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Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls

Abstract : We survey the results on no-gap second order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs [31] and [40], and discuss their further develop-ment. First, we formulate such conditions for broken extremals in the simplest prob-lem of the Calculus of Variations and then, we consider them for discontinuous con-trols in optimal control problems with endpoint and mixed state-control constraints, considered on a variable time interval. Further, we discuss such conditions for bang-bang controls in optimal control problems, where the control appears linearly in the Pontryagin-Hamilton function with control constraints given in the form of a con-vex polyhedron. Bang-bang controls induce an optimization problem with respect to the switching times of the control, the so-called Induced Optimization Problem. We show that second-order sufficient condition for the Induced Optimization Problem together with the so-called strict bang-bang property ensure second-order sufficient conditions for the bang-bang control problem. Finally, we discuss optimal control problems with mixed control-state constraints and control appearing linearly. Tak-ing the mixed constraint as a new control variable we convert such problems to bang-bang control problems. The numerical verification of second-order conditions is illustrated on three examples.
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Nikolai Osmolovskii, Helmut Maurer. Second-Order Optimality Conditions for Broken Extremals and Bang-Bang Controls. FPG 13, 2013, Krakow, Poland. ⟨hal-01101295⟩

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