Abstract
The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a first-order necessary condition for a strong local minimum) using the local maximum principle (a first-order necessary condition for a weak local minimum). In the present work, we use this method to derive second-order necessary conditions for a strong local minimum on the base of such conditions for a weak local minimum. For simplicity, we confine ourselves to the consideration of the Mayer problem with endpoint equality and inequality constraints and control inequality constraints given by a finite number of twice smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide a rather short proof of second-order necessary conditions for a strong local minimum.
Highlights
The theory of second-order optimality conditions for different types of minima in optimal control (OC) is well-developed
The idea of the proof was completely based on the reduction in the original OC problem to a nonsmooth problem of Bolza type with the subsequent application of the necessary conditions for a strong local minimum in the latter. This approach was further developed in [14], where conditions of both the first and second order for a strong local minimum were obtained for an OC problem with state constraints, Pontryagin standard dynamics, a control constraint U (t) with closed values, and a finite number of endpoint constraints
We assume that all data are twice smooth and that the gradients of active control constraints are linearly independent at any point satisfying these constraints
Summary
The theory of second-order optimality conditions for different types of minima (strong, weak and the so-called Pontryagin) in optimal control (OC) is well-developed. It is associated with the names of Bonnans, Dmitruk, Frankowska, Hestenes, Ioffe, Malanowski, Maurer, Milyutin, Osmolovskii, Warga, Zeidan, and many others. We refer the interested reader to, e.g., [1,2,3,4,5] for historical comments and bibliographical remarks. Rather complete and advanced results were obtained by the Moscow group
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