Riemann Integral of Functions R into C

In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].Miyajima Keiichi - Faculty of Engineering, Ibaraki University, Hitachi, JapanKato Takahiro - Faculty of Engineering, Graduate School of Ibaraki University, Hitachi, JapanShidama Yasunari - Shinshu University, Nagano, JapanAgnieszka Banachowicz and Anna Winnicka. Complex sequences. Formalized Mathematics, 4(1):121-124, 1993.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics, 8(1):93-102, 1999.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux's theorem. Formalized Mathematics, 9(1):197-200, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics, 9(2):281-284, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Yasunari Shidama and Artur Korniłowicz. Convergence and the limit of complex sequences. Series. Formalized Mathematics, 6(3):403-410, 1997.Murray R. Spiegel. Theory and Problems of Vector Analysis. McGraw-Hill, 1974.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Variable structure robot control systems: The RAPP approach

International audienceThis paper presents a method of designing variable structure control systems for robots. As the on-board robot computational resources are limited, but in some cases the demands imposed on the robot by the user are virtually limitless, the solution is to produce a variable structure system. The task dependent part has to be exchanged, however the task governs the activities of the robot. Thus not only exchange of some task-dependent modules is required, but also supervisory responsibilities have to be switched. Such control systems are necessary in the case of robot companions, where the owner of the robot may demand from it to provide many services.