The Non Smooth Dynamical Systems (NSDS) are a very special kind of dynamical systems characterized by their nonsmoothness in the evolution with respect to time and by a set of non smooth generalized equations. The so-called Moreau’s Sweeping Process[9, 10] is a special kind of differential inclusion with a maximal monotone operator[5] which appears to be a very nice formulation for the unilater...

The Non Smooth Dynamical Systems (NSDS) are a very special kind of dynamical systems characterized by their nonsmoothness in the evolution with respect to time and by a set of non smooth generalized equations. The so-called Moreau’s Sweeping Process[9, 10] is a special kind of differential inclusion with a maximal monotone operator[5] which appears to be a very nice formulation for the unilater...

In this paper, we consider the stabilization problem of a one-dimensional wave equation with unknown disturbance. In order to stabilize the system with disturbance, we design a distributed feedback controller by employing the idea of sliding mode control technology. For the resulted nonlinear closed-loop system, we prove its solvability by using the maximal monotone operator. Further we prove t...

This is a survey of recent results about bipotentials representing multivalued operators. The notion of bipotential is based on an extension of Fenchel’s inequality, with several interesting applications related to non associated constitutive laws in non smooth mechanics, such as Coulomb frictional contact or non-associated Drücker-Prager model in plasticity. Relations betweeen bipotentials and...

We analyze and characterize maximal monotonicity of linear relations (set-valued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most single-valued operators by Phelps and Simons and by Bauschke, Borwein and Wang. Furthermore, a description of skew linear relations in terms of the Fitzpatrick fa...

We analyze proximal methods based on entropy-like distances for the minimization of convex functions subject to nonnegativity constraints. We prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator. We also consider linearly constrained convex problems and establish a qua...

This paper deals with the design of state observers for a class of systems with set-valued right-handsides. Two main classes of set-valued mappings are considered: maximal monotone and time-invariant; or normal cones to possibly time-varying sets (not necessarily convex). Such controlled systems can be embedded in various kinds of differential inclusions with absolutely continuous and bounded v...

Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions were proposed, the first...

is not properly contained in the graph of any other monotone mapping. It is known that T is maximal iff R(I+rT ) = H for every r > 0, where R(I+rT ) = ∪ {z+rTz : z ∈ H,Tz ̸= ∅}. Monotone operators have proven to be a key class of objects in modern Optimization and Analysis; see, e.g., the books [1-7] and the references therein. Let us consider the zero point problem for a monotone operator T on ...