The discrete Euler equation on the normed linear spaceln1

Abstract

The discrete Euler equation is established as a necessary condition for the solution of discrete variational problems for both infinite and finite time sequences. The variational criterion (infinite sum) is treated as a functional on l ¹ _n, the normed linear apace of infinite sequences. The discrete Euler equation is also found for the constrained discrete variational problem, i.e. for problems in which the extremal sequences must satisfy an additional subsidiary condition. Such a condition is assumed to be of a form that is readily applicable to discrete control problems, since it includes difference equations as a special case. The discrete variational problem with finite time (finite sum criterion) is treated as a special case of the infinite time.