SUMMARYWe 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 EulerLagrange equation, that is simultaneously applicable to both shift ðqÞ-operator-based discrete-time systems and the derivative ðd=dtÞ-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.

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We develop a unified approach for the necessary conditions for optimization of open-loop control systems, starting from the basic principles of calculus of variations. The unified approach results are simultaneously applicable to both shift ðqÞ-operator-based, discrete-time systems and the derivative ðd=dtÞ-operator-based, continuous-time systems. It is shown that the optimal condition results that are now obtained separately for continuous-time and discrete-time systems can be easily obtained from the unified approach. An illustrative example is given. * Step 3: Introduction of costate vector.Next, let us introduce the costate variable kðtÞ; the augmented functionals J ra and J n ra defined as J ra ðuðtÞÞ ¼ S t f t 0 ½V r ðxðtÞ; uðtÞ; tÞ þ k 0 ðtÞff r ðxðtÞ; uðtÞ; tÞ À rxðtÞg dt ð35Þ Y. IMURA AND D. S. NAIDU 64

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