Finite-horizon bounded-control linear-quadratic ODE problem

Fix a finite horizon $T>0$, state and control dimensions $n,m\ge 1$, and the real quotient space $L^2(0,T;{\mathbb{R}}^m)$ of measurable controls modulo equality almost everywhere (The space $L^p(\mu )$ as the quotient by null functions). Let $A\in L^{\mathrm{\infty }}(0,T;{\mathbb{R}}^{n\times n})$ and $B\in L^{\mathrm{\infty }}(0,T;{\mathbb{R}}^{n\times m})$ be equivalence classes, let $x_0,x_T\in {\mathbb{R}}^n$, and let $x_d\in L^2(0,T;{\mathbb{R}}^n)$. Let $U_c\subset {\mathbb{R}}^m$ be nonempty, compact and convex. The pointwise control set is

$${\mathcal{U}}_c:=\{u\in L^2(0,T;{\mathbb{R}}^m):u(t)\in U_c\text{ for almost every }t\}\mathrm{.}$$

Let $Q\in L^{\mathrm{\infty }}(0,T;{\mathbb{R}}^{n\times n})$ and $R\in L^{\mathrm{\infty }}(0,T;{\mathbb{R}}^{m\times m})$ be symmetric almost everywhere, with $Q(t)⪰0$ and $rI⪯R(t)⪯MI$ almost everywhere for constants $0<r\le M<\mathrm{\infty }$. Let $P\in {\mathbb{R}}^{n\times n}$ be symmetric positive semidefinite. For each $u\in {\mathcal{U}}_c$, its state is the unique continuous solution of the Volterra equation

$$x_u(t)=x_0+{\int }_0^t(A(s)x_u(s)+B(s)u(s))ds\mathrm{.}$$

The same state equation is defined for every $u\in L^2(0,T;{\mathbb{R}}^m)$, and the running and terminal cost on that ambient space is

$$J(u)=\frac{1}{2}{\int }_0^T\text{\hspace{0.17em}⁣}((x_u(t)-x_d(t){)}^{T}Q(t)(x_u(t)-x_d(t))+u(t{)}^{T}R(t)u(t))dt+\frac{1}{2}(x_u(T)-x_T{)}^{T}P(x_u(T)-x_T)\mathrm{.}$$

The constrained problem minimizes this ambient functional over $\mathcal{F}$, where additional path or terminal constraints are represented by a nonempty convex weakly sequentially closed set $\mathcal{F}\subseteq {\mathcal{U}}_c$; with no additional state constraints, $\mathcal{F}={\mathcal{U}}_c$. “Weakly sequentially closed” means that every weakly convergent sequence in $\mathcal{F}$ has its weak limit in $\mathcal{F}$ (Weak convergence of nets and sequences). The definition asserts no optimizer or adjoint existence; the state-map well-definedness is supplied by The bounded measurable linear ODE has a unique affine state map ↗.