Completion of Squares for the Linear-Quadratic-Gaussian Cost

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.

Consider a linear-Gaussian state-observation model on $[0,T]$, a control dimension $k\ge1$, a control matrix assignment $B$, and cost data $Q,V,R,F$ such that additionally every $R(t)$ is positive definite; by Invertibility of Positive Definite Matrices each $R(t)^{-1}$ exists and is symmetric positive definite, and $t\mapsto R(t)^{-1}$ has continuous entries by claim 1 of Continuity and Derivative of the Inverse of a Matrix-Valued Map.

Suppose $Z$ assigns to each $t\in[0,T]$ a symmetric real $l\times l$ matrix $Z(t)$ with entries continuous in $t$, satisfying the backward Riccati equation

$$Z(t)=F+\int_t^T\Bigl(A(r)^{\top}Z(r)+Z(r)A(r)-\bigl(Z(r)B(r)+V(r)\bigr)R(r)^{-1}\bigl(Z(r)B(r)+V(r)\bigr)^{\top}+Q(r)\Bigr)\,dr\qquad(0\le t\le T),$$

with entrywise Riemann integrals of continuous functions (existing by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; degenerate intervals by the convention of Mean-Square Riemann Integral of a Family of Random Variables), the matrix product, and the transpose. Define the feedback gain

$$\Gamma(t):=-R(t)^{-1}\bigl(Z(t)B(t)+V(t)\bigr)^{\top}\qquad(0\le t\le T),$$

a real $k\times l$ matrix assignment with continuous entries.

Then for every admissible control $\alpha$ with values in $\mathbb{R}^{k}$, with controlled state $X^{\alpha}$ and cost $J[\alpha]$: the function $t\mapsto\mathbb{E}\bigl[(\alpha_t-\Gamma(t)X^{\alpha}_t)\cdot\bigl(R(t)(\alpha_t-\Gamma(t)X^{\alpha}_t)\bigr)\bigr]$ is continuous on $[0,T]$, and

$$J[\alpha]=\operatorname{tr}\bigl(Z(0)P_0\bigr)+\mathbb{E}[\xi]\cdot\bigl(Z(0)\mathbb{E}[\xi]\bigr)+\int_0^T\operatorname{tr}\bigl(Z(t)\Theta(t)\bigr)\,dt+\int_0^T\mathbb{E}\Bigl[\bigl(\alpha_t-\Gamma(t)X^{\alpha}_t\bigr)\cdot\Bigl(R(t)\bigl(\alpha_t-\Gamma(t)X^{\alpha}_t\bigr)\Bigr)\Bigr]\,dt ,$$

with the trace, the expectation, $\mathbb{E}[\xi]:=(\mathbb{E}[\xi^{1}],\dots,\mathbb{E}[\xi^{l}])$, the initial covariance matrix $P_0$ of claim 1 of The Kalman-Bucy Filter Equation and Its Solution, $\Theta$ from the model, the dot product and matrix-vector product applied componentwise to tuples, and differences of tuples formed componentwise.