Symplectic Numerical Approach for Nonlinear Optimal Control of Systems with Inequality Constraints

This paper proposes a system representation for unifying control design and numerical calculation in nonlinear optimal control problems with inequality constraints in terms of the symplectic structure. The symplectic structure is derived from Hamiltonian systems that are equivalent to Hamilton-Jacobi equations. In the representation, the constraints can be described as an input-state transformation of the system. Therefore, it can be seamlessly applied to the stable manifold method that is a precise numerical solver of the Hamilton-Jacobi equations. In conventional methods, e.g., the penalty method or the barrier method, it is difficult to systematically assign the weights of penalty functions that are used for realizing the constraints. In the proposed method, we can separate the adjustment of weights with respect to objective functions from that of penalty functions. Furthermore, the proposed method can extend the region of computable solutions in a state space. The validity of the method is shown by a numerical example of the optimal control of a vehicle model with steering limitations.