Solving Controllability Problems of Backward Ordinary Differential Systems with Matlab

摘要

Controllability problems of differential equations appear in many situations or phenomena for which one is interested in finding a mechanism for bringing a given state into a desired one. Their resolutions often involve constrained minimization problems governed by differential equations systems. This paper is particularly interested in a null controllability problem for backward differential equations systems. One develop a numerical scheme by first approximating the control space by a space of piecewise continuous functions and by transforming the controllability problem into a classical minimization problem with constraints in finite dimension space. Next, one proceed to an adapted implementation of the numerical scheme in Matlab using some of its built-in functions. One then construct a sequence of codes written in Matlab allowing to robustly compute an approximation of the null control at a lower cost. To validate the numerical approach adopted in this paper, two numerical examples are presented. The first ones concerns the controllability of a backward ordinary differential quations system and the second, the controllability of a partial differential heat equation. In both cases, the numerical results obtained are very satisfactory and show that the numerical approach with Matlab developed in this paper leads to new insights for a large class of PDE control problems.