Optimal control of differential inclusions
Michele Palladino · Spiral (Imperial College London) · 2015
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.