The Control Riccati Equation with Cross Term: Global Existence and Uniqueness, Positive Semidefiniteness, and the Lyapunov Bound

In the setting of Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions, let $a,b\in\mathbb{R}$ with $a<b$ and let $n,m\in\mathbb{N}$. Let $A,Q:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $B,V:[a,b]\to\mathcal{M}_{n\times m}(\mathbb{R})$ and $R:[a,b]\to\mathcal{M}_{m}(\mathbb{R})$ be continuous on $[a,b]$ such that for every $t\in[a,b]$, $Q(t)$ lies in the set $\mathcal{S}(n)$ of Euclidean Space and Real Matrices: Standing Notation (iv) and $R(t)\in\mathcal{S}(m)$ is positive definite, hence invertible (Invertibility of Positive Definite Matrices (i)), with inverse $R(t)^{-1}$ as in Euclidean Space and Real Matrices: Standing Notation (x). Let $\tilde A,\tilde Q:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be the functions with

$$\tilde A(t)=A(t)-B(t)R(t)^{-1}V(t)^{\top},\qquad\tilde Q(t)=Q(t)-V(t)R(t)^{-1}V(t)^{\top}\qquad\bigl(t\in[a,b]\bigr)$$

(each $\tilde Q(t)$ lies in $\mathcal{S}(n)$, by Invertibility of Positive Definite Matrices (ii), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (vii), Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (v) and Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i)). Suppose that every $\tilde Q(t)$ is positive semidefinite, let $F\in\mathcal{S}(n)$ be positive semidefinite, and let $\preceq$ be the positive semidefinite ordering on $\mathcal{S}(n)$ of Euclidean Space and Real Matrices: Standing Notation (v).

(i) The terminal value problem of The Riccati Differential Equations of Filtering and Control (ii) with data $A$, $B$, $Q$, $V$, $R$ and $F$,

$$Z'(t)=-\Bigl(A(t)^{\top}Z(t)+Z(t)A(t)-\bigl(Z(t)B(t)+V(t)\bigr)R(t)^{-1}\bigl(Z(t)B(t)+V(t)\bigr)^{\top}+Q(t)\Bigr)\quad\bigl(t\in[a,b]\bigr),\qquad Z(b)=F,$$

has exactly one solution $Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$.

In (ii) and (iii), $Z$ is the solution of (i).

(ii) For every $t\in[a,b]$, $Z(t)\in\mathcal{S}(n)$ and $Z(t)$ is positive semidefinite.

(iii) The functions $\tilde A$ and $\tilde Q$ are continuous on $[a,b]$, by Continuity and Derivative of the Inverse of a Matrix-Valued Map (i), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv), Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v) and Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals (ii). Let $\hat Z:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be the solution of the terminal value problem of The Lyapunov Differential Equation (ii) with data $\tilde A$, $\tilde Q$ and $F$,

$$\hat Z'(t)=-\bigl(\tilde A(t)^{\top}\hat Z(t)+\hat Z(t)\tilde A(t)+\tilde Q(t)\bigr)\quad\bigl(t\in[a,b]\bigr),\qquad\hat Z(b)=F$$

(it exists and is unique by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (vi), and $\hat Z(t)\in\mathcal{S}(n)$ for every $t\in[a,b]$ by The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation (viii)). Then $Z(t)\preceq\hat Z(t)$ for every $t\in[a,b]$.