Journal of Medicine, Engineering, Environmental and Physical Sciences (JOMEEPS), Vol. 1, No. 1, June-July 2023. https://klamidas.com/jomeeps-v1n1-2023-06/ |
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Uniqueness of Optimal Control for the Relative Controllability of Neutral Integro-differential Systems in Banach Spaces with Distributed Delays in the Control Paul Anaetodike Oraekie ABSTRACT A new method of approach is derived for the proof of the existence of an optimal control for Neutral Integrodifferential Systems in Banach Spaces with Distributed Delays in the Control, using the properties of optimal control energy function. The set functions upon which our studies hinged were extracted from the mild solution of the system. Use was also made of the unsymmetric Fubini theorem to establish the mild solution. Necessary and sufficient conditions for an admissible control to be an optimal control are established. Keywords: optimal control, uniqueness, controllability, controllability grammian, set functions.
INTRODUCTION: The pioneering work of Vito Volterra on the integration of the differential equations of dynamics and partial differential dynamical systems published in 1884 gave vent to the conception of integral equations of the volterra type (Oconnor and et al (2005); Oraekie (2014)). It is equally observed in Balachandran and Dauer (1989); Oraekie (2017) that the mixed initial boundary hyperbolic partial differential equation which arises in the study of Lossless transmission lines can be replaced by an associated neutral differential equation. This equivalence has been the basis of a number of investigations of the stability properties of distributed networks. (see Balachandran and Dauer (1997) and Oraekie (2015)). The problem of controllability of linear and nonlinear systems represented by ordinary differential equation in finite dimensional space has been extensively studied. Many authors have extended the controllability concept to infinite dimensional systems in Banach spaces with bounded operators (see Naito (1989)). Naito has studied the controllability of semi linear systems. Yamamoto and Park (1990) discussed the same problem for parabolic equation with uniformly bounded nonlinear term. While Chukwu and Lenhart (1991) studied the controllability of nonlinear systems in abstract spaces. Quinn and Carmichael (1984) showed that the controllability problem in Banach space can be converted into a fixed pointed theorem problem for a single-valued mapping. Balachandran (1996, 1998); Oraekie (2017) had studied the controllability and local null controllability of Sobolve type integrodifferential systems and functional differential systems in Banach spaces by using Schauders fixed point theorem. The purpose of this paper is to investigate the uniqueness of the optimal controllability of the abstract neutral functional integrodifferential systems with distributed delays in the control of the form.
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