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The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
Stability diagram classifying Poincaré maps of linear autonomous system ′ =, as stable or unstable according to their features. Stability generally increases to the left of the diagram. [ 1 ] Some sink, source or node are equilibrium points .
Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." [9] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are [9] R is a non-singular matrix and; R −1 μ < 0.
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1] [2]In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is
An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a step ), then any resulting oscillations in the output will decay at an exponential rate , and the output will tend ...
The Nyquist plot for () = + + with s = jω.. In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer Felix Strecker [] at Siemens in 1930 [1] [2] [3] and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, [4] is a graphical technique ...
In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as ...
Von Neumann stability analysis is a commonly used procedure for the stability analysis of finite difference schemes as applied to linear partial differential equations. These results do not hold for nonlinear PDEs, where a general, consistent definition of stability is complicated by many properties absent in linear equations.