The Maximum Principle for Problems with Pathwise Constraints

This chapter provides necessary conditions of optimality for dynamic optimization problems involving pathwise constraints. Attention is directed at problems in which the dynamic constraint takes the form of a controlled differential equation. Two kinds of problems are considered. In the first kind, the pathwise constraint is imposed on the state trajectories (the ‘pure state constraint’ problem) and, in the second kind, it is imposed on both state trajectories and control functions (the ‘mixed constraint’ problem). For each kind, the necessary conditions resemble the maximum principle, but modified to take account of the pathwise constraint via additional Lagrange multipliers. Concerning pure state constraint problems, we consider pathwise scalar functional inequality constraints. It might seem restrictive to limit attention to a scalar inequality constraint. But the constraint function, which is assumed merely to be upper semi-continuous and Lipschitz continuous w.r.t. to the state variable, can be constructed to embrace vector inequality constraints, set inclusion constraints and other constraints of interest. Simple examples illustrate that it is not possible, in general, to accommodate a pure state constraint by means of a simple integrable Lagrange multiplier. Instead we must do so, using a measure multiplier. The presence of a measure multiplier gives rise to a measure driven differential inclusion (differential equation in the smooth case) for the co-state trajectory, with discontinuous solutions. The discontinuous nature of the co-state trajectory is somewhat disguised in the standard formulations of the necessary conditions such as those given in this chapter, because these conditions are expressed in terms of a modified, absolutely continuous co-state trajectory, obtained by subtracting off the singular component from the ‘true’ co-state trajectory. Dealing with measure multipliers (ensuring stability of key relations under perturbations, for example) required additional analytic tools. These are provided in a separated section near the beginning of the chapter. The derivation of necessary conditions is based on introducing an integral penalty term relating to the pathwise constraint. With the help of Ekeland’s theorem, we find a minimizer to a perturbed version of the penalized problem, close to the original minimizer. The necessary conditions for the perturbed problem are then interpreted as a perturbation version of the desired conditions. Passage to the limit, as the parameters controlling the perturbations vanish, completes the derivation. We also address the degeneracy issue associated with necessary conditions for pure state constraint problems. This is connected with the fact that, for certain problems of interest, in which the pure state constraint is active at either of the end-times, the standard necessary conditions are trivial, in the sense that they are satisfied by all admissible state trajectory/control pairs. Following on from work of Arutyunov and Aseev, we show, under additional hypotheses, that this form of degeneracy can be eliminated. We adopt a general formulation of the mixed state constraint problem which captures, as special cases, other formulations appearing in the literature, including formulations involving combined functional inequality/equality constraints and set inclusion relations. We seek necessary conditions in which, typically, the pathwise constraint is accommodated by an absolutely continuous Lagrange multiplier. The hypotheses we must impose to justify employing multipliers of this nature, which require some kind of stability of the set of controls satisfying the mixed constraint as the state varies, are violated for pure state constraint problems. Thus necessary conditions for mixed constrained problems do not subsume those for pure state constraints. Following Clarke and de Pinho, we derive conditions for mixed constraint problems by introducing a differential inclusion which incorporates the control constraint, the mixed constraint and the dynamic constraint. The desired necessary conditions are obtained by applying the generalized Euler Lagrange conditions of Chap. 8 to the resulting differential inclusion problem. We consider also formulations of both pure state and mixed constraint problems, in which the end-times are included among the choice variables (‘free time’ problems). In each case, we provide the extra necessary conditions associated with the free end-times.