There are three kinds: stable, critical and unstable, which correspond to asymptotic stability, stability and instability in the sense of Lyapunov. There are two common mathematical models for linear systems: ① Higher-order differential equations, where x(i) represents the first derivative of X and ai is the scalar coefficient. ② First-order differential equation, where a is n×n constant matrix. The stability theorems of linear systems represented by these two mathematical models are given below. ① Stability theorem of linear systems of higher order differential equations. If the characteristic roots of the first equation above, that is, the characteristic equation λ n+a1λ n-1+…+an-1λ+an = 0, the system is stable; If there is a zero root or a pair of imaginary roots and the other roots have negative real parts, the system is critical; In other cases, the system is unstable. In order to avoid seeking roots, the stability of the system can be judged directly by the coefficients of the equation. There are algebraic criteria: A. leonid hurwicz criterion and E. J. Lao Si test. ② Stability theorem of first order linear equations. If the characteristic roots of the second equation above, that is, the roots of the characteristic equation det [λ ι-a] = 0 all have negative real parts, then the system is stable; If there is a root of positive real part, the system is unstable; If the algebraic multiplicity of a root with zero real part is equal to its geometric multiplicity, and all other roots have negative real parts, this is the critical case; If the algebraic multiplicity of multiple roots with zero real part is greater than the geometric multiplicity, the system is unstable.