Unified approach for Euler–Lagrange equation arising in calculus of variations

We address the development of a unified approach for the necessary conditions for optimization of a functional arising in calculus of variations. In particular, we develop a unified approach for the Euler–Lagrange equation, that is simultaneously applicable to both shift (q)-operator-based discrete-time systems and the derivative (d/dt)-operator-based continuous-time systems. It is shown that the Euler–Lagrange results that are now obtained separately for continuous- and discrete-time systems can be easily obtained from the unified approach. An illustrative example is given.