Discrete Approximations of a Controlled Sweeping Process

The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhe- dral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of equality and inequality types. It makes challenging and difficult their anal- ysis and optimization. In this paper we establish some existence results for the sweeping process under consideration and develop the method of discrete approximations that allows us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts

Multidimensional play operators with arbitrary BV inputs

In this paper we provide an integral variational formulation for a vector play operator where the inputs are allowed to be arbitrary functions with (pointwise) bounded variation, not necessarily left or right continuous. We prove that this problem admits a unique solution, and we show that in the left continuous and right continuous cases it reduces to the well known existing formulations

Convex valued geodesics and applications to sweeping processes with bounded retraction

In this paper we provide a formulation for sweeping processes with arbitrary locally bounded retraction, not necessarily left or right continuous. Moreover we provide a proof of the existence and uniqueness of solutions for this formulation which relies on the reduction to the $1$-Lipschitz continuous case by using a suitable family of geodesics for the asymmetric Hausdorff-like distance called "excess".Comment: arXiv admin note: text overlap with arXiv:2005.08966, arXiv:1707.0976

A Feedback-Based Regularized Primal-Dual Gradient Method for Time-Varying Nonconvex Optimization

This paper considers time-varying nonconvex optimization problems, utilized to model optimal operational trajectories of systems governed by possibly nonlinear physical or logical models. Algorithms for tracking a Karush-Kuhn-Tucker point are synthesized, based on a regularized primal-dual gradient method. In particular, the paper proposes a feedback-based primal-dual gradient algorithm, where analytical models for system state or constraints are replaced with actual measurements. When cost and constraint functions are twice continuously differentiable, conditions for the proposed algorithms to have bounded tracking error are derived, and a discussion of their practical implications is provided. Illustrative numerical simulations are presented for an application in power systems

A Feedback-Based Regularized Primal-Dual Gradient Method for Time-Varying Nonconvex Optimization

This paper considers time-varying nonconvex optimization problems, utilized to model optimal operational trajectories of systems governed by possibly nonlinear physical or logical models. Algorithms for tracking a Karush-Kuhn-Tucker point are synthesized, based on a regularized primal-dual gradient method. In particular, the paper proposes a feedback-based primal-dual gradient algorithm, where analytical models for system state or constraints are replaced with actual measurements. When cost and constraint functions are twice continuously differentiable, conditions for the proposed algorithms to have bounded tracking error are derived, and a discussion of their practical implications is provided. Illustrative numerical simulations are presented for an application in power systems