On an approach to solution to optimal control problems on the classes ofpiecewise constant, piecewise linear, and piecewise given functions

Optimal control problems for objects described by an ordinary differential equations systemon the classes of piecewise constant, piecewise linear, and piecewise given control functions areconsidered in the paper. Three types of problems are investigated, that depend on the various conditionsimposed of controls: 1) controls belong to a class of piecewise constant functions:u(t) = vj = const, t∈ [τj−1,τj), vj∈ Er, j=1,…,L; 2) controls belong to a class of piecewise linearfunctions: u(t) = 1jc (t−τj−1)+ 2jc , t∈ [τj−1,τj), 1, 2j jc c ∈Er, j=1,…,L; 3) controls belong to a class ofpiecewise given functions: u(t) = ( )11Mjm m jmc t−=Σ ϕ −τ , t∈ [τj−1,τj), jmc ∈Er, m = 1,…,M, j=1,…,L.Piecewise constant values of the coefficients participating in the expression of controls and,what is more important, the boundaries of constancy intervals of these values are optimized. Thenecessary optimality conditions and formulas for the functional gradient in the space of the optimizedparameters that allow us to use numerical methods of first order finite dimensional optimizationfor solving the optimal control problem are obtained in this paper. Results of numerical experimentsare given.

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