document: Approximate relaxed descent method for optimal control problems We consider an optimal control problem for systems governed by ordinary differential equations with control constraints . Since no convexity assumptions are made on the data , the problem is reformulated in relaxed form . The relaxed state equation is discretized by the implicit trapezoidal scheme and the relaxed controls are approximated by piecewise constant relaxed controls . We then propose a combined descent and discretization method that generates sequences of discrete relaxed controls and progressively refines the discretization . Since here the adjoint of the discrete state equation is not defined , we use , at each iteration , an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation and the integral involved by appropriate trapezoidal schemes . It is proved that accumulation points of sequences constructed by this method satisfy the strong relaxed necessary conditions for optimality for the continuous problem . Finally , the computed relaxed controls can be easily approximated by piecewise constant classical controls
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