It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. Since it is equivalent to another lemmas used in optimization and control theory, such as Yakubovich's S-lemma,[1] Finsler's lemma has been given many proofs and has been widely used, particularly in results related to robust optimization and linear matrix inequalities.
The following statements are equivalent:[2] When the matrix L is indefinite, replacing strict inequalities with non-strict ones still maintains the equivalence between the statements of Finsler's lemma.
However, if L is not indefinite, additional assumptions are necessary to ensure equivalence between the statements.
The following statements, which are also referred as Finsler's lemma in the literature, are equivalent:[4] There is also a variant of Finsler's lemma for quadratic matrix inequalities, known as matrix Finsler's lemma, which states that the following statements are equivalent for symmetric matrices Q and L belonging to R(l+k)x(l+k):[5][6] under the assumption that
It is interesting to note that if the strict inequalities are changed to non-strict inequalities, the equivalence does not hold anymore: only the second statement imply the first statement.
[8] Finsler's lemma also generalizes for matrices Q and B depending on a parameter s within a set S. In this case, it is natural to ask if the same variable μ (respectively X) can satisfy
[9] The matrix variant of Finsler lemma has been applied to the data-driven control of Lur'e systems[5] and in a data-driven robust linear matrix inequality-based model predictive control scheme.
[10] Finsler's lemma can be used to give novel linear matrix inequality (LMI) characterizations to stability and control problems.
[4] The set of LMIs stemmed from this procedure yields less conservative results when applied to control problems where the system matrices has dependence on a parameter, such as robust control problems and control of linear-parameter varying systems.
By the use of Finsler's lemma, it is possible to derive a sufficient condition for universal stabilizability in terms of a differential linear matrix inequality.