Ordinary Differential Equations on a Compact Interval: Standing Notation

(i) This setting extends Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions: its conventions and notation are in force.

(ii) Let $a,b,c\in\mathbb{R}$ with $a<c\le b$ and $n,p,q\in\mathbb{N}$, let $F$ be a function $[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ or a function $[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$, and let $\xi\in\mathbb{R}^{n}$, respectively $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$. The initial value problem on $[a,c]$ with right-hand side $F$ and initial value $\xi$, respectively $\Xi$, is that of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i), respectively The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii); its solutions on $[a,c]$ are those of Solution of an Initial Value Problem on $[a,c]$ (i), respectively Solution of an Initial Value Problem on $[a,c]$ (ii), and, for $c=b$, its solutions are those of Solution of an Initial Value Problem on $[a,c]$ (iii). Background: Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation, Gronwall's Lemma (Integral Form), The Picard–Lindelöf Theorem on a Closed Interval: Existence, Uniqueness and Continuous Dependence for Globally Lipschitz Systems, A Priori Bounded Initial Value Problems with Locally Lipschitz Right-Hand Side Have Exactly One Solution on the Whole Interval and Time Reversal: Terminal Value Problems as Initial Value Problems.

(iii) For $a<b$, $n\in\mathbb{N}$ and a continuous function $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$, $\Phi_{A}$ is the state-transition matrix of $A$ on $[a,b]$. Background: Linear Systems with Continuous Coefficients: Continuity, Global Lipschitz Bound and Well-Posedness of the Initial Value Problem, Basic Properties of the State-Transition Matrix and Variation of Constants for Linear Systems.

(iv) For $n\in\mathbb{N}$ and $M\in\mathcal{M}_{n}(\mathbb{R})$, $e^{M}$ is the matrix exponential of $M$. Background: Powers of a Square Matrix, The Exponential Series of a Real Square Matrix: Convergence, a Norm Bound and the Exponential Map, The State-Transition Matrix of a Constant Coefficient Matrix Is a Matrix Exponential and Properties of the Matrix Exponential.

(v) The Lyapunov differential equation and the backward Lyapunov differential equation are those of The Lyapunov Differential Equation (i) and The Lyapunov Differential Equation (ii), and the filter and control Riccati differential equations those of The Riccati Differential Equations of Filtering and Control (i) and The Riccati Differential Equations of Filtering and Control (ii), with their solutions as defined there. Background: The Lyapunov Differential Equation: Well-Posedness, Representation, Symmetry, Positive Semidefiniteness, Comparison and the Backward Equation, The Filter Riccati Equation: Global Existence and Uniqueness, Symmetry, and the Lyapunov Bound and The Control Riccati Equation with Cross Term: Global Existence and Uniqueness, Positive Semidefiniteness, and the Lyapunov Bound.