This paper uses the fixed‐point theorem to generate a set of sufficient conditions which confine the solutions of weak nonlinear differential equations to some low‐order stable integral manifold. Furthermore, it is shown that these conditions can be weakened to a great extent if the initial values of the differential equations are confined to a bounded region.
A method is proposed for finding the integral manifold in an analytic form such as a Fourier series and a new calculation algorithm which applies the Newton method and other techniques. Next, the obtained results are applied to weak nonlinear optimal regulator problems and sufficient conditions are derived for the existence of a nonlinear feedback control law which meets the necessary conditions for optimality. Simulations using the algorithm exhibited favorable results.
In this paper, we use the fixed point theorem to generate a set of sufficient conditions which confine the solutions of weak non-linear differential equations to a stable integral manifold. The obtained conditions are used in the solution of a weak non-linear optimal regulator problem. The necessary conditions for optimal control laws are expressed by the weak non-linear differential equations with state variable x integral manifold. The method for finding an approximate solution of p(x) in an analyltic form such as a Fourier series is introduced, and a new calculation algorithm is presented which applies the Newton method and other techniques. Simulations using the algorithm exhibit favorable results.
SUMMARYIn conventional digital image processing, topological properties have been studied only for selected types of neighborhoods such as a 4-or 8-pixels connection. This paper analyzes properties of a finite topological space by defining it as a topological space with no restriction on the shape of a neighborhood. This leads to the identificationof topological properties which are independent of the shape of a neighborhood and can be applied to image processing using neighborhoods other than a 4-or 8-connection. Since a finite topological space can treat only a single neighborhood this cannot be applied to image processing which uses multiple neighborhoods simultaneously. A finite topological space has been extended to a formal topological space havingmultiple neighborhoods, and its properties are analyzed in this paper.This theory is then applied to image processing.It has been difficult to detect scratches on a surface with patterns, such as a hard disk or a hairline-finished metal, by using a conventional digital topology because the scratches often consist of many small blocks and their images are stained by noise. The properties obtained through the formal topology have successfully been applied to these problems, and its effectiveness has been confirmed.
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