Stability and invariance results for a class of non-monotone set-valued Lur’e dynamical systems

Abstract

In this paper, we analyse the well-posedness, stability and invariance results for a class of non-monotone set-valued Lur’e dynamical system which has been widely studied in control and applied mathematics. Many recent researches deal with the case when the set-valued part is the sub-differential of some proper, convex, lower semicontinuous function in order to use the nice properties of maximally monotone operators. But in practice, particularly in electronics, there are some devices such as diac, silicon controller rectifier (SCR) that their voltage-current characteristics are not monotone but only locally hypo-monotone. This fact motivates us to write the paper which is organized as follows: firstly, the existence and uniqueness of solutions are proved by using Filippov’s method and local hypo-monotonicity; then, the stability analysis and generalized LaSalle’s invariance principle are presented. The theoretical results are supported by numerical simulations for some examples in electronics. Our methology is based on non-smooth and variational analysis.

Keywords:

• well-posedness
• stability analysis
• LaSalle’s invariance principle
• hypo-monotone
• Filippov’s solutions
• electronic devices

AMS Subject Classifications:

• 37N35
• 37N40
• 45M10
• 37KJ55
• 46N10