Construction of solutions in certain differential games with phase constraints

Computation of Value Functions in Nonlinear Differential Games with State Constraints

“…Since about fifteen years, several groups are working on appropriate numerical methods (see e.g. [17,18,19,20,21]), but only few three-dimensional problems are solved numerically. The following example deals with a very famous unsolved problem.…”

Recent Advances in Mobile Robotics

“…Backward procedures for the construction of the solvability sets at given time and by given time are intensively developed (Grigor'eva et al, 2005), (Mikhalev & Ushakov, 2007) for control problems and differential games. Elements of the backward constructions are included in one or another form (Sethian, 1999), (Cristiani & Falcone, 2009) into grid methods for solving differential games.…”

On reachability analysis for nonlinear control systems with state constraints

“…Different approaches for computing the reachable sets, including those for systems with state constraints, are presented in [14,17,16,6,13,1,3,11,7,8]. The method of removing state constraints in the construction of reachable sets for differential inclusions was proposed in [12], where the tube of trajectories of the differential inclusion with convex state constraint was approximated by the solutions of a family of differential inclusions without constraints depending on a matrix "penalty" parameter.…”

“…Consider the solution x ε (t) of system (6) corresponding to the control u(·) ∈ U. Since f ε (x, u) is a convex combination of the vectors f (x, u) and f (x,ū(x)) belonging to the convex set f (x, U ), we see thatẋ ε (t) ∈ f (x ε (t), U ) a.e.…”