Page - 151 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 57 Specialized methods and algorithms General methods and algorithms direct methods and algorithms Euler method Ritz-Galerkin’s method special methods of nonlinear programming Methods and algorithms of optimal control indirect methods and algorithms methods of reduction to finite-dimensional problems methods of state- space search methods of gradient notion in control space gradient methods of unconstrained optimization methods of gradient projection gradient methods based on penalty functions methods and algorithms based on Bellman’s principle of optimality methods and algorithms of variations in a state space methods and algorithms for two-point boundary problems methods of successive approximations methods of perturbation theory methods based on necessary conditions of optimal control methods of state- space search combined methods methods and algorithms for linear- problems of optimal control approximate analytical methods and algorithms methods and algorithms for linear time-minimization problems methods and algorithms for linear problems with quadratic objective functions methods and algorithms for quasi-linear systems methods and algorithms for system with weak controlability averaging approximate methods and algorithms \ \ U\ \ U\ U Figure2.Optimalcontrol computationalmethods. Newton’s method and its modifications. For the first iteration of the method, an initial approximationψ(T0) = ψ(0)(T0) is set. Then, the successive approximations are received from theformula: ψ(r)(T0)=ψ(r−1)(T0)− [ ∂ρ(r−1)(Tf) ∂ψ(r−1)(T0) ] ρ(r−1)(Tf) (11) whereρ(r)(Tf)=a−x(r−1)(Tf), r=1,2, . . . . Theoperationofpartialderivation,beingperformedatall iterations, is rathercomplicated.Here thederivativematrix: Π˜= ( ∂ρ(r−1)(Tf) ∂ψ(r−1)(T0) ) (12) shouldbeevaluated. Thismatrixcanbereceivedeitherviafinite-difference formulasorviaavariational integration of thesystem.Modificationof themethodcanalsobeused. The followingmethod is the simplest. Theformula (11) is substitutedfor: ψ(r)(T0)=ψ(r−1)(T0)−γ(r)Π˜−1ρ(r−1)(Tf) (13) 151