Optimization of a Stability Bound
Copyright policy )Optimization of a Stability Bound
The paper deals with the following question. Let \(A\in \mathbb{R}^{n\times n}\) be an asymptotically stable matrix. What is the ``largest'' \(\Delta A\) such that \(A+\Delta A\) remains stable? Letting \(V= x^ T Bx\) be the Lyapunov function for \(\dot x= Ax\), where \(B= B^ T>0\), and using one of the two Lyapunov equations \(A^ T B+ BA+ C=0\) or \(AB+ BA^ T+ C=0\), respectively, in which \(C= C^ T>0\), the author states four alternative upper bounds for \(\Delta A\). To discuss optimal choice for \(B\) or \(C\) and the ``best'' of the four bounds, some numerical examples are presented.
asymptotically stable matrix, Lyapunov function, Lyapunov equations, Perturbations of ordinary differential equations, Norms of matrices, numerical range, applications of functional analysis to matrix theory, robust stability, Stability of solutions to ordinary differential equations, Robust stability
asymptotically stable matrix, Lyapunov function, Lyapunov equations, Perturbations of ordinary differential equations, Norms of matrices, numerical range, applications of functional analysis to matrix theory, robust stability, Stability of solutions to ordinary differential equations, Robust stability
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