DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 2, Number 3, Xxxx 1996 pp. 297–305
CONTINUITY OF ADMISSIBLE TRAJECTORIES FOR STATE CONSTRAINTS CONTROL PROBLEMS
M. Arisawa and P.-L. Lions
CEREMADE , URA CNRS 749, Universit´e Paris-Dauphine Place du Marchal de Lattre de Tassigny
75775 Paris Cedex 16, France
I. Introduction and Main ResultWe consider here deterministic control prob-lems with state constraints that is probprob-lems where admissible controls are those which keep the state of the system in a given region for all times. We prove that, given an initial state and an admissible control for that state, it is possible to construct admissible controls for all initial states such that the control and the corresponding trajectory are Lipschitz (in convenient norms) with respect to the initial condition.
In order to be more specific, we need some notation. Let C be the set in which the state of the system is required to stay. We assume that C is the closure of a bounded smooth (C1,1 for instance...) open set Ω in IRN
: various extensions are possible that we shall briefly mention but we insist on that case in order to simplify the presentation. The state of the controlled system is the solution of the following ordinary differential equation
˙
X = g(X, a) for t ≥ 0 , X(0) = x ∈ C , (1) where g(x, a) ∈ C(IRN × A) is Lipschitz continuous in x ∈ IRN uniformly in a ∈ A and A is a compact metric space representing the set of possible values of the controls. In equation (1), a denotes an arbitrary measurable function from [0, ∞) into A and is the control. Obviously, given a and x, the state of the system X ∈ C([0, ∞); IRN) is uniquely determined by (1). We now define admissible controls for an initial condition x in C by the set denoted by Axof those controls a for which the
corresponding solution X remains in C for all t ≥ 0Ax =
n
a(t) meas.
a(t) ∈ A a.e , X(t) ∈ C for all t ≥ 0o. We shall always assume that the following natural condition holds
∃ ν > 0 , ∀x ∈ ∂Ω , ∃ a ∈ A , g(x, a) · n(x) < −ν < 0 (2) where n denotes the unit outward normal. This condition, as is well-known - see for instance H.M. Soner [12] -, implies that Ax6= φ for all x ∈ C.
Our main result, stated below, states that if y ∈ C , b ∈ Ay and Y denotes the
corresponding optimal trajectory, then we can find, for all > 0 and for all x ∈ C, some admissible control a ∈ Ax such that we have
|X(t) − Y (t)| + Z t
0
ω(a(s), b(s))ds ≤ C eλ0t( + |x − y|), for all t ≥ 0. (3)