Lay summary
It is proposed to address the Lyapunov function construction from the perspective of iteratively selecting, deforming, and transforming equivalence sets of level surfaces of sub-Lyapunov functions.
Stated differently, we proceed from the simple to the more complex, starting with a small-sized state space (say 1 dimensional) and gradually increasing the synthesis until the full-sized original system is attained.
Iteratively reducing the state space is not a new idea, however, and it appears at the core of many numerical methods applied to linear systems, such as for either pole placement (Miminis-Paige algorithm1 and the Nichols-Van Dooren), or finding linear time invariant Lyapunov functions (using the Hessenberg-Shur).
In the nonlinear setting this idea of reducing the system to a staircase form and then constructing the Lyapunov function from this representation is not commonplace.
There is nevertheless a formal calculus approach based on algebraic geometry that, to a certain extent, goes along this line of thought. The idea is cast in the polynomial multivariable setting. The reason for this is that for linear systems, only quadratic polynomial forms need to be considered.
Therefore, restricting the class of systems to those for which multivariate polynomials are used (as building blocks for the Lyapunov function construction) is a good start. This handles a relatively large class of nonlinear systems. Moreover, it is well adapted through the use of algebraic ideal basis construction such as Gröbner basis. The link to our proposal comes from the fact that Gröbner basis construction heavily relies on Dikson’s Lemma. This lemma is used for testing that a particular monomial belongs to a monomial ideal. The test is a staircase comparison between the particular monomial and those defining the ideal, i.e. if it lies below the staircase it does not belong to the ideal. Nevertheless, there is yet no clear way on how to use such tools so as to address the level set modification required. Especially, the main difficulty, as opposed to classical Gröbner basis usage for systems of polynomial equations is that, in this case, we deal with nonlinear polynomial inequalities, instead of equalities.