### Optimal Control Problems with Mixed and Pure State Constraints

A. Boccia, M. D. R. de Pinho, R. B. Vinter
2016 SIAM Journal of Control and Optimization
This paper provides necessary conditions of optimality for optimal control problems, in which the pathwise constraints comprise both "pure" constraints on the state variable and "mixed" constraints on control and state variables. The proofs are along the lines of earlier analysis for mixed constraint problems, according to which Clarke's theory of "stratified" necessary conditions is applied to a modified optimal control problem resulting from absorbing the mixed constraint into the dynamics;
more » ... nto the dynamics; the difference here is that necessary conditions which now take into account the presence of pure state constraints are applied to the modified problem. Necessary conditions are given for a rather general formulation of the problem containing both forms of the constraints, and then these are specialized to problems having special structure. While combined pure state and mixed control/state problems have been previously treated in the literature, the necessary conditions in this paper are proved under less restrictive hypotheses and for novel formulations of the constraints. together with "complementary slackness" conditions. (These conditions are "restricted" in the sense that the Hamiltonian is maximized along the optimal state trajectoryx(.), not over U (t) but over the smaller set U (t) ∩ {u : φ 1 (t,x(t), u ) ≤ 0 and φ 2 (t,x(t), u ) = 0}.) Implicit in the derivation of this modified PMP is the notion that, under a suitable "constraint qualification," the mixed constraints can be absorbed into the dynamic constraint in such a way that the hypotheses on the dynamic constraint invoked in the original PMP continue to be satisfied; application of the original PMP to the reformulated optimal control problem, with an "absorbed" dynamic constraint, yields the desired, modified, necessary conditions. It might be thought that the theory of necessary conditions for pure state constraints "h(t, x(t)) ≤ 0" could simply be subsumed into that for mixed state constraints by setting φ 1 (t, x, u) = h(t, x). But this is not possible (if we are to use the proof techniques referred to above) because, when φ 1 (t, x, u) does not depend on u(.), the constraint qualification is violated, in consequence of which the mixed constraint cannot be used to eliminate control variable components and thereby generate an equivalent optimal control problem to which standard versions of the PMP can be applied. This is the reason why "pure" state constraints have been treated separately from "mixed control/state constraints," and it accounts for the fundamentally different nature of modifications to the PMP that have been derived for optimal control problems with pure state constraints: the modified PMP, in the pure state constraints case, is formulated in terms of a "measure multiplier" and a, possibly discontinuous, costate trajectory q(.) that is of bounded variation. Papers rigorously treating pure state constraints and involving a discontinuous costate trajectory originated in the 1960s and 1970 with the independent work of Dubovitskii and Milyutin [10] and Warga [20] ; for references to earlier literature, see, for example, [19] . Necessary conditions to cover the combined occurrence of the two types of constraints were derived by Dmitruk [11] and other members of the Dubovitskii-Milyutin school (see [12] for an overview of this work) and also by Makowski and Neustadt [14] . A breakthrough in the development of new tools for tackling a variety of differently structured optimal control problems, with nonsmooth data, was the publication of Clarke's paper [4] , the centerpiece of which was "stratified" necessary conditions for optimal control problems, whose dynamic constraint took the form of a differential inclusion. Clarke's paper introduced a new, and very useful, way of capturing MIXED AND PURE STATE CONSTRAINTS 3063 the requisite Lipschitz continuity-like properties of the differential inclusion for the validation of the Euler-Lagrange inclusion, and related necessary conditions, namely the "bounded slope" hypothesis. Subsequently, Clarke and de Pinho [5] examined the implications of these tools for "mixed constraint problems." The authors provided necessary conditions for very general formulations of mixed constraint problems by showing that these conditions could be reduced to the optimal control problems treated in [4], by absorbing the mixed constraints into the dynamic constraint in such a way that the bounded slope hypothesis continued to be satisfied. The results in [5] improve on many earlier-derived mixed constraint conditions, as described in detail in [5, sect. 8]-in some respects even when attention is restricted to problems with smooth data. However the presence of pure state constraints is excluded from the necessary conditions in [5, pp. 4503-4504]: . . . the bounded slope condition excludes unilateral state constraints. . . It is well-known that in the presence of such constraints, necessary conditions of the type given. . . fail, and that their appropriate extensions involve measures and adjoint arcs p that are discontinuous. The aim of our paper is to provide extensions of the necessary conditions for mixed control/state constraints problems of [5] to allow also for pure state constraints. A key tool is the set of stratified necessary conditions of [1], which generalize the main necessary conditions in Clarke's paper, to allow for unilateral state constraints. The proof technique for deriving necessary conditions and applying them to optimal control problems with both mixed control/state and pure state constraints is to reduce the problems to ones involving pure state constraints alone by absorbing the mixed constraints into the dynamic constraint and applying the stratified pure-state-constraint necessary conditions of [1]. The necessary conditions for combined mixed control/state and pure state constraints in this paper reduce to the main necessary conditions in [5] and [6] for mixed constraints alone (both in a general setting and when the mixed constraint has explicit representations including those of [6]), following removal of the pure state constraint. Clarke and de Pinho [5, sect. 8] give details of how their necessary conditions improve on earlier necessary conditions, with regard to hypotheses on the mixed state constraint data. These improvements are all the more evident in this paper, since the framework is broadened to include pure state constraints. On the other hand, this paper extends the earlier necessary conditions involving both mixed constraints and pure state constraints in [11] and [14] by allowing nonsmooth data, by adopting a very general formulation of the mixed constraints of the form (x(t), v(t)) ∈ S(t, w(t)) involving the controls (u = (v, w)) and states x in place of a collection of functional equality and inequality constraints, and by permitting general endpoint constraints. This paper treats only optimal control problems with mixed constraints that are "regular" in the sense that they separate into a constraint satisfying the bounded slope condition, a pure control constraint, and a pure state constraint. It fails to provide conditions for problems with mixed constraints that do not decompose in this way, for example, |x| 2 + |u| 2 ≤ 1. Necessary conditions for such problems have been studied by Milyutin and coworkers. For an overview, references to this work, and some open questions, see [12] . We mention that the "stratified" necessary conditions in this paper are expressed in terms of an arbitrary radius multifunction R(t), in place of the ballsẋ(t) + r(t) B involving the radius function r(t), as in [5] . This extra degree of generality in these necessary conditions for the general formulation of the optimal control problem of