Strong Local Optimality for Generalized L1 Optimal Control Problems

Abstract

In this paper, we analyze control affine optimal control problems with a cost functional involving the absolute value of the control. The Pontryagin extremals associated with such systems are given by (possible) concatenations of bang arcs with singular arcs and with zero control arcs, that is, arcs where the control is identically zero. Here, we consider Pontryagin extremals given by a bang-zero control-bang concatenation. We establish sufficient optimality conditions for such extremals, in terms of some regularity conditions and of the coerciveness of a suitable finite-dimensional second variation.

Singular extremals inL1-optimal control problems: sufficient optimality conditions

“…This section is devoted to its proof. The proof of the strong local optimality of the reference trajectory follows the same lines of ( [10], Thm. 4.1) (see also [30]), thus we are just recalling the main arguments.…”

“…Indeed, as already stressed, the flow generated by the maximised Hamiltonian, appearing in (3.2a)-(3.2b), depends on how many times the function ψ changes sign along the reference trajectory, as each of these points bears a non-smoothness point. This issue has been accurately treated in [10], and the same computations carried out there could extend with no modifications to the current problem. Thus, due to the complexity introduced by the presence of a singular arc, and in order to simplify the presentation, we make the following assumption.…”

“…We stress that this assumption does not cause any loss of generality; the result (Thm. 7.1) holds true also if we drop it, provided that the other assumptions are satisfied and that the second variation and the maximised flow are suitably computed, according to the rules given in [10].…”

“…L 1 -minimisation problems, that is, optimal control problems aiming to minimise the L 1 norm of the control, have shown to model quite effectively fuel consumption optimisation problems in engineering [7,8,26,27] and some systems appearing in neurobiology [5]. In a recent paper dealing with the problem of minimising the fuel consumption of an academic vehicle [6], a generalised version of this problem is studied: the cost to be minimised is the absolute work, modelled as the integral of the absolute value of the control, weighted by the absolute value of a function dependent on the state (such kind of problems have been called generalised L 1 optimal control problems in [10]).…”

“…They show that, according to the values of the parameters, the optimal extremal trajectories are given by the concatenations of bang-zero-bang arcs or of bang-singular-zero-bang arcs. Inspired by this result, we look for sufficient optimality conditions for such extremals: in [9,10], we focus on extremals made by a concatenation of bang-zero-bang arcs; here we consider the case of a concatenation of bang-singular-zero-bang arcs and provide an adequate set of sufficient conditions.…”