Nonlinear Optimal Control by Monte-Carlo Methods Applied to Path Integrals

Itami, Teturo

doi:10.5687/iscie.16.637

Cited by 3 publications

(7 citation statements)

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“…In this section, we first review the Hamilton-Jacobi theory of feedback optimization from the point of view of canonical dynamics [1], where we describe dynamics by a state variable x, its canonical conjugate momentum p x , and a Hamiltonian H. Subsequent to this, we apply quantum mechanical wave theory [1] to a Hamiltonian operator Ĥ x that we give as a linear operator representation of the Hamiltonian H. Full use of linearity of the wave equation will allow a path integral representation [4,5] of the system.…”

Section: Path Integral Of Nonlinear Optimal Feedback Controlmentioning

confidence: 99%

“…We can use stationary phase approximation to calculate path integral representation (34). In addition to using studies [4,5] preceding this report, on the phase of the wave function, we also calculate the absolute value of the wave function. The path integral (34) of the wave function is the 2N-dimensional version of a scalar integral where Π and H R correspond to x and a, respectively.…”

Section: Stationary Phase Approximation Of the Path Integralmentioning

confidence: 99%

“…They only set the function as independent on H R − R(x, t F ; H R ) = e phase approximation. We explicitly show how to calculate the absolute value that we have not presented in earlier studies [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…After we derive the stationary phase approximation to the path integral in Section 3, we take the limit of an infinitesimal time interval ε → 0 in Section 4. Combination of these results leads to an explicit formula not only of the phase [4,5] but also of the absolute value of the wave function. Section 5 gives a summary and discussion.…”

Section: Introductionmentioning

confidence: 99%

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Quantum fluctuation in affine optimal control systems

Itami

2007

Electrical Engineering Japan

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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 ...

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Section: Path Integral Of Nonlinear Optimal Feedback Controlmentioning

confidence: 99%

Section: Stationary Phase Approximation Of the Path Integralmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum fluctuation in affine optimal control systems

Itami

2007

Electrical Engineering Japan

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“…Incidentally, we have recently found some research studies 8,9) where concepts of quantum mechanics are applied to the optimal control problem. These are based on similar ideas to that of the authors of this article.…”

Section: Introductionmentioning

confidence: 99%

Design Tool Using a New Optimization Method Based on a Stochastic Process

Yoshida

Yamaguchi

Ishikawa

2007

Trans. Japan Soc. Aero. S Sci.

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Conventional optimization methods are based on a deterministic approach since their purpose is to find out an exact solution. However, such methods have initial condition dependence and the risk of falling into local solution. In this paper, we propose a new optimization method based on the concept of path integrals used in quantum mechanics. The method obtains a solution as an expected value (stochastic average) using a stochastic process. The advantages of this method are that it is not affected by initial conditions and does not require techniques based on experiences. We applied the new optimization method to a hang glider design. In this problem, both the hang glider design and its flight trajectory were optimized. The numerical calculation results prove that performance of the method is sufficient for practical use.

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Fluctuation in Quantum Mechanics of Affine Optimal Control System

Itami¹

2006

IEEJ Trans.EIS

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Nonlinear Optimal Control by Monte-Carlo Methods Applied to Path Integrals

Cited by 3 publications

References 0 publications

Quantum fluctuation in affine optimal control systems

Quantum fluctuation in affine optimal control systems

Design Tool Using a New Optimization Method Based on a Stochastic Process

Fluctuation in Quantum Mechanics of Affine Optimal Control System

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