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HJSD: Hamilton-Jacobi on Stratified Domains HJSD is a software for the numerical solution of a class of optimal control problems defined on stratified domains in two and three dimensions. More precisely, the state space \(\mathbb{R}^N\) (\(N=2, 3\)) admits a stratification given by a disjoint union of locally flat \(k\)-dimensional submanifolds \((\mathbf{M}^k)_{k=0,\dots, N}\) where the controlled dynamics and running cost may have discontinuities, and the value function \(u\) of the problem satisfies the following stratified Hamilton-Jacobi equation: \[H(x,u(x),Du(x)):=\max_{(k,j)\in L(x)}\,\sup_{\alpha\in A^{k,j}}\left\{-b^{k,j}(x,\alpha)\cdot Du(x)+c^{k,j}u(x)-\ell^{k,j}(x,\alpha)\right\}=0\qquad x\in\mathbf{R}^N,\] where the \(k\)-dimensional submanifold is composed of the union of its connected components \(\mathbf{M}^k=\cup_{j=1}^{J(k)} \mathbf{M}^{k,j}\) with \(J(k)\in\mathbb{N}\), \(L(x)=\left\{(k,j):\, x\in \overline{\mathbf{M}^{k,j}}\right\}\) and we denote by \(A^{k,j}\), \(b^{k,j}\), \(c^{k,j}\), \(\ell^{k,j}\) respectively the control set, the controlled dynamics, the discount factor and the running cost on \(\mathbf{M}^{k,j}\). For further details refer to: Cacace, S., Camilli, F., Approximation of the value function for optimal control problems on stratified domains, preprint arxiv:2207.06892. HJSD will be soon available for free download! HJSD takes in input \(\texttt{.hjsd}\) files, which are simple text files containing all the relevant information for the definition of the stratification and of the corresponding control problems. Then it computes an approximation of the stratified viscosity solution of the problem, and writes the results in a \(\texttt{vtk}\) file, ready for the visualization in \(\texttt{Paraview}\). The output file contains the geometry of the stratification, the value function and the corresponding vector field of the optimal dynamics, which can be also used to compute the optimal trajectories starting from arbitrary points (directly in \(\texttt{Paraview}\), by means of the \(\texttt{StreamTracerWithCustomSource}\) filter). A sample test in 3D Consider the stratification of \(\mathbb{R}^3\) obtained combining a horizontal square (including its sides and corners) with a vertical segment (including its end-points). The speed function for the controlled dynamics is \(5\) on \(\mathbf{M}^1\cup \mathbf{M}^2\) and \(1\) on \(\mathbf{M}^3=\mathbb{R}^3\setminus(\overline{\mathbf{M}^0\cup\mathbf{M}^1\cup\mathbf{M}^2})\) (no dynamics on \(\mathbf{M}^0\)!)  A flat stratification of \(\mathbb{R}^3\) The parameters of the control problem are such that: - the top end-point of the segment is a target for the optimal dynamics - the other points in \(\mathbf{M}^0\) have no influence on the solution - \(\mathbf{M}^1\cup \mathbf{M}^2\) attracts the optimal trajectories due to its higher speed 
 Solution iso-surfaces and optimal trajectories Optimal paths to the target |