Neighboring feasible trajectories and value functions for state constrained control problems

¹ Istituto Nazionale di Alta Matematica “F. Severi”, Unità di ricerca presso l’Università di Padova, Italy
² Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, via Trieste 63, 35121 Padova, Italy

^* Corresponding author: colombo@math.unipd.it

Received: 22 April 2024
Accepted: 12 August 2025

Abstract

The inward pointing condition (IPC) for a control system whose state x is constrained in a smooth set C requires that at each point of the boundary of C the intersection between the dynamics and the interior of the tangent cone of C at x be nonempty. Thanks to IPC, for every system trajectory x̅(·) on an interval [0, T], possibly violating the constraint by an amount d, one can construct a new system trajectory x(·) that both satisfies the constraint and whose distance from x̅(·) is bounded by a quantity proportional to d. When IPC does not hold, the construction of such a trajectory is not possible in general. This paper is devoted to a state constrained dynamics that is affine and symmetric with respect to the control and possibly contains an uncontrolled drift, that is not too large with respect to the controlled vector fields. We formulate an inward pointing condition involving Lie brackets of the dynamics’ vector fields and, by implementing a suitable “rotating” control strategy, we construct a constrained trajectory whose distance from the reference one x̅ is bounded by a quantity proportional to √d. As an application, we establish the continuity up to the boundary of the value function of related optimal control problems with finite and infinite horizon. The main result also removes two assumptions that were used in the preceding paper [Colombo et al., SIAM J. Control Optim. 60 (2022) 3326–3357] and allows to prove, in the case of a driftless dynamics, that the mere inward pointing conditions involving the Lie bracket is enough to obtain the continuity of value functions.