On the quadratic optimal control problem for Volterra integro- differential equations (Q1062269)

The paper deals with the synthesis of the following optimal control problem: minimize \[ J(u)=\int^{T}_{0}\{| u(t)|^ 2+| x(t)|^ 2\}dt \] over all \((u,x)\in L^ 2(0,T)\times C(0,T)\) subject to the condition \[ \dot x(t)=\int^{t}_{0}K(t- s)x(s)ds+u(t),\quad x(0)=x_ 0. \] Using the abstract theory of ordinary differential equations, under appropriate hypothesis on the kernel K, a feedback law is given.