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 Milyutin and Osmolovskii (Calculus of Variations and Optimal Control. Translations of Mathematical Monographs. American Mathematical Society, Providence, 1998) and Osmolovskii and Maurer (Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM Series Design and Control, vol. DC 24. SIAM Publications, Philadelphia, 2012), and discuss their further development. First, we formulate such conditions for broken extremals in the simplest problem of the Calculus of Variations and then, we consider them for discontinuous controls 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 convex 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 ensures 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. Taking 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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