We consider a multilevel defense system on a given direction. This is a particular model of terminal discrete optimal control that may be solved by the gradient descent method. The main difficulty is the nondifferentiability of the Lipschitzian functions in the right-hand sides of the equation of motion and their derivatives with respect to all variables, which leads to an ill-posed problem when applying the classical results on differentiability of the terminal function and constructing its gradient from the conjugate system. A method is proposed for the solution of the problem by averaging the right-hand sides in combination with the stochastic gradient projection method. The study develops Germeier's defenseattack model by allowing for a multilevel defense structure, which in general leads to the synthesis problem for a discrete optimal control system.Keywords: discrete optimal control, objective-function differentiability conditions, conjugate system, structure of the objective function gradient, averaging of the right-hand sides, structure of the derivatives of the averaged functions, combined gradient projection method and Polyak method, randomization of the combined gradient descent method, stochastic gradient of the average problem, almost-sure convergence of the randomized procedure.

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