## Summary

In the first two sections of this paper existence theorems are proved for optimal control problems described either by Fredholm systems of integral equations (with control appearing nonlinearly) on a given (possibly infinite) measure space, eventually in presence of eigenvalues, or by Urysohn systems, both with pointwise and integral constraints on controls and states. Controllability is assumed throughout the paper. In the last section necessary conditions are proved for optimal controls of Fredholm systems (a minimum principle in pointwise, or « differential » form), and theorems (that seem to be new also for the corresponding ordinary differential control systems) are deduced about bang-bang and regularization properties of the optimal control (s) (which turns out to be a continuous function except for a finite number of points, if positivity and semicontinuity assumptions hold about the data): in some particular cases the optimal control will be piecewise constant. All these results are independent on the known ones.

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## Additional information

Lavoro eseguito nell'ambito del Centro di Matematica e Fisica Teorica del C.N.R. presso l'Università di Genova.

Entrata in Redazione il 6 maggio 1971.

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### Cite this article

Zolezzi, T. Esistenza e condizioni necessarie per controlli ottimi di sistemi definiti da equazioni integrali.
*Annali di Matematica* **91, **97–118 (1971) doi:10.1007/BF02428816

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