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In this paper the necessary conditions of optimality in the form of maximumprinciple are derived for a very general class of variational problems. Thisclass includes problems with any optimization criteria and constraints that canbe constructed by combining some basic types (differential equation, integralequations, algebraic equation, differential equations with delays, etc). Foreach problem from this class the necessary optimality conditions are producedby constructing its Lagrange function $R$ and then by dividing its variablesinto three groups denoted as $u(t)$, $x(t)$ and $a$ correspondingly. $a$ areparameters which are constant over time. The conditions of optimality statethat a non-zero vector function of Lagrange multipliers exists such that on theoptimal solution function $R$ attains maximum on $u$, is stationary on $x$, andthe integral of $R$ over the control period $S$ can't be improved locally.Similar conditions are also obtained for sliding regimes. Here solution isgiven by the limit of maximizing sequence on which the variables of the secondgroup are switching with infinite frequency between some (basic) values.

Maximum Principle for variational problems with scalar argument