Optimal control with state-space constraint I
SIAM Journal on Control and Optimization
Viability theory
A New Formulation of State Constraint Problems for First-Order PDEs
SIAM Journal on Control and Optimization
Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics
Journal of Scientific Computing
Numerical Discretization of Boundary Conditions for First Order Hamilton--Jacobi Equations
SIAM Journal on Numerical Analysis
Anti-Dissipative Schemes for Advection and Application to Hamilton-Jacobi-Bellmann Equations
Journal of Scientific Computing
A method to construct viability kernels for nonlinear control systems
ACC'09 Proceedings of the 2009 conference on American Control Conference
An Efficient Data Structure and Accurate Scheme to Solve Front Propagation Problems
Journal of Scientific Computing
A Discontinuous Galerkin Solver for Front Propagation
SIAM Journal on Scientific Computing
HJB approach for motion planning and reachabilty analysis
Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
Optimal Trajectories of Curvature Constrained Motion in the Hamilton---Jacobi Formulation
Journal of Scientific Computing
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This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for its approximation. Since this value function is discontinuous, usual discretization schemes (such as finite differences) would provide a poor approximation quality because of numerical diffusion. Hence, we investigate the Ultra-Bee scheme, particularly interesting here for its anti-diffusive property in the transport of discontinuous functions. Although currently there is no available convergence proof for this scheme, we observed that numerically, the experiments done on several benchmark problems for computing viability kernels and capture basins are very encouraging compared to the viability algorithm, which fully illustrates the relevance of this scheme for numerical approximation of viability problems.