Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Abstract

Sampled-data control systems have steadily been gaining interest for their applications in Engineering and Automation. In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. In particular we obtain a nonpositive averaged Hamiltonian gradient condition associated to an adjoint vector being a function of bounded variations. As a well-known challenge, theoretical and numerical difficulties may arise in optimal permanent control problems with state constraints due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the trajectory in contact with the boundary of the restricted state space). However, in our case of sampled-data controls, we find that, under certain general hypotheses, the optimal trajectory only contacts the running inequality state constraints at most at the sampling times. In that context the adjoint vector only experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Hence, taking advantage of this bouncing trajectory phenomenon, we are able to implement a numerical indirect method which we use to solve three simple examples.

Pontryagin maximum principle for state constrained optimal sampled-data control problems on time scales

“…Finally a maximal velocity minimization problem for the harmonic oscillator with sampled-data control is numerically solved in Section 4.2 using a shooting method based on the necessary conditions provided in Theorem 3.7. As in the recent work [18], a bouncing trajectory phenomenon is observed (see Figs. 1 and 2).…”

“…2.6). More recently Bourdin and Dhar have extended in [18] the previous work to the state constrained case. In that context the authors have observed and studied a particular behavior of the trajectories with respect to the state constraints, called bouncing trajectory phenomenon.…”

“…Inherent to this behavior, the singular component of the costate vector vanishes and its discontinuities are reduced to a finite number of jumps, which turns out very useful for numerical simulations based on shooting methods. We refer to Sections 4 and 5 of [18] for details and discussion. We conclude this paragraph by mentioning the work [17] in which the optimization of the sampling times (in the state constraint-free case) has also been explored in view of a PMP formulation.…”

“…and with inequality state constraints (see, e.g., [18]); -general optimal sampled-data control problems on time scales without inequality state constraint (see, e.g., [21]).…”

“…Note that the optimal trajectory drawn in Figure 1 exhibits a bouncing trajectory phenomenon. This particular behavior has recently been highlighted and studied in [18], which deals with a class of state constrained continuous-time optimal sampled-data control problems. In that context, under some hypotheses (see [18], Sect.…”