Linear control problems with total differential equations without convexity
Author:
M. B. Suryanarayana
Journal:
Trans. Amer. Math. Soc. 200 (1974), 233249
MSC:
Primary 49A35
MathSciNet review:
0355729
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Neustadt type existence theorems are given for optimal control problems described by DieudonnéRashevsky 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.
 [1]
Dorothy
L. Bernstein, Existence Theorems in Partial Differential
Equations, Annals of Mathematics Studies, no. 23, Princeton University
Press, Princeton, N. J., 1950. MR 0037440
(12,262c)
 [2a]
Lamberto
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 0220130
(36 #3196)
 [2b]
Lamberto
Cesari, Existence theorems for multidimensional Lagrange
problems, J. Optimization Theory Appl. 1 (1967),
87–112. MR
0226463 (37 #2053)
 [2c]
Lamberto
Cesari, Existence theorems for optimal controls of the Mayer
type, SIAM J. Control 6 (1968), 517–552. MR 0243400
(39 #4722)
 [2d]
Lamberto
Cesari, Optimization with partial differential equations in
DieudonnéRashevsky form and conjugate problems, Arch. Rational
Mech. Anal. 33 (1969), 339–357. MR 0240695
(39 #2040)
 [2e]
Lamberto
Cesari, Existence theorems for abstract multidimensional control
problems, J. Optimization Theory Appl. 6 (1970),
210–236. MR 0271806
(42 #6687)
 [2f]
Lamberto
Cesari, Convexity of the range of certain integrals, SIAM J.
Control 13 (1975), 666–676. MR 0380595
(52 #1494)
 [3]
, Problems of optimization, SpringerVerlag (to appear).
 [4]
L.
Cesari and M.
B. Suryanarayana, Closure theorems without seminormality
conditions, J. Optimization Theory Appl. 15 (1975),
441–465. Existence theory in the calculus of variations and optimal
control. MR
0365278 (51 #1531)
 [5]
J. B. DiazLopez, 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 Jr., On Filippov’s implicit functions
lemma, Proc. Amer. Math. Soc. 18 (1967), 41–47. MR 0208590
(34 #8399), http://dx.doi.org/10.1090/S0002993919670208590X
 [7]
Charles
B. Morrey Jr., Multiple integrals in the calculus of
variations, Die Grundlehren der mathematischen Wissenschaften, Band
130, SpringerVerlag New York, Inc., New York, 1966. MR 0202511
(34 #2380)
 [8]
Lucien
W. Neustadt, The existence of optimal controls in the absence of
convexity conditions, J. Math. Anal. Appl. 7 (1963),
110–117. 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]
M.
B. Suryanarayana, Existence theorems for optimization problems
concerning linear, hyperbolic partial differential equations without
convexity conditions, J. Optimization Theory Appl. 19
(1976), no. 1, 47–61. Existence theorem issue. MR 0428155
(55 #1183)
 [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. 115132. MR 36 #3196. MR 0220130 (36:3196)
 [2b]
 , Existence theorems for multidimensional Lagrange problems, J. Optimization Theory Appl. 1 (1967), 87112. MR 37 #2053. MR 0226463 (37:2053)
 [2c]
 , Existence theorems for optimal controls of Mayer type, SIAM J. Control 6 (1968), 517552. MR 39 #4722. MR 0243400 (39:4722)
 [2d]
 , Optimization with partial differential equations in DieudonnéRashevsky form and conjugate problems, Arch. Rational Mech. Anal. 33 (1969), 339357. MR 39 #2040. MR 0240695 (39:2040)
 [2e]
 , Existence theorems for abstract multidimensional control problems, J. Optimization Theory Appl. 6 (1970), 210236. 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, SpringerVerlag (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. DiazLopez, 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), 4147. 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, SpringerVerlag, 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), 110117. 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), 130147. 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:
http://dx.doi.org/10.1090/S0002994719740355729X
PII:
S 00029947(1974)0355729X
Keywords:
Existence without convexity,
linear control problems,
bang bang phenomenon,
implicit function theorem,
necessary conditions,
multipliers
Article copyright:
© Copyright 1974
American Mathematical Society
