This book characterizes the open-loop and closed-loop solvability for time-delayed linear quadratic optimal control problems. Different from the existing literature, in the current book, we present a theory of deterministic LQ problems with delays which has several new features:

Our system is time-varying, with both the state equation and cost functional being allowed to include discrete and distributed delays, both in the state and the control. We take different approaches to discuss the unboundedness of the control operator.

The open-loop solvability of the lifted problem is characterized by the solvability of a system of forward-backward integral evolution equations and the convexity condition of the cost functional. Surprisingly, the adjoint equations involve some coupled partial differential equations, which is significantly different from that in the literature, where, the adjoint equations are all some anticipated backward ordinary differential equations.

The closed-loop solvability is characterized by the solvability of three equivalent integral operator-valued Riccati equations and two equivalent backward integral evolution equations which are much easier to handle than the differential operator-valued Riccati equations used in the literature to study similar problems.

The closed-loop representation of open-loop optimal control is presented through three equivalent integral operator-valued Riccati equations.