Abstract
For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.
Highlights
In [1], we studied the following nonparametric calculus of variations problem, denoted by ( P), which consists in minimizing a functional of the form
The main novelty of the work in [24] is precisely the removal of continuity of the proposed optimal controls in the main sufficiency theorem of that paper
The main properties of the new sufficiency theorems of this paper can be outlined as follows: given an admissible process which needs to be neither continuous nor piecewise continuous but only essentially bounded, the pieces of the new sufficiency results of this article are two crucial first-order sufficient conditions involving the Hamiltonian of the problem, the classical transversality condition, an essential symmetric inequality which arises from the properties of the original algorithm used to prove the main theorem of the article, a similar condition of the necessary condition of Legendre–Clebsch, the positivity of the second variation on a cone of critical directions, and three conditions involving some Weierstrass excess functions
Summary
In [1], we studied the following nonparametric calculus of variations problem, denoted by ( P), which consists in minimizing a functional of the form. The main novelty of the work in [24] is precisely the removal of continuity of the proposed optimal controls in the main sufficiency theorem of that paper This proof has been generalized in [25] to optimal control problems containing equality restrictions depending on the controls and on the time and the states. The main properties of the new sufficiency theorems of this paper can be outlined as follows: given an admissible process which needs to be neither continuous nor piecewise continuous but only essentially bounded, the pieces of the new sufficiency results of this article are two crucial first-order sufficient conditions involving the Hamiltonian of the problem, the classical transversality condition, an essential symmetric inequality which arises from the properties of the original algorithm used to prove the main theorem of the article, a similar condition of the necessary condition of Legendre–Clebsch, the positivity of the second variation on a cone of critical directions, and three conditions involving some Weierstrass excess functions.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
