Conclusion

We showed that the SDP problems, which is one of the greatly important problems in LMI problems, are reduced to quantifier elimination problems by Sylvester's criterion and presented the concrete procedure to solve them by using QE.

The efficiency of the method proposed here depends on deeply that of QE procedure. Due to the great theoretical and practical complexity of QE for the elementary theory of the real closed field, currently the method is practically applicable to moderate size problems. However, compared with existing (numerical) methods for SDP and ESDP problems the QE based method has many merits as follows;

• Non-convex case can be dealt with,

• Parametric case can be dealt with,

• Easy to handle the domain restricted case and parameter uncertainty,

• Produce exact answer and check the infeasibility exactly.

Furthermore, it is important and necessary to examine the practically effective usage of QE for not only LMI & BMI but also other control problems hereafter.