A feedback control for a linear-quadratic optimal-control problem in the case of the legendre degenerate condition

A feedback control for five linear-quadratic optimal-control problems (the problem with a quadratic or linear-quadratic functional with the fixed left end and free right one, the problem with fixed ends, the periodical problem, and the regulation problem on an infinite interval) connected with the minimization of the functional phi(x(T)) + 1/2 integral(0)(T) ([x(t), W(t)x(t)] + 2[x(t), S(t)u(t)] + [u(t), R(t)u(t)])dt, in the case when the operator R(t) is symmetrical and nonnegative definite for any t is an element of [0, T], is obtained.