Optimal control with state-space constraint I
SIAM Journal on Control and Optimization
The Bellman equation for minimizing the maximum cost
Non-Linear Analysis
Viability theory
Level sets of viscosity solutions: some applications to fronts and rendez-vous problems
SIAM Journal on Applied Mathematics
Pursuit Differential Games with State Constraints
SIAM Journal on Control and Optimization
An anti-diffusive scheme for viability problems
Applied Numerical Mathematics - Numerical methods for viscosity solutions and applications
Anti-Dissipative Schemes for Advection and Application to Hamilton-Jacobi-Bellmann Equations
Journal of Scientific Computing
Ellipsoidal Techniques for Reachability Under State Constraints
SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
On reachability and minimum cost optimal control
Automatica (Journal of IFAC)
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Because of their importance in many applications, questions of path planning and reachability analysis for nonlinear dynamical systems have been studied extensively in the control theory. Here we focus on the cases when the controlled systems are constrained to evolve in a certain known set (e.g avoidance of obstacles). We study general framework based on an optimal control approach and on solving Hamilton-Jacobi (HJ) equations. This approach provides a very efficient tool for treating many cases encountered in real applications and can be extended to general situations including moving targets and/or obstacles problems, dynamical systems under uncertainties, or differential games. The relevance of the method will be shown on some numerical examples (motion planning with obstacle avoidance).