The Equivalence of Extremals in Different Representations of Unbounded Control Problems

Document Type


Publication Date



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 (relaxed or unrelaxed) differential inclusions by introducing a new independent variable (which is a function of the old state and control functions). These differential inclusions can have different “compact control” representations depending on both the choice of the new independent 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 compared. It is proved that, for a representation that is Lipschitzian in the state variables, the extremals corresponding to different choices of the independent variable are in a one-to-one correspondence, with the corresponding state functions having the same images. If different representations that correspond to different choices of the independent variable and parametrization are compared, then the one-to-one correspondence applies to the sets of “Lojasiewicz extremals” (that is, state functions that remain extremal for every control that generates them) provided the representations are, except for a scalar factor, continuously differentiable in the state variables and satisfy certain “nondegeneracy” conditions. The latter results rely heavily on a theorem of S. Lojasiewicz, Jr., on the equivalence of extremals, which we generalize in certain respects.