Optimal control of differential inclusions
File(s)
Author(s)
Palladino, Michele
Type
Thesis
Abstract
The thesis concerns some recent advances on necessary conditions for optimal control
problems, paying particular attention to the case in which the velocity constraint is
expressed in terms of a multifunction.
In the first part of the thesis we have explored the link which arises between
relaxation and first order necessary conditions. Relaxation is a widely used regularization
procedure in optimal control, involving the replacement of velocity sets by
their convex hulls, to ensure the existence of a minimizer. It turns out that some
pathological situations arise in which the costs of relaxed and original problems do
not coincide (infimum gap conditions). In this case, we cannot obtain approximate
solution of the optimal control problem of interest. In particular, we show how necessary
conditions expressed in terms of Fully Convexified Hamiltonian Inclusion are
a↵ected by the presence of an infimum gap. Applications of these results are showed
also in the case in which the velocity constraint is expressed in terms of controlled
di↵erential equations.
In the second part of the thesis we study the regularity of the Hamiltonian
along the optimal trajectory for problems with state constraint. Two applications
of these properties are demonstrated. One is to derive improved conditions which
guarantee the nondegeneracy of necessary conditions of optimality, in the form of
a Hamiltonian inclusion. The other application is to derive new, less restrictive,
conditions under which minimizers in the calculus of variations have bounded slope.
problems, paying particular attention to the case in which the velocity constraint is
expressed in terms of a multifunction.
In the first part of the thesis we have explored the link which arises between
relaxation and first order necessary conditions. Relaxation is a widely used regularization
procedure in optimal control, involving the replacement of velocity sets by
their convex hulls, to ensure the existence of a minimizer. It turns out that some
pathological situations arise in which the costs of relaxed and original problems do
not coincide (infimum gap conditions). In this case, we cannot obtain approximate
solution of the optimal control problem of interest. In particular, we show how necessary
conditions expressed in terms of Fully Convexified Hamiltonian Inclusion are
a↵ected by the presence of an infimum gap. Applications of these results are showed
also in the case in which the velocity constraint is expressed in terms of controlled
di↵erential equations.
In the second part of the thesis we study the regularity of the Hamiltonian
along the optimal trajectory for problems with state constraint. Two applications
of these properties are demonstrated. One is to derive improved conditions which
guarantee the nondegeneracy of necessary conditions of optimality, in the form of
a Hamiltonian inclusion. The other application is to derive new, less restrictive,
conditions under which minimizers in the calculus of variations have bounded slope.
Version
Open Access
Date Issued
2015-02
Date Awarded
2015-07
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Vinter, Richard
Sponsor
European Commission
Grant Number
264735-SADCO
Publisher Department
Electrical and Electronic Engineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)