Free End-Time Problems

This chapter provides necessary conditions of optimality for free end-time dynamic optimization problems, that is problems in which the left and right end-times are included among the choice variables. Minimum time problems, in which the aim is to drive the state from an initial state to a target set in the state space in minimum time, are important examples of such problems. We shall see that the earlier derived necessary conditions for fixed end-point problems (Clarke’s nonsmooth maximum principle when the dynamic constraint is a controlled differential equation and either the generalized Euler Lagrange condition or a condition expressed in terms of the Hamiltonian for differential inclusion problems) can be supplemented by extra conditions to take account of the enlargement of the set of choice variables to include the end-times. It turns out that these extra conditions are boundary conditions on the Hamiltonian evaluated along the minimizing state and co-state trajectories. But here we have a problem because, for problems with measurable time dependence, the Hamiltonian is only almost everywhere defined and conditions involving point evaluation of the Hamiltonian at the optimal end-times require interpretation. The necessary conditions featuring in this chapter fall into two groups. For problems in the first group, the data is assumed to be Lipschitz continuous w.r.t. time; in this situation the Hamiltonian (evaluated along the minimizing state and co-state trajectories) is Lipschitz continuous. So the boundary conditions on this function can be interpreted in the obvious way. For this group of problems, necessary conditions (including the classically interpreted extra boundary conditions on the Hamiltonian), can be derived by using a family of re-parameterizations of the time variable to reduce the free end-time problem to a fix end-time problem, to which we apply fixed end-time necessary conditions from earlier chapters. For problems in the second group we allow the data to be merely measurable w.r.t. time. Here it is still possible to derive free end-time necessary conditions. But now the boundary conditions are interpreted as conditions involving the ‘essential values’ (strictly speaking, the super and sub essential values) of the Hamiltonian. Essential values associated with a given function are set-valued functions that reduce to a point-valued function coinciding with the original function, when the original function is continuous. They provide a means of interpreting boundary conditions on the Hamiltonian, because essential values are invariant under changes to the original function on a nullset. To prove the necessary conditions, for this group of problems, we build up the proofs in stages, providing proofs of the necessary conditions under progressively less stringent hypotheses. A simple variation of the end-times in the first state establishes the required boundary condition (expressed in terms of essential values). Robustness properties of essential values then ensure that these boundary conditions are retained as we progress through all the stages. In this chapter we make use of new, refined concepts of ‘essential value’ that allow for more precise forms of the necessary conditions than appears in the earlier literature.