×

Strong optimality for a bang-bang trajectory. (English) Zbl 1020.49021

Summary: We give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the \(C^{0}\) topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.

MSC:

49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49K15 Optimality conditions for problems involving ordinary differential equations
58E25 Applications of variational problems to control theory
PDFBibTeX XMLCite
Full Text: DOI