## Existence of finitely optimal solutions for infinite-horizon optimal control problems.(English)Zbl 0579.49004

We investigate the existence of finitely optimal solutions for the Lagrange problem of optimal control defined on $$[0,+\infty)$$ under weaker convexity and seminormality hypotheses than those of previous authors. The notion of finite optimality has been introduced into the literature as the weakest of a hierarchy of types of optimality that have been defined to permit the study of Lagrange problems, arising in mathematical economics, whose cost functionals either diverge or are not bounded below.
Our method of proof requires us to analyze the continuous dependence of finite-interval Lagrange problems with respect to a prescribed terminal condition. Once this is done, we show that a finitely optimal solution can be obtained as the limit of a sequence of solutions to a sequence of corresponding finite-horizon optimal control problems. Our results utilize the convexity and seminormality hypotheses which are now classical in the existence theory of optimal control.

### MSC:

 49J15 Existence theories for optimal control problems involving ordinary differential equations 49J45 Methods involving semicontinuity and convergence; relaxation 93C15 Control/observation systems governed by ordinary differential equations
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### References:

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