Jacobi conditions and the Riccati equation for a broken extremal
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.
scientific article; zbMATH DE number 1526520