Reachability and Controllability Analysis

Origin. Lev Pontryagin's reachability sets (1950s-1960s); Nikolai Krasovskii's differential game theory and pursuit-evasion analysis (1960s-1970s); foundational to Soviet control theory.

Mechanism. The reachable set from a state is the set of all states the system can reach under some admissible control. The controllable set to a target is the set of all states from which the target can be reached. A system is controllable if every state can reach every other state. The method shifts the question from "what is the optimal control?" to "what states are reachable at all?" — separating possibility from optimality.

Procedure. Define the state space, the dynamics, and the control constraints. From the current state, compute the forward reachable set: all states reachable in one time step under any control. Iterate to compute the N-step reachable set. For controllability to a target, compute backward: what states can reach the target in one step? Intersect the forward and backward sets. If the intersection is empty, the target is unreachable from the current state within N steps.

Applies to. Safety analysis (can the system reach a dangerous state?), mission planning (can the objective be achieved?), pursuit-evasion games, and any problem where feasibility must be established before optimality.

Limitations. Reachable sets grow exponentially in dimension and are computationally intractable for systems above four or five dimensions without approximation. The analysis also assumes known dynamics; model uncertainty makes the computed reachable set unreliable. The boundary of the reachable set is where small model errors produce large conclusion errors.