The Linear-Quadratic-Gaussian Cost Functional

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 $k\ge1$, and a control matrix assignment $B$ as in Controlled State and Controlled Observations in the Linear-Gaussian Model.

Cost data for this system is a name for a tuple of assignments $Q$ (real $l\times l$) and $R$ (real $k\times k$) of symmetric matrices to each $t\in[0,T]$ and an assignment $V$ (real $l\times k$) to each $t\in[0,T]$, all with entries continuous in $t$, together with a symmetric real $l\times l$ matrix $F$.

For an admissible control $\alpha$ with values in $\mathbb{R}^{k}$ and its controlled state $X^{\alpha}$, the linear-quadratic-Gaussian cost of $\alpha$ is the real number

$$J[\alpha]:=\int_0^T\mathbb{E}\Bigl[X^{\alpha}_t\cdot\bigl(Q(t)X^{\alpha}_t\bigr)+2\,X^{\alpha}_t\cdot\bigl(V(t)\alpha_t\bigr)+\alpha_t\cdot\bigl(R(t)\alpha_t\bigr)\Bigr]\,dt+\mathbb{E}\Bigl[X^{\alpha}_T\cdot\bigl(F X^{\alpha}_T\bigr)\Bigr],$$

with the expectation, the dot product and matrix-vector product applied componentwise to tuples of random variables, and the Riemann integral.

Well-definedness. Each expectation above is defined and finite, and the integrand is a continuous function of $t$, by claims 1-2 of Expected Bilinear Forms: Trace Formula and Mean-Square Continuity, the component families of $X^{\alpha}$ and $\alpha$ being mean-square continuous; the integral therefore exists by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval.