Stabilizability in optimal control

Abstract

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with (p0, W)regulated cost, for some continuous, state-dependent function W and some constant p0 > 0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W (z)/p0. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W , called p0-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a p0-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with (p0, W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

Stabilizability in optimization problems with unbounded data

“…Assume that f , l satisfy (H.1-2). Then a locally bounded feedback K : R n \ C → U sample stabilizes the original problem (14)- (15) to C with (p 0 , W )regulated cost for some p 0 ≥ 0, if and only if it sample stabilizes the rescaled problem (16)- (17) to C with (p 0 , W )-regulated cost.…”

“…For every z ∈ R n \ C with d(z) ≤ R and every partition π = (t k ) of [0, +∞) with diam(π) ≤ δ, let us consider an arbitrary π-sampling process (x 0 , x, u) from z for the original problem (14)- (15). Let [0, T − ) be the maximal definition interval of x and let us definê…”

“…Following [15], we define Euler trajectories and costs as locally uniform limits of sampling trajectories and costs.…”

“…Sufficient stabilizability conditions in optimal control. In this section we provide sufficient conditions for the sample, Euler, and weak Euler stabilizability with regulated cost of the original problem (14)- (15). Such conditions rely on the existence of a p 0 -Minimum Restraint function.…”

“…Minimum Restraint functions were first introduced in [19], where the existence of a function of this type was shown to guarantee global asymptotic controllability to a set, with regulated cost (see also [17]). The problem of defining a stabilizing feedback law with regulated cost through the use of a Minimum Restraint function has been addressed only recently in [15], just in the case of bounded data. The extension to unbounded dynamics is not achieved by refining the techniques already used.…”