Clarke Subgradient Research Articles

A class of history-dependent systems of evolution inclusions with applications

This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework of evolution triples of spaces. The multivalued terms are of the Clarke subgradient or of the convex subdifferential form. Using a surjectivity result for multivalued maps and a fixed point argument for a history-dependent operator, we prove that the system has a unique solution. We conclude with two examples of an evolutionary differential variational–hemivariational inequality and of a dynamic frictional contact problem in mechanics, which illustrate the abstract results.

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems**Project supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant agreement No. 823731 CONMECH, and National Science Center of Poland under Preludium Project No. 2017/25/N/ST1/00611. It is also supported by the Natural Science Foundation of Guangxi under Grant No. 2018JJA110006, and

This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge’s fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p -Laplace operator and an implicit obstacle.

Evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications

In this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion with weakly–weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to the non-stationary Navier–Stokes equation with nonmonotone and multivalued frictional boundary conditions.

On definable multifunctions and Łojasiewicz inequalities

We investigate several possibilities of obtaining a Łojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets is definable when composed with definable multifunctions. This allows us to obtain Łojasiewicz-type inequalities for definable multifunctions obtained from Clarke's subgradient or the tangent cone. The paper ends with a Łojasiewicz-type subgradient inequality in the spirit of Bolte–Daniilidis–Lewis–Shiota or Phạm.

On the approximate controllability for some impulsive fractional evolution hemivariational inequalities

In this paper, we study the approximate controllability for some impulsive fractional evolution hemivariational inequalities. We show the concept of mild solutions for these problems. The approximate controllability results are formulated and proved by utilizing fractional calculus, fixed points theorem of multivalued maps and properties of generalized Clarke subgradient under some certain conditions.

Improved Sensitivity Relations in State Constrained Optimal Control

Sensitivity relations in optimal control provide an interpretation of the costate trajectory and the Hamiltonian, evaluated along an optimal trajectory, in terms of gradients of the value function. While sensitivity relations are a straightforward consequence of standard transversality conditions for state constraint free optimal control problems formulated in terms of control-dependent differential equations with smooth data, their verification for problems with either pathwise state constraints, nonsmooth data, or for problems where the dynamic constraint takes the form of a differential inclusion, requires careful analysis. In this paper we establish validity of both `full' and `partial' sensitivity relations for an adjoint state of the maximum principle, for optimal control problems with pathwise state constraints, where the underlying control system is described by a differential inclusion. The partial sensitivity relation interprets the costate in terms of partial Clarke subgradients of the value function with respect to the state variable, while the full sensitivity relation interprets the couple, comprising the costate and Hamiltonian, as the Clarke subgradient of the value function with respect to both time and state variables. These relations are distinct because, for nonsmooth data, the partial Clarke subdifferential does not coincide with the projection of the (full) Clarke subdifferential on the relevant coordinate space. We show for the first time (even for problems without state constraints) that a costate trajectory can be chosen to satisfy the partial and full sensitivity relations simultaneously. The partial sensitivity relation in this paper is new for state constraint problems, while the full sensitivity relation improves on earlier results in the literature (for optimal control problems formulated in terms of Lipschitz continuous multifunctions), because a less restrictive inward pointing hypothesis is invoked in the proof, and because it is validated for a stronger set of necessary conditions.

Nonsmooth identification of mechanical systems with backlash-like hysteresis

Backlash-like hysteresis is one of the nonsmooth and multi-valued nonlinearities usually existing in mechanical systems. The traditional identification method is quite difficult to be used to model the systems involved with such complex nonlinearities. In this paper, a nonsmooth recursive identification algorithm for the systems with backlashlike hysteresis is proposed. In this method, the concept of Clarke subgradient is introduced to approximate the gradients at nonsmooth points and the so-called bundle method is used to obtain the optimization search direction in nonsmooth cases. Then, a recursive algorithm based on the idea of bundle method is developed for parameter estimation. After that, the convergence analysis of the algorithm is investigated. Finally, simulation results to validate the proposed method on a simulated mechanical transmission system are presented.

Optimality and duality for the multiobjective fractional programming with the generalized ( F, ρ) convexity

A class of multiobjective fractional programmings (MFP) are first formulated, where the involved functions are local Lipschitz and Clarke subdifferentiable. In order to deduce our main results, we give the definitions of the generalized ( F, ρ) convex class about the Clarke subgradient. Under the above generalized convexity assumption, the alternative theorem is obtained, and some sufficient and necessary conditions for optimality are also given related to the properly efficient solution for the problems. Finally, we formulate the two dual problems (MD) and (MD1) corresponding to (MFP), and discuss the week, strong and reverse duality.

Lipschitz functions with prescribed derivatives and subderivatives

In general it is difficult to construct Lipschitz functions which are not directly built up from either convex or distance functions. One impediment to such constructions is that outside of the real line it is difficult to find anti-derivatives. The main result of this paper provides, under suitable circumstances, a technique for constructing such anti-derivatives. More precisely, we show that if $f_{1}$, $f_{2}$,$\dots$, $f_{n}$ are continuously differentiable real-valued locally Lipschitz functions defined on a non-empty open subset $A$ of a separable Banach space $X$, then there exists a real-valued locally Lipschitz function $g$ defined on $A$ such that at each point $x\in A$ the Clarke subgradient of $g$ at $x$ equals $co\{\nabla f_{1}(x), \nabla f_{2}(x),\dots, \nabla f_{n}(x)\}$. This same construction also shows that for any finite family $\{T_{1},T_{2},\dots, T_{n}\}$ of maximal cyclically monotone mappings from $A$ into non-empty subsets of $X^{*}$, there exists a real-valued locally Lipschitz function $g$ defined on $A$ such that at each point $x\in A$ the Clarke subgradient of $g$ at $x$ equals $co\{ T_{1}(x), T_{2}(x),\dots, T_{n}(x)\}$. Moreover, we show that $g$ is convex if and only if $T_{1}=T_{2}=\cdots=T_{n}$.