Because of their importance in applications ranging from engineering to biology and economics, questions
of reachability, viability and invariance have been studied extensively in the dynamics and control literature.
Most recently, the study of these concepts has received renewed attention through the study of safety problems
in hybrid systems. Reachability computations have been used in this context to address problems in the:
air traffic management systems, safety of ground transportation systems, flight control systems, etc.
Reachability is a crucial concept in the study of dynamical systems. Roughly speaking, a state,
is called reachable if the system find its way to this state while moving along one of its trajectories. In the presence
of inputs (control and/or disturbance) this basic notion gives arise to a number of different concepts:
- Viability: If only control inputs are available, one may want to select control trajectories so that the evolution of the state
starting in a given set stays in the set.
- Reachability: If only control inputs are available, one may want to select control trajectories so that the evolution of the state
starting in given set reachs a certain set.
- Invariance: If only disturbances are present, one may want to select state trajectories so that the evolution of the state
starting in a given set stays in the set.
- Robust Reachability: If both control and disturbance inputs are available, the reachability problem can be thought of as a pursuit-evasion game,
where the controller wins if it can keep the system from entering a "bad" subset of the state space, called the capture set, while the
disturbance wins if it can drive the state into the bad set.
- Stochastic reachability: In the presence of stochastic disturbance, reachability can thought of as characterizing the probability of the state
reaching certain set. If control inputs are available, one may want to select a control policy to maximize or minimize this probability
Our research focuses on the investigation of the reachability problem for the followings cases:
