Backward-forward reachable set splitting for state-constrained differential games

Abstract

This paper is about a set-based computing method for solving a general class of two-player zero-sum Stackelberg differential games. We assume that the game is modeled by a set of coupled nonlinear differential equations, which can be influenced by the control inputs of the players. Here, each of the players has to satisfy their respective state and control constraints or loses the game. The main contribution is a backward-forward reachable set splitting scheme, which can be used to derive numerically tractable conservative approximations of such two player games. In detail, we introduce a novel class of differential inequalities that can be used to find convex outer approximations of these backward and forward reachable sets. This approach is worked out in detail for ellipsoidal set parameterizations. Our numerical examples illustrate not only the effectiveness of the approach, but also the subtle differences between standard robust optimal control problems and more general constrained two-player zero-sum Stackelberg differential games.

On the distributed backward reachability problem for large scale systems

“…In what follows, we briefly provide an idea on how to cast the theory for this case, while a thorough study will be provided in a dedicated paper. First of all, dynamics (8a) take the form (20) with matrices A ij , B ij and E i of suitable dimensions, and with disturbance d i (t) ∈ D i ⊂ R vi belonging to a bounded polytope. Also, X i and U i in (8c) are polytopes while for (8b) inequality constraints L il (⋅) ≤ 0 will be adopted, with L il (⋅) linear affine expressions.…”

“…The reader interested in fundamental results about reachability analysis is referred to [17,23,8,7,6,34,42,2]. In particular, reachability has been studied and exploited as a fundamental tool for evaluating or enforcing state space invariance of control systems [8,27,9,41,28] or reach-avoid set control and differential games [34,18,30,10,20]. A renewed interest in the (both forward and backward) reachability problem is witnessed by more recent literature too, where this tool is exploited in the derivation of dynamical systems abstraction techniques and symbolic control approaches for the verification of fundamental properties such as safety or for the enforcement of formal logics specifications [44,39,46,47,49,48,37,45,16,11,31,32].…”

Approximate Reachability for Koopman Systems Using Mixed Monotonicity

“…This forms a 'tube' of trajectories starting from the initial conditions on the state space. Such reachability analysis has been used in optimal control [2], safety-critical systems [3], autonomous navigation [4], cybersecurity [5], and differential games [6].…”

“…The reachability problem is solved in one of three ways: Hamilton-Jacobi (HJ) reachability, verification methods, and numerical approximation methods. Optimal control or game theoretic formulations of reachability often use HJ partial differential equations [2], [6], [7]. HJ reachability often deals with the computation of reach (or avoid, or reach-avoid) sets as level sets to get games of degree from the initial HJ formulation of games of kind [1], [8].…”