ABSTRACT

As the right hand side of Eq. (eq:nuova) is continuous, also the left hand side can be continuously extended to X . So we can assume x E domA. In this case, the Riccati equation (18) takes the form

(A x ,P x) + {Px ,Ax) + {x ,Q x ) - (D * A * P x ,D * A * P x ) = Q Vx E dom A . Now let us consider the form of the optimal control u(t; Xo) for t > 0. A

dynamical programming argument shows that its restriction to [T, +oo) is op­ timal for the cost evaluated on [T, +oo), so that the optimal control is given in feedback form by:

*+(*; x0) = D*A*P x+(t; x0) . (2 0 ) This completes the arguments of the present paper.