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New existence results for optimal controls in the absence of convexity: The importance of extremality. (English) Zbl 0813.49004
The author determines existence results for optimal control problems without convexity. The model is such that the objective function contains separate integral terms for the control and state. The integrand for the state term is assumed to be concave in the state. There are also direct and indirect final cost functionals. The direct cost functional is also concave in the state. The trajectories are generated by a representation operator that is continuous, and affine in the term representing the control operator. The author shows that this is general enough to cover dynamical systems of the form $\dot y(t)= B(t) y(t)+ b(t, u(t)).$ One of the important points leading to the proofs of these results, is that using the methods followed by the author, the optimal relaxed control function is a convex combination of relaxations of ordinary control functions, where the convexity parameters, $$\lambda_ i$$, are independent of $$t$$. The number of the $$\lambda_ i$$ is bounded above by the dimensions of the problem. The same linear relationship is inherited by the corresponding trajectories.
Results for existence without convexity by other authors are related to the ones in this paper. The author also applies these results to a variational problem with no dynamical system.

##### MSC:
 49J15 Existence theories for optimal control problems involving ordinary differential equations 49J27 Existence theories for problems in abstract spaces 49J45 Methods involving semicontinuity and convergence; relaxation
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