The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form

Fix parameters as in The Ising Population Data (i), adopt the Ising population data with those parameters, and adopt The Ising Population Model Instantiates the Data of the Fluctuation Theory, The Even-Split Equilibrium of the Ising Population Model is a Stationary Mean-Field Triple and is Optimal and The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity, with the matrices $\mathcal{E}_{t}$, $\Theta^{\star}_{t}$, $\tilde{D}_{t}$, $E_{t}$, $\mathsf{B}_{t}$, $Q_{t}$, $V_{t}$, $R_{t}$, $\hat{F}$ and the number $\tilde{d}>0$ defined there. Write $\exp$ for the exponential function and $\sqrt{\cdot}$ for the nonnegative square root, and put

$$\Delta=\sqrt{1+\chi\psi},\qquad \rho=\frac{\Delta-1}{\Delta+1},\qquad \Gamma=\sqrt{4+2\tilde{d}},\qquad \rho_{\Pi}=\frac{\Gamma-2}{\Gamma+2},$$

so that $\Delta>1$, $\Gamma>2$ and $\rho,\rho_{\Pi}\in(0,1)$. Integrals over compact intervals are Lebesgue integrals of continuous integrands. The symbol $\Delta$ here is a positive real number and is unrelated to the probability simplex $\Delta^{2}$, and $\rho$, $\rho_{\Pi}$ are real numbers, unrelated to any metric. Then the following hold.

1. (The scalar control Riccati equation.)(i) There is exactly one map $z:[0,T]\to\mathbb{R}$ with continuous values such that

$$z_{t}=\int_{[t,T]}\Bigl(-4z_{r}-8\chi\,z_{r}^{2}+\tfrac{1}{2}\psi\Bigr)\,dr\qquad(t\in[0,T]),$$

and it is given by

$$z_{t}=\frac{\Delta-1}{4\chi}\cdot\frac{1-\exp\bigl(-4\Delta(T-t)\bigr)}{1+\rho\,\exp\bigl(-4\Delta(T-t)\bigr)}\qquad(t\in[0,T]).$$

It satisfies $z_{T}=0$ and $0\le z_{t}<\dfrac{\Delta-1}{4\chi}$ for every $t\in[0,T]$, is nonincreasing in $t$, and obeys

$$0\ \le\ \frac{\Delta-1}{4\chi}-z_{t}\ \le\ \frac{\Delta-1}{2\chi}\,\exp\bigl(-4\Delta(T-t)\bigr).$$

2. (The scalar filter Riccati equation.)(ii) There is exactly one map $p:[0,T]\to\mathbb{R}$ with continuous values such that

$$p_{t}=\int_{[0,t]}\Bigl(-4p_{r}-2\tilde{d}\,p_{r}^{2}+1\Bigr)\,dr\qquad(t\in[0,T]),$$

and it is given by

$$p_{t}=\frac{1}{\Gamma+2}\cdot\frac{1-\exp(-2\Gamma t)}{1+\rho_{\Pi}\exp(-2\Gamma t)}\qquad(t\in[0,T]).$$

It satisfies $p_{0}=0$ and $0\le p_{t}<\dfrac{1}{\Gamma+2}$ for every $t\in[0,T]$, is nondecreasing in $t$, and obeys

$$0\ \le\ \frac{1}{\Gamma+2}-p_{t}\ \le\ \frac{2}{\Gamma+2}\,\exp(-2\Gamma t).$$

3. (The control Riccati family and hypothesis (H2).)(iii) Define $Z_{t}=z_{t}\,vv^{\top}+\mu\,(T-t)\,\mathsf{n}\mathsf{n}^{\top}$ for $t\in[0,T]$. Every $Z_{t}$ is symmetric, the entries of $Z$ are continuous in $t$, $Z_{T}=\hat{F}=0$, and for all $\gamma,\delta\in\{1,2\}$ and $t\in[0,T]$,

$$Z^{\gamma\delta}_{t}=\hat{F}^{\gamma\delta}+\int_{[t,T]}\Bigl(E_{r}^{\top}Z_{r}+Z_{r}E_{r}-W_{r}R_{r}^{-1}W_{r}^{\top}+Q_{r}\Bigr)^{\gamma\delta}dr,\qquad W_{r}=Z_{r}\mathsf{B}_{r}+\tfrac{1}{2}V_{r} .$$

Equivalently, the entries of $Z$ satisfy $Z^{\gamma\delta}_{t}=Z^{\gamma\delta}_{0}+\int_{[0,t]}\dot{z}^{\gamma\delta}(r)\,dr$ with the continuous densities $\dot{z}^{\gamma\delta}(r)=-\bigl(E_{r}^{\top}Z_{r}+Z_{r}E_{r}-W_{r}R_{r}^{-1}W_{r}^{\top}+Q_{r}\bigr)^{\gamma\delta}$ and the terminal condition $Z_{T}=\hat{F}$. Hence hypothesis (H2) of Completion of Squares and A Priori Control Bound for the Fluctuation Cost holds with this family $Z$. Moreover

$$W_{t}=-z_{t}\,vv^{\top},\qquad \Xi_{t}:=W_{t}R_{t}^{-1}W_{t}^{\top}=8\chi\,z_{t}^{2}\,vv^{\top}\qquad(t\in[0,T]).$$

4. (The Kalman covariance.)(iv) The data $A=\mathcal{E}$, $C=\Theta^{\star}$, $D=\tilde{D}$ on $[0,T]$ and the zero matrix as the initial value $P_{0}$ satisfy the hypotheses of The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound, and the unique solution $\Pi$ of the Kalman covariance Riccati equation that the theorem furnishes for them is

$$\Pi_{t}=p_{t}\,vv^{\top}\qquad(t\in[0,T]).$$

5. (The two pairings.)(v) For every $t\in[0,T]$,

$$\sum_{\gamma=1}^{2}\sum_{\delta=1}^{2}Z^{\gamma\delta}_{t}\,\Theta^{\star\gamma\delta}_{t}=4\,z_{t},\qquad \sum_{\gamma=1}^{2}\sum_{\delta=1}^{2}\Xi^{\gamma\delta}_{t}\,\Pi^{\gamma\delta}_{t}=32\,\chi\,z_{t}^{2}\,p_{t},$$

neither depending on $\mu$, and the map $t\mapsto4z_{t}+32\chi z_{t}^{2}p_{t}$ is continuous on $[0,T]$.