Nonlinear Analysis: Hybrid Systems 1 (2007) 510–526 www.elsevier.com/locate/nahs Generalized solutions in systems with active unilateral constraints ✩ Boris M. Miller a,∗ , Joseph Bentsman b a School of Mathematical Sciences, Monash University, Clayton, 3800, Victoria, Australia b Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA Received 16 March 2006; accepted 21 March 2006 Abstract The concept of the generalized solution that admits well-posed representation of controlled complex behavior in systems with active unilateral phase constraints is proposed. Based on this concept, the definition of the generalized solution for this class of problems is introduced that encompasses Zeno type behavior and sliding modes along the constraint boundary. The general representation of such solutions in terms of nonlinear differential equations with a measure is derived. The latter is shown to solve a long-standing problem of providing unique extensibility of a trajectory beyond accumulation points in systems with Zeno-type behavior. An example is given, showing that the representation proposed completely captures Zeno-type behavior and provides unique extensibility of solutions without the need to truncate infinite sequences and/or switch system coefficients depending on system motion relative to the generalized coordinates of the accumulation point. c  2006 Elsevier Ltd. All rights reserved. Keywords: Generalized solutions; Phase constraints; Differential equations with a measure 1. Introduction In recent years there has been a significant interest in modeling and control of systems characterized by interaction with actuated unilateral constraints (cf. [1,10–13,42]), such as, for example, mechanical systems with impacts. This interest is motivated by a wide range of applications that involve motions ultra-fast in comparison to the natural system dynamics, including vibro-impact mechanics [1], robotics [16], micro-electro-mechanics [8], power electronics [17,23,25], and, in general, hybrid systems with discrete transitions such as mobile sensor networks [18]. In all these applications, system dynamics can be viewed as being affected by the boundary of a constraint impinging upon a system and exerting on it an impulsive actuation capable of almost abruptly changing system velocity. Therefore, the constraints in these applications become the additional means of control that have to be incorporated both into the system design and the controller synthesis framework. The latter observation, however, significantly complicates optimal controller synthesis for this class of systems. Indeed, the control signals involved act through the constraint boundary as an impulsive feedback localized at some surface. This feature gives rise to a very particular ✩ This work is supported by NSF grants CMS-0000458, CMS-0324630, and ECS-0501407, and by Russian Basic Research Foundation Grant N 05-01-00508. ∗ Corresponding author. E-mail addresses: boris.miller@sci.monash.edu.au (B.M. Miller), jbentsma@uiuc.edu (J. Bentsman). 1751-570X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2006.03.004