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The logarithmic norm. History and modern theory

2006; Springer Science+Business Media; Volume: 46; Issue: 3 Linguagem: Inglês

10.1007/s10543-006-0069-9

1572-9125

Gustaf Söderlind,

Stability and Control of Uncertain Systems

In his 1958 thesis Stability and Error Bounds, Germund Dahlquist introduced the logarithmic norm in order to derive error bounds in initial value problems, using differential inequalities that distinguished between forward and reverse time integration. Originally defined for matrices, the logarithmic norm can be extended to bounded linear operators, but the extensions to nonlinear maps and unbounded operators have required a functional analytic redefinition of the concept. This compact survey is intended as an elementary, but broad and largely self-contained, introduction to the versatile and powerful modern theory. Its wealth of applications range from the stability theory of IVPs and BVPs, to the solvability of algebraic, nonlinear, operator, and functional equations.