THE EQUIVALENCE OF EXTREMALS IN DIFFERENT REPRESENTATIONS OF UNBOUNDED CONTROL-PROBLEMS

Control problems defined by ordinary differential equations with right -hand sides that are unbounded functions of the control variables are considered. These problems can be reformulated in terms of bounded (re laxed or unrelaxed) differential inclusions by introducing a new indep endent variable (which is a function of the old state and control func tions). These differential inclusions can have different ''compact con trol'' representations depending on both the choice of the new indepen dent variable and on the different parametrizations of the set-valued right-hand sides. The extremals of different (relaxed or unrelaxed) '' compact control'' representations of such unbounded problems are compa red. It is proved that, for a representation that is Lipschitzian in t he state variables, the extremals corresponding to different choices o f the independent variable are in a one-to-one correspondence, with th e corresponding state functions having the same images. If different r epresentations that correspond to different choices of the independent variable and parametrization are compared, then the one-to-one corres pondence applies to the sets of ''Lojasiewicz extremals'' (that is, st ate functions that remain extremal for every control that generates th em) provided the representations are, except for a scalar factor, cont inuously differentiable in the state variables and satisfy certain ''n ondegeneracy'' conditions. The latter results rely heavily on a theore m of S. Lojasiewicz, Jr., on the equivalence of extremals, which we ge neralize in certain respects.