Adjoint identity for the cost difference of two admissible pairs

Let $t_0<t_1$, let the dynamics $F$ be jointly continuous, and let $(u_0,x_0)$ be an admissible pair on $[t_0,t_1]$ with $u_0$ essentially bounded. Let $F_x(x_0(s),u_0(s))$ and $\ell_x(x_0(s),u_0(s))$ be operator-valued and functional-valued coefficient paths along the pair, assumed integrable on $[t_0,t_1]$; in the application they are the state derivatives of the dynamics and of the running cost $\ell$, but only their integrability is used. Let $\lambda$ be absolutely continuous on $[t_0,t_1]$ with $\lambda(t_1)=0$ and satisfy the adjoint equation

for almost every $t\in(t_0,t_1)$. Then for every admissible pair $(u,x)$ with the same initial state and an essentially bounded control,

where $J(u,x)=\int_{t_0}^{t_1}\ell(x(s),u(s))\,ds$ is the running cost, $H(x,u,\lambda)=\langle\lambda,F(x,u)\rangle+\ell(x,u)$ is the Hamiltonian, and every integrand is evaluated at the time $s$.

This exact identity is the integration by parts behind Luenberger's equation (9). The function $s\mapsto\langle\lambda(s),x(s)-x_0(s)\rangle$ is absolutely continuous and vanishes at both endpoints, and the adjoint equation converts its derivative into the linearized-state term. The identity expresses the cost difference as a Hamiltonian difference plus a term of first order in $x-x_0$, without any optimality assumption, and is the starting point of every needle-variation argument for the minimum principle.

Formalization Note. The essential bounds on the controls and the joint continuity of $F$ guarantee that the Hamiltonian paths are integrable. The coefficient paths $F_x$ and $\ell_x$ enter only through the adjoint equation and their integrability; no differentiability of $F$ or $\ell$ is assumed.