© 1988 by Institute of Mathematics and its Applications
Systems with Time Delay in the Calculus of Variations: The Method of Steps
Centro de Investigación en Matemáticas A.C. A.P. 402, Guanajuato, Gto., 36000—Mexico
In a recent article (see [8]), we derived necessary and sufficient conditions for minima for the fixed-endpoint problem in the calculus of variations involving a constant delay in the phase coordinates. These conditions are expressed, explicitly, in terms of the first and second variations. The vanishing of the first variation is characterized in terms of an extended Euler's equation, just as for delay-free problems, but the characterization of the conditions on the second variation remained unsolved. In this paper we convert, through the ‘method of steps’, the delay problem into one without delay. Although this problem will not have fixed-endpoint constraints, it allows us to introduce, in a natural way, the concept of ‘conjugate sequence’; this solves the main difficulty for delay problems, namely, to give initial conditions for existence and uniqueness of solutions of the Hamiltonian system (which is a difference–differential system with both advanced and retarded arguments). The conditions on the second variation are then characterized by an extra condition which is based exclusively on a solution of a given matrix Riccati equation.