The calculation of discriminating kernel based on viability kernel and reachability
Han and Gao Advances in Difference Equations
The calculation of discriminating kernel based on viability kernel and reachability
Yanli Han 0 1
Yan Gao 1
0 School of Mathematics and Information Science, Henan Polytechnic University , Jiaozuo, 454000 , China
1 School of Management, University of Shanghai for Science and Technology , 516 Jungong Road, Shanghai, 200093 , China
We discuss the calculation of discriminating kernel for the discrete-time dynamic game and continuous-time dynamic game (namely differential game) using the viability kernel and reachable set. For the discrete-time dynamic game, we give an approximation of the viability kernel by the maximal reachable set. Then, based on the relationship between viability and discriminating kernels, we propose an algorithm of the discriminating kernel. For the differential game, we compute an underapproximation of the viability kernel by the backward reachable set from a closed target. Then, we put forward an algorithm of the discriminating kernel using the relationship of the discriminating and viability kernels. This means that the victory domain can be computed because it is computed by the discriminating kernel. The novelty is that we give two algorithms of the discriminating kernel for a dynamic game that contains two control variables, not one control variable as in differential inclusion.
dynamic game; viability kernel; discriminating kernel; nonsmooth analysis; reachability
1 Introduction
As an important part of control theory, game theory pours attention into economics,
social, political science, and other behavioral sciences. Game theory aims to help us
understand situations in which decision makers interact. A dynamic game usually consists of two
players, the pursuer and the evader, with conflicting goals. Each player attempts to
control the states of the system so as to achieve his goal. Although dynamic games are closely
related to optimal control problems, there is a little difference between the two: there is a
single control input u(t) and a single criterion to be optimized in an optimal control
problem, and dynamic game theory generalizes this to two control inputs u(t), v(t) and two
criteria. In [], quantitative and qualitative differential game problems are discussed using
set-valued analysis and viability theory. In the case of a two-player differential game, the
value function is computed by determining the discriminating kernel for the game. In [],
a two-player zero-sum differential game with incomplete information on the initial state
is investigated. In [], a two-player zero-sum differential game with infinitely many
initial positions and without Isaacs condition is proposed. By optimal transportation theory
and stochastic control, there exists a value of the game with such random strategies. In
[], a bounded discriminating domain for linear pursuit-evasion differential game is
studied. For a constraint set K , the discriminating kernel Disc(K ) is the largest subset of the
discriminating domain K .
Viability theory is used to study stability, reachability, and dynamic games. The research
of such questions for differential inclusions has started with the pioneering works of Aubin
[]. A presentation of viability kernels and capture basins of a target viable in a constrained
subset satisfying tangential conditions or duality and normal conditions is provided in [].
In [], a method to construct viability kernels is given. In [], an algorithm suited to the
identification of specific trajectories or to the computation of viability kernels associated
with delayed dynamics is proposed. In [], based on the proximal normal cone, the method
to verify the viability of approximate viable set for continuous-time and discrete-time
linear systems is given. The problem of viable controller design is formulated as a problem of
linear inequalities. In [], an algorithm that computes the approximating viability kernel
of a discrete-time system is proposed. In [], it is shown that determining the viability
of a polytopic set expressed by a convex hull of finitely many points can be transformed
into verifying the viability criteria at vertices without the assumption that the input set is
a polytope, which is needed in the existing criteria.
Reachability analysis is an essential problem of control systems. The goal of
reachability analysis is to compute the set of reachable states in the state space for a given model
and a set of initial states. In [], the notions of maximal and minimal reachability are
introduced. The reachability analysis of a linear control system is discussed in []. The
main contribution is that its sets of initial states and inputs are given by arbitrary convex
compact sets represented by their support functions. In [], an efficient and scalable
maximal reachability technique to compute the continual reachable set is introduced. At the
same time, an approximation of this set based on ellipsoidal techniques (...truncated)