An optimal control problem for a linear evolution equation in a Hilbert space X is considered, \(dx/dt=Ax(t)+Bu\), \(x(0)=x_ 0\) where A is the infinitesimal generator of a \(C_ 0\) semigroup, \(B\in {\mathcal L}(V;X)\), V is the Hilbert space of controls. The cost function includes final and distributed in time observations \[ J(u)=M(x(T))+\int^{T}_{0}(Q(x(t))+(1/2)<Ru(t),\quad u(t)>)dt \] where M and Q are \(C^ 2\) convex functionals.
First the existence and uniqueness of an optimal control is proved and the first order necessary optimality conditions are given in the form of an integral equation. Then a quasi-Riccati operator equation is introduced. A normal solution of that equation is a nonlinear operator P(t,x) such that (1) P(t,\(\cdot)\) is a gradient operator (2) the corresponding closed-loop state equation has a global mild solution, and (3) having certain regularities. Any such normal solution yields a closed-loop optimal control. Then the nonlinear integral equation giving the optimal adjoint state at time s knowing the state at time t (t\(\leq s)\) is studied. Its solution mapping is denoted by K(s,t,x). Results on the existence, uniqueness and regularity of K are given. Finally, it is shown that \(P(t,x)=K(t,t,x)\) is a normal solution of the quasi-Riccati equation. An extension to differential games is proposed.