Completion of Squares and A Priori Control Bound for the Fluctuation Cost

Adopt the full setting of the \reftext{thm:n-agent-cost-expansion-2026a}{second-order expansion of the $N$-agent cost}: the \reftext{def:n-agent-fluctuation-processes-2026a}{fluctuation processes} $\mathfrak{s}_t$, $\mathfrak{a}_t$ of a \reftext{def:n-agent-controlled-dynamics-2026a}{solution} about a \reftext{def:mean-field-trajectory-pair-2026a}{mean-field trajectory pair} $(S,A)$, the \reftext{def:c2-transition-rate-extension-2026a}{extension} $(U,\bar{\beta})$ of $\beta$ with derivative bound $K$ and \reftext{def:extended-aggregate-state-drift-2026a}{extended aggregate state drift} $\bar{b}$, the \reftext{def:c2-population-cost-extension-2026a}{extension} of $(L,G)$ - whose open set we here write $U_c$, so the extension is $(U_c,\bar{L},\bar{G})$, freeing the letter $V$ - a \reftext{def:stationary-mean-field-triple-2026a}{stationary co-state} $P$, the \reftext{def:fluctuation-lqg-cost-2026a}{fluctuation linear-quadratic cost} with Hessian coefficients $H_{ij}(t)$ and $F_{\gamma\delta}$, the \reftext{def:n-agent-cost-2026a}{$N$-agent cost} $J^N[h]$, the \reftext{def:mean-field-cost-2026a}{mean-field cost} $J^{MF}=J^{MF}[(S),(A)]$, the quantity $\zeta_N$, and the remainder $R_N$ of that theorem, under its hypothesis $\mathcal{A}=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}_t|^2]\,dt<\infty$, with $\mathbb{E}$ the \reftext{def:expectation-variance-2026a}{expectation} and $|\cdot|$ the Euclidean norm (\reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} to the origin). Let $\Theta$ be the \reftext{def:aggregate-fluctuation-covariance-2026a}{aggregate fluctuation covariance} of $\beta$, and let $\Lambda=l(B+K)\sqrt{l(l+m)}$ and $g_s=\sqrt{N}(b(\Sigma_s,\alpha_s)-b(S_s,A_s))$ be as in the \reftext{lem:fluctuation-state-moment-bound-2026a}{a priori second-moment bound}, $b$ being the \reftext{def:aggregate-state-drift-2026a}{aggregate state drift} of $\beta$. For real matrices with any index ranges we use entry notation: $x\cdot My=\sum_{p,q}M^{pq}x^py^q$ over the matching index ranges (extending the square-matrix convention of the \reftext{lem:fluctuation-weighted-second-moment-2026a}{weighted second-moment evolution lemma}), $(MM')^{pq}=\sum_{r}M^{pr}M'^{rq}$, and $(M^T)^{qp}=M^{pq}$. Define, for $t\in[0,T]$:

where the sans-serif $\mathsf{B}_t$ is distinct from the rate bound $B$, and

\textbf{Hypotheses.} \textbf{(H1)} There is a real $r>0$ with $a\cdot R_ta\ge r|a|^2$ for every $t\in[0,T]$ and $a\in\mathbb{R}^m$; by (H1) and conclusion (a) below, whose proof uses only (H1), each $R_t$ is invertible. \textbf{(H2)} There is a family $Z=(Z_t)_{t\in[0,T]}$ of symmetric real $l\times l$ matrices, continuously differentiable in integral form as in the \reftext{lem:fluctuation-weighted-second-moment-2026a}{weighted second-moment evolution lemma}, whose densities are

and which satisfies the terminal condition $Z_T=\hat{F}$ (a backward Riccati equation).

Define $u_t=\mathfrak{a}_t+R_t^{-1}W_t^T\mathfrak{s}_t$ and $e_s=g_s-E_s\mathfrak{s}_s-\mathsf{B}_s\mathfrak{a}_s$. Then:

\textbf{(a) (Coefficients.)} $Q_t$, $R_t$, and $\hat{F}$ are symmetric; for every $t$ and every $(x,a)\in\mathbb{R}^l\times\mathbb{R}^m$, writing $z=(x,a)\in\mathbb{R}^{l+m}$,

under (H1) each $R_t$ is \reftext{def:positive-semidefinite-matrix-2026a}{symmetric positive definite}, hence \reftext{lem:pd-inverse-2026a}{invertible}; and all entries of $E_t,\mathsf{B}_t,Q_t,V_t,R_t,R_t^{-1},W_t$, and $R_t^{-1}W_t^T$ are continuous in $t$ (using the \reftext{lem:matrix-inverse-continuity-2026a}{continuity of the matrix inverse}), hence \reftext{lem:continuous-compact-interval-bounded-2026a}{bounded}: fix reals $C_Z$ and $C_K$ with $|Z^{\gamma\delta}_t|\le C_Z$ and $|(R_t^{-1}W_t^T)^{j\gamma}|\le C_K$ for all indices and $t$.

\textbf{(b) (Linearization error.)} With $c_e=\tfrac{3}{2}\,l^{3/2}(l+m)\,K$, at every point of $[0,T]\times\Omega$,

\textbf{(c) (Completion of squares.)} All integrals and expectations below are finite, and

\textbf{(d) (A priori control bound.)}

moreover $\mathcal{A}\le2\int_{[0,T]}\mathbb{E}[|u_s|^2]ds+2\,m\,l\,C_K^2\int_{[0,T]}\mathbb{E}[|\mathfrak{s}_s|^2]ds$, and, with $\hat{\Lambda}=\Lambda\sqrt{1+2mlC_K^2}$, for every $t\in[0,T]$,