Differential-algebraic inclusions with maximal monotone operators

Abstract

Abstract-The term differential-algebraic inclusions (DAIs) not only describes the dynamical relations using set-valued mappings, but also includes the static algebraic inclusions, and this paper considers the problem of existence of solutions for a class of such dynamical systems described by the inclusionfor a symmetric positive semi-definite matrix P ∈ R n×n , and a maximal monotone operator M : R n ⇒ R n . The existence of solutions is proved using the tools from the theory of maximal monotone operators. The class of solutions that we study in the paper have the property that, instead of the whole state, only P x is absolutely continuous and unique. This framework, in particular, is useful for studying passive differential-algebraic equations (DAEs) coupled with maximal monotone relations. Certain class of irregular DAEs are also covered within the proposed general framework. Applications from electrical circuits are included to provide a practical motivation.

Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems

“…𝑡 ≥ 0 0 ≤ 𝜆(𝑡) ⊥ 𝑤 (𝑡) = 𝐶𝑥 (𝑡) + 𝐷𝜆(𝑡) + 𝐹 (𝑡) ≥ 0, for all 𝑡 ≥ 0 with 𝑥 (𝑡) ∈ ℝ 𝑛 , 𝜆(𝑡) ∈ ℝ 𝑚 , 𝐴, 𝐵, 𝐶, 𝐷 constant matrices of appropriate dimensions, 𝑃 ∈ ℝ 𝑛×𝑛 has rank 𝑝 < 𝑛. Dissipativity is a fundamental property in Systems and Control [25,87,88]. The material in this article relies on a result in [28,32,51] on a special Weierstrass form for passive descriptor variable systems [25,Section 3.1.7] which possess a minimal state-space realization [28,Theorem 3.1]. The passivity of the quintuple (𝑃, 𝐴, 𝐵, 𝐶, 𝐷) means the passivity of the operator 𝜆 ↦ → 𝑤, see Appendix d. Assuming that 𝐸 (𝑡) = 0 and 𝐹 (𝑡) = 0, passivity and minimality, this special form writes as (see [51] [32, Equation ( 16)] [28, Proposition A.3]):…”

“…The LMI in (d.2) holds. Due to the passivity and the complementarity conditions which imply 𝜆(𝑡) ⊤ 𝑤 (𝑡) = 0 for all times, it follows that for all 𝑡 ≥ 0 one obtains 𝑉 (𝑥 1 (𝑡), 𝑥 2 (𝑡)) [32] and [28, Proof of Theorem 5.1], with 𝑋 1 ≻ 0 if the system is strongly SPR and minimal [28,Theorem 5.4], see Appendix d. In the latter case 𝑥 1 (•) is bounded, 𝐴 1 is Hurwitz 1 , D ≻ 0 and 𝐵 3 is full row rank (⇒ 𝑛 2 ≤ 𝑚) [28,Proposition A.4].…”

“…The LDCS in (2.2) are suitably transformed in [32] to enable their analysis. The transformation may be seen as a kind of relative degree reduction, involving the multiplier derivative.…”

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…Time-delays in f (t, x) are taken into account in [318]. Further extensions of (2.16) are proposed in [154], as:…”

“…Existence or non-existence of accumulation of events (Zeno states) is analysed in [298]. Extensions of LCS may be found in [135,136,154]. For NLCS (2.24), local analytic solutions are used in [549], global AC or right-continuous LBV (hence admitting state jumps) solutions in [143], see also [550] for a class of cone NLCS with solutions in W 1,2 ([a, b]); R n ) and integrable λ.…”