Seminari di Equazioni Differenziali e Applicazioni: Relaxed Optimal Control Problems - Optimal control problem with stratified state constraints

Lunedì 19 Maggio 2014, ore 11:30 - Aula 1BC45 - Michele Palladino - Cristopher Hermosilla



Lunedì 19 Maggio 2014 alle ore 11:30 in aula 1BC45, Michele Palladino (Università di Padova) terrà un seminario dal titolo "Relaxed Optimal Control Problems".

Relaxation is a regularization procedure used in optimal control, involving the replacement of velocity sets by their convex hulls, to ensure the existence of a minimizer. It can be an important step in the construction of sub-optimal controls for the original, unrelaxed, optimal control problem (which may not have a minimizer), based on obtaining a minimizer for the relaxed problem and approximating it. In some cases the infimum cost of the unrelaxed problem is strictly greater than the infimum cost over relaxed state trajectories; there is a need to identify such situations because then the above procedure fails. Following on from earlier work by Warga, we explore the relation between, on the one hand, non-coincidence of the minimum cost of the optimal control and its relaxation and, on the other, abnormality of necessary conditions (in the sense that they take a degenerate form in which the cost multiplier set to zero). For optimal control problems in which the dynamic constraint is formulated as a differential inclusion, we show that a local minimizer which is not also a relaxed local minimizer is an abnormal extremal, in the sense that it satisfies an abnormal form of the Hamiltonian inclusion in which the cost multiplier is zero. We also show that a relaxed local minimizer that is not also a local minimizer is a relaxed abnormal extremal. We discuss the extent to which the existence of an infimum gap is also manifested through the existence of abnormal extremals, also for optimal control problems in which the dynamic constraint is formulated as a differential equation with control.

A seguire, Cristopher Hermosilla (ENSTA Paris) terrà un seminario dal titolo "Optimal control problem with stratified state constraints".

This talk is mainly concerned with the problem of characterizing the value function for a class of optimal control problem with state constraints. The principal feature of the approach we present is that the state constraints are assumed to be decomposable into a locally finite number of smooth embedded submanifold. In particular, this allows us to treat problems on network and on sets with empty interior. The value function is then characterized in the proximal sense by means of an appropriated HJB equation which depends upon the stratification of the state constraints. This is a joint work with Hasnaa Zidani.

Rif. Int. M. Bardi, E. Feleqi, F. Ancona

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