Optimal control problems in delay differential equation

In this dissertation we are concerned with optimal control problems whose costs are quadratic and whose states are governed by linear delay equations and general boundary conditions. The basic new idea of this dissertation is to introduce a new class of linear operators in such a way that the state equation subject to a starting function can be viewed as an inhomogeneous boundary value problem in the new linear operator equation. In this way we avoid the usual semigroup theory treatment to the problem and use only linear operator theory. Specifically, the state x(t) and boundary conditions have the following form:(UNFORMATTED TABLE OR EQUATION FOLLOWS) $$\eqalign{&\rm u(\cdot)\in {\bf U}\cr&\cases{\rm\dot x(t){-}A\sb1(t)x(t){-}A\sb2(t)x(t-\tau)=B(t)u(t),\ &\rm t \in \lbrack 0, \rm t\sb1\rbrack\cr\cr\rm\quad x(t) = B(t)u(t),&\rm t \in \lbrack-\tau,0)\cr}\cr&\rm\int\sbsp{-\tau}{t\sb1}(f(t)x(t) + g(t)u(t))\ dt = r,\cr}$$ (TABLE/EQUATION ENDS)where U is a convex subset of ${\bf L}\sb2(\lbrack{-}\tau,\rm t\sb1\rbrack\: {\bf R}\sp{n}),$ r is a constant vector, and A$\sb1$(t), A$\sb2$(t), B(t), f(t), g(t) are appropriate real matrix-valued functions of t $\in \lbrack{-}\tau, \rm t\sb1\rbrack.$ The cost functional has the form:(UNFORMATTED TABLE OR EQUATION FOLLOWS) $$\eqalign{{\bf J}\rm(u,x)&=\rm\int\sbsp{-\tau}{t\sb1}(\vert Uu\vert\sp2+\vert Wx\vert\sp2)dt + \vert{\bf F}\sb1(u,x)\vert\sp2\cr\rm where\qquad\qquad\qquad\cr 1)\ {\bf F}\sb1\rm(u,x)&=\rm\int\sbsp{-\tau}{t\sb1}(f\sb{11}\sp\*(t)x(t) + f\sb{12}\sp\*(t)u(t))\ dt.\cr}$$ (TABLE/EQUATION ENDS)where 2) U(t), W(t), f$\sb{11}$(t), f$\sb{12}$(t) are appropriate real matrix-valued functions of $\rm t \in \lbrack {-}\tau, t\sb1\rbrack.$

We develop an existence theory for optimal controls over convex sets and also consider a feedback synthesis for a certain class of optimal control problems.