# Transformation of quadratic forms to perfect squares for broken extremals

N. P. Osmolovskii, F. Lempio:
**Transformation of quadratic forms to perfect squares for broken extremals**

*Set-Valued Analysis*
**10**
(2-3),
209 - 232,
2002

DOI:
10.1023/A:1016588116615

MR Nummer:
1926381

Zentralblattnummer:
1050.49016

Keywords: broken extremals, perfect squares, Riccati equation, sufficient optimality conditions.

Mathematics Subject Classification Code: 49K15

## Abstract:

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 Θ 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 corresponding Riccati equation satisfying some jump condition at each point of the set Θ, 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.

## Contents:

1. | Introduction |

2. | The Simplest Problem of the Calculus of Variations |

3. | General Problem of the Calculus of Variations |