الفهرس 
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Abstract
Optimal control prpoblerms occur modren applied science in rather different fields. techno-mechnical systems, in economy and ecological studies, in medicine and life-science, from their mathematical nature, optimal control can be charascterized as a branch of optimization in function space, or infinite-dimensional optimization.
The main goal of this thesis is deduce the differentation and integration matrices for legender polynomiasls, and using their matrices to solve optimal control problems governed by some differential problemes, this procedure depends on a new spectral approximiation for any continous function, its finite integrals, derivatives and obtained numerical solutions of initial and boundary value problems. the round off erros resulting during computing the entries of the first three differentiation matrices are discussed. compare all results with when using differeniation matrices discussed .compare all results with using differentation and integration matrives for chebyshev polynimials.