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Transformation of quadratic forms to perfect squares for broken extremals. (English) Zbl 1050.49016
Summary: In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set $$\Theta$$ of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to the corresponding Riccati equation satisfying some jump condition at each point of the set $$\Theta$$, then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.

##### MSC:
 49K15 Optimality conditions for problems involving ordinary differential equations
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