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This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint <italic>G</italic>(<italic>u</italic>) ≤ 0, given by a <italic>C</italic>2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set <italic>U</italic>, defined by the inequality <italic>G</italic>(<italic>u</italic>) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient <italic>H</italic><italic>u</italic> of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.
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This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0, given by a C2-function. A similar second-order condition was obtained earlier by the author for a strong minimum in a much more general problem. In the present paper, we would like to take a narrower perspective than before and thus provide shorter and simpler proofs. In addition, the paper uses the first and second order tangents to the set U, defined by the inequality G(u) ≤ 0. The main difficulty of the proof, clearly shown in the paper, refers to the set, where the gradient Hu of the Hamiltonian is small, but the condition of quadratic growth of the Hamiltonian is satisfied. The paper can be valuable for self-explanation and provides a basis for extensions.

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Control and Cybernetics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

This paper is devoted to a sufficient second-order condition for a weak local minimum in a simple optimal control problem with one control constraint G(u) ≤ 0,...

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Publié en ligne: 12 août 2022

Pages: 151 - 169

Reçu: 01 févr. 2022

Accepté: 01 avr. 2022

DOI: https://doi.org/10.2478/candc-2022-0012

Mots clés
critical cone, quadratic form, first and second order tangents, second order optimality condition, weak local minimum, inequality control constraint, Pontryagin’s maximum principle

© 2022 Nikolai P. Osmolovskii, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A second-order sufficient condition for a weak local minimum in an optimal control problem with an inequality control constraint

Publié en ligne: 12 août 2022

Pages: 151 - 169

Reçu: 01 févr. 2022

Accepté: 01 avr. 2022

DOI: https://doi.org/10.2478/candc-2022-0012

Mots cléscritical cone, quadratic form, first and second order tangents, second order optimality condition, weak local minimum, inequality control constraint, Pontryagin’s maximum principle

© 2022 Nikolai P. Osmolovskii, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
critical cone, quadratic form, first and second order tangents, second order optimality condition, weak local minimum, inequality control constraint, Pontryagin’s maximum principle