In this thesis , analysis and optimal control of multi-delay systems with piecewise constant delay functions have been investigated . Approximation methods for solving the problem of analysis of linear multi-delay systems based on Legendre hybrid functions and Chebyshev hybrid functions are proposed . Two upper bounds for Legendre hybrid functions and also for Chebyshev hybrid functions are established . Composite interpolation functions based on Legendre-Gauss-Lobatto nodes and Chebyshev-Gauss-Lobatto nodes are introduced . The main contribution of this work is to derive the necessary conditions of optimality for this class of delayed control problems . We also obtain a set of simple conditions that commute the operations of discretization and dualization . This means that the nonlinear control problems can be solved efficiently and accurately without developing the necessary conditions of optimality . Moreover , the adjoint variables can be obtained from the Lagrange multipliers of optimization problem that is associated to the discretized control problem .