UNIVERSITY JOSEPH FOURIER – GRENOBLE 1 T H E S I S To

Abstract: We present a set of methods for the verification and control of continuous and hybrid systems, based on the use of individual trajectories. In the first part, we specify the class of the systems considered and their properties. We start from continuous systems governed by ordinary differential equations to which we add inputs and discrete events, thus constituting a class of hybrid dynamical systems. The second part is devoted to the verification problem and is based on reachable sets computations. We study how a finite number of trajectories can cover the infinite set of the states reachable by the system. We show that by using a sensitivity analysis w.r.t. initial conditions, an over-approximation of the reachable set can be obtained. We deduce from it an algorithm which, by an iterative and hierarchical selection of the trajectories, finds quickly a bad behavior or proves that none exists. The third part is concerned with optimal control and is based on approximate dynamic programming techniques. A cost is defined for each trajectory, and the inputs minimizing this cost are deduced from a value function defined on the state-space and which we represent by using a function approximator. We use the experience provided by test trajectories to improve this approximation. Lastly, we use the results of the second part to select these trajectories in coherence with the local generalization properties of the function approximator and in order to restrict the exploration of the state-space to limit the computational cost. We present a set of methods for the verification and control of continuous and hybrid systems, based on the use of individual trajectories. In the first part, we specify the class of the systems considered and their properties. We start from continuous systems governed by ordinary differential equations to which we add inputs and discrete events, thus constituting a class of hybrid dynamical systems. The second part is devoted to the verification problem and is based on reachable sets computations. We study how a finite number of trajectories can cover the infinite set of the states reachable by the system. We show that by using a sensitivity analysis w.r.t. initial conditions, an over-approximation of the reachable set can be obtained. We deduce from it an algorithm which, by an iterative and hierarchical selection of the trajectories, finds quickly a bad behavior or proves that none exists. The third part is concerned with optimal control and is based on approximate dynamic programming techniques. A cost is defined for each trajectory, and the inputs minimizing this cost are deduced from a value function defined on the state-space and which we represent by using a function approximator. We use the experience provided by test trajectories to improve this approximation. Lastly, we use the results of the second part to select these trajectories in coherence with the local generalization properties of the function approximator and in order to restrict the exploration of the state-space to limit the computational cost.