Weierstrass–Erdmann type conditions for a broken extremal in the calculus of variations and optimal control

Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

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Received: 20 July 2025
Accepted: 15 April 2026

Abstract

Necessary first-order conditions for a strong local minimum in the simplest problem of the calculus of variations, as well as in the Mayer-type optimal control problem with a control constraint u ∈ U are discussed. In the simplest problem, these conditions pertain to the break point of the extremal, and in the Mayer problem, to the control discontinuity point. Like the classical Weierstrass-Erdmann conditions, our conditions follow from the Weierstrass condition (respectively, from the Pontryagin maximum principle). In the simplest problem, a piecewise smooth extremal with a single discontinuity point of the derivative is considered. According to the Weierstrass-Erdmann conditions, a jump in the derivative is possible only if a non-strict inequality in the Weierstrass condition is satisfied, namely, at the point where this inequality is realized as an equality. The same is true for a control jump and the maximum principle. All possible cases of this situation are analyzed, including "control jumps" at the ends of the time interval. The obtained conditions are new both for the calculus of variations and for optimal control.