Abstract
A two-dimensional optimal control problem is considered with a performance index depending upon a scalar parameter. For the zero value of the parameter one has the well known Fuller problem with the chattering phenomenon involving infinitely many switching points in finite time (Fuller, 1963). A maximum principle based preliminary analysis shows that such a regime takes place up to a certain critical value of the parameter, after which one has a two-switching regime involving a first order singular arc. The optimality of the above mentioned regimes is proved using dynamic programming. A group-invariant analysis of the Bellman equation, similar to that in (Wonham, 1963) reveals the structure of the smooth Bellman function involving several unknown constants which can be found numerically from an algebraic system of equations. From the same system one gets the coefficients of the switching, curve which consists of two generally nonsymmetric half-parabolas, as in (Marchall, 1975).
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