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

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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.

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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