This paper proposes a novel algorithm that characterizes the robust capture basin and the discriminating kernel for constrained nonlinear systems with uncertainties based on viability theory. For nonlinear systems with constrained inputs and bounded uncertainties, the viability kernel is the largest set of states possessing a possibility to be viable in a set, and the capture basin is the largest set of states possessing a possibility to reach a target in a finite time, and keeping viable in a set before reaching the target. However, in the viability theory, both control and uncertainty in a parameterized system are considered as parameters:the discriminating kernel and the proposed robust capture basin link viability theory with robust control, which take both control and uncertainties into account. For the constrained uncertain nonlinear systems, the discriminating kernel is the largest set of states that is robust invariant in a set with proper control, and the robust capture basin is the largest set of states reaching their target in finite time with proper control despite of uncertainties and keeping viable in a set before reaching the target. Furthermore, we map all the states to optimal regulatory control such that the systems are regulated by a regulation map. To compute the robust capture basin and the discriminating kernel, we use interval methods to provide guaranteed solutions. The proposed algorithms in this paper approximate an outer approximation of the minimum reachable target and inner approximations of the robust capture basin and the discriminating kernel in a guaranteed way.

This paper proposes an optimal distributed control protocol for multi-vehicle systems with an unknown switching communication graph. The optimal distributed control problem is formulated to differential graphical games, and the Pareto optimum to multi-player games is sought based on the viability theory and reinforcement learning techniques. The viability theory characterizes the controllability of a wide range of constrained nonlinear systems; and the viability kernel and the capture basin are the pillars of the viability theory. The capture basin is the set of all initial states, in which there exist control strategies that enable the states to reach the target in finite time, while remaining inside a set before reaching the target. In this regard, the feasible learning region is characterized for the reinforcement learner. In addition, the approximation of the capture basin provides the learner with prior knowledge. Unlike the existing works that employ the viability theory to solve control problems with only one agent and differential games with only two players, the viability theory, in this paper, is utilized to solve multiagent control problems and multi-player differential games. The distributed control law is composed of two parts: the approximation of the capture basin and reinforcement learning, which are computed off-line and on-line respectively. The convergence properties of the parameters' estimation errors in reinforcement learning are proved, and the convergence of the control policy to the Pareto optimum of the differential graphical game is discussed. The guaranteed approximation results of the capture basin are provided, and the simulation results of the differential graphical game are provided for multi-vehicle systems with the proposed distributed control policy.

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