Quantum fluctuation in affine optimal control systems

Abstract

SUMMARYQuantum mechanical wave functions are shown to approximate optimal feedback laws of affine control systems, when we set the absolute values of the terminal wave functions positive and with no singular dependence on a control constant H R , which is similar in position to the action constant % introduced by Planck to explain quantum phenomena. Calculation of the wave functions makes use of the path integral representation that we approximate at stationary phase. The phases of the wave functions approximate in H R → 0 to Hamilton-Jacobi value functions, because quantum mechanical fluctuation vanishes in the limit. It is simple to take the terminal absolute value function that meets the condition of having no singularity at H R = 0. The terminal absolute value function without any dependence on the constant H R apparently satisfies the no-singularity condition. Although we restrict ourselves to scalar systems, generalization to systems with higher dimensionality is straightforward. Key words: nonlinear optimal feedback control; linear wave equation; path integral; stationary phase approximation. IntroductionThere have so far been various algorithms and software tools for treating linear systems in mathematical science and technology. Many researchers are still engaged in the development of such linear tools. We therefore have various ideas that allow us to calculate nonlinear control systems within a framework of linear theories. As one such linear theory, the author has developed a quantum mechanical theory of nonlinear optimal control that fully utilizes complex wave equations [1]. The linearity of the wave equations allows us to apply methods that have been developed for linear systems, such as eigenvalue analysis, the path integral method, random walk simulation, and so on. Let us consider an affine nonlinear system with control cost quadratic in control inputs. The real part of the complex wave equation is an extension of the conventional Hamilton-Jacobi equation with an additional term that causes quantum mechanical fluctuation. A control constant H R that has a similar role to Planck's constant % [2] has been brought into our quantum mechanical control theory. It is generally considered that in macroscopic systems, where the constant % is negligible, classical mechanics approximately describes the systems. _ _as a combination of the absolute value R(x, t; H R ) ≡ |ψ(x, t; H R )| of the wave function ψ(x, t; H R ) and its first and second partial derivatives with respect to the spatial variable x. It is necessary to clarify conditions when the absolute value R(x, t; H R ) has no singularity at H R = 0. In optimal control schemes, designers introduce a terminal cost at terminal time t F as one of the control specifications. This terminal cost defines a time boundary condition of a phase of the wave function in our quantum mechanical control theory. On the contrary, we can set a boundary condition R(x, t F ; H R ) freely. What we show in this report is: If the terminal absolute value R(x, t F ; H R ) is ...