (12:262c)
  • [2a] L. Cesari, Existence theorems for multidimensional problems of optimal control, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P. R., 1965), Academic Press, New York, 1967, pp. 115-132. MR 36 #3196. MR 0220130 (36:3196)
  • [2b] -, Existence theorems for multidimensional Lagrange problems, J. Optimization Theory Appl. 1 (1967), 87-112. MR 37 #2053. MR 0226463 (37:2053)
  • [2c] -, Existence theorems for optimal controls of Mayer type, SIAM J. Control 6 (1968), 517-552. MR 39 #4722. MR 0243400 (39:4722)
  • [2d] -, Optimization with partial differential equations in Dieudonné-Rashevsky form and conjugate problems, Arch. Rational Mech. Anal. 33 (1969), 339-357. MR 39 #2040. MR 0240695 (39:2040)
  • [2e] -, Existence theorems for abstract multidimensional control problems, J. Optimization Theory Appl. 6 (1970), 210-236. MR 42 #6687. MR 0271806 (42:6687)
  • [2f] -, Convexity of the range of certain integrals, SIAM J. Control (to appear). MR 0380595 (52:1494)
  • [3] -, Problems of optimization, Springer-Verlag (to appear).
  • [4] L. Cesari and M. B. Suryanarayana, Closure theorems without seminormality conditions, J. Optimization Theory Appl. (to appear). MR 0365278 (51:1531)
  • [5] J. B. Diaz-Lopez, Approximation of generalized solutions by usual solutions in the problems of optimization, Ph. D. Thesis, University of Michigan, Ann Arbor, Mich., 1971.
  • [6] E. J. McShane and R. B. Warfield, On Filippov's implicit function lemma, Proc. Amer. Math. Soc. 18 (1967), 41-47. MR 34 #8399. MR 0208590 (34:8399)
  • [7] C. B. Morrey, Multiple integrals in the calculus of variations, Die Grundlehren der math. Wissenschaften, Band 130, Springer-Verlag, New York, 1966. MR 34 #2380. MR 0202511 (34:2380)
  • [8] L. Neustadt, The existence of optimal control in the absence of convexity conditions, J. Math. Anal. Appl. 7 (1963), 110-117. MR 27 #4714. MR 0154768 (27:4714)
  • [9a] M. B. Suryanarayana, Optimization with hyperbolic partial differential equations, Ph. D. Thesis, University of Michigan, Ann Arbor, Mich., 1969.
  • [9b] M. B. Suryanarayana, Necessary conditions for optimization problems with hyperbolic partial differential equations, SIAM J. Control 11 (1973), 130-147. MR 0317136 (47:5684)
  • [9c] -, Existence theorems for optimization problems concerning linear hyperbolic partial differential equations without convexity conditions, J. Optimization Theory Appl. (to appear). MR 0428155 (55:1183)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49A35

Retrieve articles in all journals with MSC: 49A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0355729-X
Keywords: Existence without convexity, linear control problems, bang bang phenomenon, implicit function theorem, necessary conditions, multipliers
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society