## Reachability Analysis and Efficient Computation for Nonlinear/Hybrid Systems

Approximate algorithms that can solve the Hamilton-Jacobi-Isaacs (HJI) PDE for game theory or Hamilton-Jacobi-Bellman (HJB) PDE for optimal control (or optimization problems in general) in real-time. For approximation, we use a polytopic approximation method.

Numerical algorithms that can efficiently solve necessary conditions, HJB, and HJI equations for various optimization problems including optimal control, game theory, etc. With our best knowledge, this method is a new algorithm based on transformation not on either gradients or grids which most currently available algorithms are based on. For transformation, we use the D-Transformation which is a differential transformation while Laplace and Fourier transformations are integral transformations. By using this methods, we solve a set of algebraic equations in the transformed domain rather than a set of (partial) differential equations (Figure 1).

Figure 1. Numerical simulations of the optimal state trajectory and optimal control law for a two-mode hybrid system optimal control problem by using the differential transformation based numerical algorithms

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