Graph-Based Algorithms for Boolean Function Manipulation
IEEE Transactions on Computers
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations
SIAM Journal on Numerical Analysis
Approximate reachability with BDDs using overlapping projections
DAC '98 Proceedings of the 35th annual Design Automation Conference
The fast construction of extension velocities in level set methods
Journal of Computational Physics
Improved approximate reachability using auxiliary state variables
Proceedings of the 36th annual ACM/IEEE Design Automation Conference
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Motion of curves in three spatial dimensions using a level set approach
Journal of Computational Physics
High-frequency wave propagation by the segment projection method
Journal of Computational Physics
Counterexample-guided choice of projections in approximate symbolic model checking
Proceedings of the 2000 IEEE/ACM international conference on Computer-aided design
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
HSCC '98 Proceedings of the First International Workshop on Hybrid Systems: Computation and Control
Series of Abstractions for Hybrid Automata
HSCC '02 Proceedings of the 5th International Workshop on Hybrid Systems: Computation and Control
Safety verification of conflict resolution manoeuvres
IEEE Transactions on Intelligent Transportation Systems
Controllers for reachability specifications for hybrid systems
Automatica (Journal of IFAC)
On the adaptive control of a class of SISO dynamic hybrid systems
Applied Numerical Mathematics
Parameter Synthesis in Nonlinear Dynamical Systems: Application to Systems Biology
RECOMB 2'09 Proceedings of the 13th Annual International Conference on Research in Computational Molecular Biology
On the adaptive control of a class of SISO dynamic hybrid systems
Applied Numerical Mathematics
Proceedings of the 14th international conference on Hybrid systems: computation and control
Morphological Control Problems with State Constraints
SIAM Journal on Control and Optimization
Computing the viability kernel using maximal reachable sets
Proceedings of the 15th ACM international conference on Hybrid Systems: Computation and Control
External estimates of the reachability sets of nonlinear controlled systems
Automation and Remote Control
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In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.