GIP integrators for matrix Riccati differential equations. (English) Zbl 1337.65074
Summary: Matrix Riccati Differential Equations (MRDEs) are initial value problems of the form:
\[
X' = A_{21} - XA_{11} + A_{22} X - XA_{12} X,\qquad X(0) = X_0.
\]
These equations arise frequently throughout applied mathematics, science, and engineering. It can happen that even when the \(A_{ij}\) are smooth functions of \(t\) or constant, the solution \(X\) may have a singularity or even infinitely many singularities.
This paper shows several classes of numerical algorithms, which we call GIP integrators, that can solve for \(X\) past its singularities. Furthermore, none of the algorithms require knowledge of the placement or even existence of singularities in \(X\). Also, it is shown how embedded Runge-Kutta methods can be used to construct GIP integrators to not only approximate \(X\) past singularities but also provide for error estimation to allow efficient time stepping. Finally, several examples are shown to validate the theory.
This paper shows several classes of numerical algorithms, which we call GIP integrators, that can solve for \(X\) past its singularities. Furthermore, none of the algorithms require knowledge of the placement or even existence of singularities in \(X\). Also, it is shown how embedded Runge-Kutta methods can be used to construct GIP integrators to not only approximate \(X\) past singularities but also provide for error estimation to allow efficient time stepping. Finally, several examples are shown to validate the theory.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 93C05 | Linear systems in control theory |
| 15A24 | Matrix equations and identities |
Keywords:
matrix Riccati differential equation; singularity; GIP integrator; Runge-Kutta; generalized inverse propertyReferences:
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