Approximation of Nonstationary Optimal Feedback Law using Quantum Mechanical Eigenvalue Analysis
Nonlinear optimal feedback law subject to a terminal cost is approximated using quantum mechanical eigenvalue analysis. The optimal feedback law is governed by nonstationary Hamilton-Jacobi equation with a final condition. Numerical time integration in the backward direction must be done along a characteristic curve of the system. Compatibility under such constraints has restricted practical calculations to the system with dimensionality of only 1 or 2. For approximating optimal feedback with a terminal cost, we propose a quantum mechanical algorithm applicable to systems of arbitrary dimensions. The nonlinear optimal control system is represented by a complex-valued linear wave equation, where a time derivative of the system is connected with linear Hamiltonian operator H. We then apply eigenvalue analysis to this operator H. The value function applied to calculation of optimal feedback is approximated using terms, each of which are a product of eigenfunction φK (x) and time exponential function characterized by corresponding eigenvalue EK. The proposed method thus needs no time integration in the backward time direction. Simulation studies are performed for systems with a state variable of either 1 or 2 dimensions. The new algorithm has an advantage over conventional calculations in that we can fully make use of storage and development of eigenvalue analysis tools.