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Second order sufficient conditions for time-optimal bang-bang control. (English) Zbl 1068.49015
This paper develops a second-order sufficient optimality condition for optimal control problems with control appearing linearly. Since second order sufficient optimality conditions amount to testing the positive definiteness of a certain quadratic form on the so-called critical cone or subspace, a well-known numerical recipe allows the conversion of the quadratic form to a perfect square under the strict Legendre-Clebsch condition. In spite of this device has been performed in a number of practical examples and has played a crucial role in sensitivity analysis of parametric control problems, it is not applicable to the optimal control problems considered in the present paper. Actually the time-optimal bang-bang controls with given initial and terminal state is studied here. Thus, the aim is to derive second order sufficient optimality conditions in a form that is also suitable for practical verification. This is carried out through a detailed study of the critical subspace and an adaptation of the Riccati approach to bang-bang controls.

49K15 Optimality conditions for problems involving ordinary differential equations
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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