Metric regularity in optimal control

Abstract

The talk is devoted to certain extensions of the properties of strong metric regularity and sub-regularity of mappings, focusing on mappings associated with optimal control problems. Namely, for control constrained optimal control problems, the first order necessary optimality conditions can be recast as an inclusion 0 ∈ 𝐹(𝑦)(often in the form of a variational inequality), where 𝑦 includes the state, co-state and control variables, and 𝐹 is a mappingbetweensubsetsofBanachspaces(theso-called optimality mapping). Various types of regularity properties of the optimality mapping are of interest, but often the standard definitions of regularity have to be modified in order to capture in a relevant way the mappings into question. The regularity properties we discuss in the talk are the so called Strong Metric sub-Regularity (SMsR) and Strong Metric Regularity (SMR), where, however, two metrics are used in each of the domain and image spaces. Validity of these two regularity properties of the optimality mapping have been established for various optimal control problems. In the talk we focus on problems that satisfy the Legendre-Clebsch condition and onaffine optimal control problems, which will be considered separately. Twoapplications will be briefly presented: (i) existence of Lipschitz continuous optimal feedback control; (ii) convergence with error estimates of discretization and Newton-like methods for optimal control. The talk is partly based on results obtained in the papers given in the reference list below.