Dynamic optimization for reachability problems

This paper uses dynamic programming techniques to describe reach sets and related problems of forward and backward reachability. The original problems do not involve optimization criteria and are reformulated in terms of optimization problems solved through the Hamilton–Jacobi–Bellman equations. The reach sets are the level sets of the value function solutions to these equations. Explicit solutions for linear systems with hard bounds are obtained. Approximate solutions are introduced and illustrated for linear systems and for a nonlinear system similar to that of the Lotka–Volterra type.