Second Order Optimality Conditions for Controls with Continuous and Bang-Bang Components
Second order necessary and sufficient optimality conditions for bang-bang control problems in a very general form have been obtained by the first author. These conditions require the positive (semi)-definiteness of a certain quadratic form on the finite-dimensional critical cone. In the present paper we formulate a generalization of these results to optimal control problems where the control variable has two components: a continuous unconstrained control appearing nonlinearly and a bang-bang control appearing linearly and belonging to a convex polyhedron. Many examples of control of this kind may be found in the literature.
keywordsbang-bang control Pontryagin minimum principle second order necessary and sufficient conditions critical cone quadratic form strengthened Legendre condition
- H. Maurer, N.R. Osmolovskii. Second order optimality conditions for bang-bang control problems. Control & Cybernetics 32:555–584, 2003.Google Scholar
- N.R Osmolovskii. High-order necessary and sufficient conditions for Pontryagin and bounded-strong minima in the optimal control problems. Dokl Akad. Nauk SSSR, Ser. Cybernetics and Control Theory 303: 1052–1056, 1988; English transl., Sov. Phys. Dokl. 33, N. 12:883–885, 1988.MATHMathSciNetGoogle Scholar
- N.R Osmolovskii. Second order conditions for broken extremal. In: Calculus of variations and optimal control. (Technion 1998), A. Ioffe, S. Reich and I. Shafir, eds., Chapman and Hall/CRC, Boca Raton, Florida, 198–216, 2000.Google Scholar