On the Linear Quadratic Minimum-fuel Problem

E.I.Verriest and F.L.Lewis have presented in [1] a new method to approach the minimum-time control of linear continous-time systems avoiding the Bang-Bang control. Their method relied on the optimization of a cost including time energy and precisions terms. Then, N.Elalami and N.Znaidi [2], extended these results to the discrete-time linear systems. The objective of this work is to propose an approach for minimal-fuel problem where the term of consumption is increased by an energetic term in order to avoid Bang-off-Bang control and singular intervals . Indeed we consider the equation of Hamilton-Jacobi-Bellman(HJB) relating to the problem of minimal consumption. And by making use of a nonlinear programming problem on a partition of R, the solution of the minimum-fuel problem, according to the Riccati matrix, is reduced to the resolution of a system of differential equation.