Adjoint Energy Identity for the Kalman Covariance Riccati Equation

Adopt the setting and notation of the Kalman covariance Riccati existence theorem: real numbers $a<b$, a natural number $k\ge1$, assignments $A$, $C$ and $D$ of real matrices with $k$ rows and $k$ columns to each $t\in[a,b]$, all entries continuous in $t$, with every $C(t)$ and every $D(t)$ positive semidefinite, a positive semidefinite $P_{0}$, and the unique assignment $P$ of real $k\times k$ matrices with continuous entries such that

every $P(t)$ being symmetric by that theorem. Integrals of matrix- or vector-valued functions are taken entrywise as Riemann integrals of continuous functions, which exist by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval; products are the matrix product and the matrix-vector product, $(\cdot)^{\top}$ is the transpose, and $x\cdot y$ is the dot product on the Euclidean space $\mathbb{R}^{k}$. Continuity of a real-valued function on an interval is understood as in that theorem. Let $s\in[a,b]$ and $x\in\mathbb{R}^{k}$.

Then the following hold.

1. (Adjoint path.) There is exactly one assignment $\lambda$ of a vector $\lambda(u)\in\mathbb{R}^{k}$ to each $u\in[a,b]$, with continuous components, such that $\lambda(s)=x$ and

2. (Injection profile.) The assignment $\psi$ defined by $\psi(u)=P(u)\lambda(u)$ has continuous components and satisfies

3. (Energy identity.) $x\cdot\psi(s)=x\cdot\bigl(P(s)x\bigr)$; the function

is continuous and nonnegative on $[a,b]$; the real number $\lambda(a)\cdot\bigl(P_{0}\lambda(a)\bigr)$ is nonnegative; and