  In this work we approach the dual optimal reach-safe control problem using
sparse approximations of Koopman operator. Matrix approximation of Koopman
operator needs to solve a least-squares (LS) problem in the lifted function
space, which is computationally intractable for fine discretizations and high
dimensions. The state transitional physical meaning of the Koopman operator
leads to a sparse LS problem in this space. Leveraging this sparsity, we
propose an efficient method to solve the sparse LS problem where we reduce the
problem dimension dramatically by formulating the problem using only the
non-zero elements in the approximation matrix with known sparsity pattern. The
obtained matrix approximation of the operators is then used in a dual optimal
reach-safe problem formulation where a linear program with sparse linear
constraints naturally appears. We validate our proposed method on various
dynamical systems and show that the computation time for operator approximation
is greatly reduced with high precision in the solutions.
