On a class of inverse optimal control problems

Four distinct, though closely related, inverse optimal control problems are considered. Given a closed, convex setU in a real Hilbert spaceX and an elementu ₀ inU, it is desired to find all functionals of the form (u,Ru) such that (i)R is a self-adjoint positive operator and (u,Ru) is minimized over the setU at the pointu ₀, (ii)R is self-adjoint, positive definite and (u,Ru) is minimized overU atu ₀, (iv)R is self-adjoint, positive definite and (u,Ru) is uniquely minimized overU atu ₀. The interrelationships among the sets of solutions of these problems are pointed out. Necessary and sufficient conditions which explicitly characterize the solutions to each of these problems are derived. The question of existence of a solution (namely, Given a particular setU and a particular elementu ₀, under what conditions does there exist an operatorR having certain required properties?) is discussed. The results derived are illustrated by an example.