Jacobi conditions and the Riccati equation for a broken extremal
type: Item
instance of: scholarly article
title: Jacobi conditions and the Riccati equation for a broken extremal
zbMATH Open document ID: 0968.49016
DOI: 10.1007/BF02673843
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-11-07T00:00:00Z
review text: Normally, the classical computation of variations dealing with second order necessary and sufficient condition is used for the case of smooth extremal. In the paper there is proved that the classical second-order conditions formulated as a conjugate point and Riccati equation may be generalized to the case of broken extremal. The subject of this work is the following simple controlled object: \[ \dot{x}(t)=u(t), \text{ almost everywhere} \] \[ x(0)=b_{0}, \quad x(T)=b_{1} \] \[ (x(t), u(t),t) \in Q, \text{ almost everywhere} \] where \(x : [0, T] \rightarrow \mathbb R^n\) is an absolutely continuous function, \(x(\cdot)\in W^{1,1}\left([0,T], R^{n}\right)\) and \(u(t):[0,T]\rightarrow \mathbb R^n\) is bounded and measurable, \(u(t) \in L^{\infty}\left([0,T], \mathbb R^n\right)\). The results, although received in classical theoretical way, may be used in practical applications of optimal control theory.
reviewed by: Władysław Hejmo
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 1526520
zbMATH Keywords: calculus of variations
zbMATH Keywords: optimal control theory
zbMATH Keywords: second-order conditions
zbMATH Keywords: conjugate point
zbMATH Keywords: Riccati equation
zbMATH Keywords: broken extremal
author: Nikolai P. Osmolovskii
author: Frank Lempio
MaRDI profile type: Publication
cites work: Q4129369
cites work: Jacobi-type conditions for the problem of Bolza with inequalities
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cites work: Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment)
cites work: A NECESSARY AND SUFFICIENT CONDITION FOR A STRONG EXTREMUM IN A DEGENERATE PROBLEM OF THE CALCULUS OF VARIATIONS
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description: scientific article; zbMATH DE number 1526520