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

Adopt the full setting of the second-order expansion of the $N$-agent cost: the fluctuation processes $\mathfrak{s}_t$, $\mathfrak{a}_t$ of a solution about a mean-field trajectory pair $(S,A)$, the extension $(U,V,\bar{\beta})$ of $\beta$ with derivative bound $K$ and its extended aggregate state drift $\bar{b}$, the extension of $(L,G)$ - whose open set, written $W$ in the cost expansion theorem, we here write $U_c$, so the extension is $(U_c,\bar{L},\bar{G})$, freeing the letter $W$; the time-indexed matrices $V_t$ and $W_t$ below are unrelated to the control-side open set $V$ of $(U,V,\bar{\beta})$ - a stationary co-state $P$, the fluctuation linear-quadratic cost with Hessian coefficients $H_{ij}(t)$ and $F_{\gamma\delta}$, the $N$-agent cost $J^N[h]$, the mean-field cost $J^{MF}=J^{MF}[(S),(A)]$, the quantity $\zeta_N$, and the remainder $R_N$ of that theorem, under its hypotheses - the control set $\mathcal{A}$ being convex and $\mathcal{A}_2=\int_{[0,T]}\mathbb{E}[|\mathfrak{a}_t|^2]\,dt<\infty$ - with $\mathbb{E}$ the expectation and $|\cdot|$ the Euclidean norm (Euclidean distance to the origin). Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $\Theta$ be the 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 a priori second-moment bound, $b$ being the 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 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

Hypotheses. (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. (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 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:

(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 symmetric positive definite, hence 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 continuity of the matrix inverse), as are the entries of $Z_t$ by (H2); all are hence bounded on $[0,T]$ by the extreme value theorem: 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 $t$.

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

(c) (Completion of squares.) All integrals and expectations below are finite, and

(d) (A priori control bound.)

moreover $\mathcal{A}_2\le2\int_{[0,T]}\mathbb{E}[|u_s|^2]ds+2\,m\,l\,C_K^2\int_{[0,T]}\mathbb{E}[|\mathfrak{s}_s|^2]ds$, and, with $\hat{\Lambda}=\Lambda\sqrt{1+2mlC_K^2}$, for every $t\in[0,T]$,