Numerical Construction of Viable Sets for Autonomous Conflict Control Systems

Abstract

A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. N. Krasovskii and A. I. Subbotin. The viability kernel is constructed as the limit of sets generated by a Pontryagin-like backward procedure. This method is implemented in the framework of a level set technique based on the computation of limiting viscosity solutions of an appropriate Hamilton-Jacobi equation. To fulfill this, the authors adapt their numerical methods formerly developed for solving time-dependent Hamilton-Jacobi equations arising from problems with state constraints. Examples of computing viability sets are given.

Aircraft Control During Cruise Flight in Windshear Conditions: Viability Approach

“…Property 3 is the consequence of the previous two properties. The first part of property 4 is proved in [9], and the last part follows from the continuous dependency of the solution sets of the inclusions (29) and (30) on the initial state and from the condition G = int G. The last two properties are true due to the following climes: the minimax u-stability property implies the maximin one, and they are equivalent if the saddle point condition holds.…”

“…Property 1 follows from properties of the differential inclusions (29) and (30); see, e.g., [9] for the proof. Property 2 immediately follows from the definition of u-stability.…”

“…The proposition deals with both types of u-stability. Proposition 1 (See [9] for the proof) Let W be a maximal u-stable subset of…”

“…The rate of the angle of attack, α K , and the rate of the thrust setting, δ T , are considered as controls, whereas W x and W h are regarded as disturbances; see formula (9). The following constraints are imposed on the controls and disturbances:…”

“…In other words, the second player uses the so-called feedback counter-strategies. The value function of this game is computed using a stable grid algorithm reported in [9,10]. Each level set of the value function converges to a viability (discriminating) kernel as the backward time tends to infinity.…”