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Testing stability by quantifier elimination. (English) Zbl 0886.65087

This paper is concerned with the stability of solutions of a variety of pure or mixed problems arising from ordinary differential equations, partial differential equations and difference equations. If linearization techniques are used then stability often reduces to requiring the zeros of a characteristic polynomial to lie in the unit disk and be simple on the boundary, which can be determined by the Routh-Hurwitz criterion.
These aspects, however, can be viewed as a quantifier elimination problem which, due to the Theorem of Tarski and Seidenberg, is always solvable in a finite number of steps. Unfortunately, the complexity of this algorithm is such that it is impractical to implement. Thus the quantifier elimination by partial algebraic decomposition algorithm is used to solve non-trivial problems analytically using various computer algebra tools. Although this approach is a very powerful representation, it is still the case that many problems of practical importance are not solvable using these techniques.

MSC:

65L07 Numerical investigation of stability of solutions to ordinary differential equations
68W30 Symbolic computation and algebraic computation
39A11 Stability of difference equations (MSC2000)
35B35 Stability in context of PDEs
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
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