Regional Analysis Approximations and Simulations

This work is a part of mathematics applied to the theoretical automatic. The concepts of regional analysis, internal and boundary, of the gradient were introduced for systems governed by hyperbolic equations. Exactly, it consist to observe the gradients of the initial state and the initial speed on an internal or boundary region of the geometric domain on which the system is defined. The main idea of the introduction of this concept is the directly reconstruction of the initial gradients on a part without the need to reconstruct its state or its speed. That is motivated by many reasons attached to real problems. Various properties and characteristics are established, in particular in relation to the structure of the sensors. By use of the extending of the HUM method, we have established explicit expressions of the gradients of the initial state and the initial speed on the internal region, which are exploitable from a numerical point of view. After, we have extended this study to the boundary case. Algorithms for numerical implementation were developed and illustrated by various examples and which attest the success of the approach considered.