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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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