Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

@article{Donchev2007DiscreteAR,
  title={Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces},
  author={T. Donchev and E. Farkhi and B. Mordukhovich},
  journal={Journal of Differential Equations},
  year={2007},
  volume={243},
  pages={301-328}
}
We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modified one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main… Expand
Implicit Euler approximation and Optimization of one-sided Lipschitzion differntial inclusions
This paper concerns the study of the generalized Bolza problem governed by differential inclusions satisfying the so-called "relaxed one-sided Lipschitzian" (ROSL) condition with respect to the stateExpand
Discrete Approximations and Optimization of Evolution Inclusions
The paper studies discrete approximations of nonconvex valued evolution inclusions with the right-hand side satisfying Kamke condition which is more general than the Lipschitz one and more convenientExpand
Optimal Control of Semilinear Unbounded Evolution Inclusions with Functional Constraints
TLDR
This paper constructs a sequence of discrete approximations to the optimal control problem for evolution inclusions and proves that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Expand
The optimality principle for second-order discrete and discrete-approximate inclusions
This paper deals with the necessary and sufficient conditions of optimality for the Mayer problem of second-order discrete and discrete-approximate inclusions. The main problem is to establish theExpand
Optimal control of a nonconvex perturbed sweeping process
TLDR
The paper establishes the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data. Expand
Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space
This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on theExpand
Discrete Approximations and Optimal Control of Nonsmooth Perturbed Sweeping Processes
TLDR
The main goal of this paper is developing the method of discrete approximations to derive necessary optimality conditions for a class of constrained sweeping processes with nonsmooth perturbations, and employing advanced tools of second-order variational analysis to establish novel results for original nonsm Smooth sweeping control problems that include extended Euler-Lagrange and maximization conditions for local minimizers. Expand
Optimal control of nonconvex integro-differential sweeping processes
This paper is devoted to the study, for the first time in the literature, of optimal control problems for sweeping processes governed by integro-differential inclusions of the Volterra type withExpand
Optimization of a Perturbed Sweeping Process by Constrained Discontinuous Controls
TLDR
A refined method of discrete approximations is developed with establishing its well-posedness and strong convergence to attack optimal control problems described by a controlled version of Moreau's sweeping process governed by convex polyhedra, where measurable control actions enter additive perturbations. Expand
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process while using controls to determine theExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 25 REFERENCES
Variational Analysis of Evolution Inclusions
TLDR
The approach and results developed in the paper make a bridge between optimal control/dynamic optimization and constrained mathematical programming problems in infinite-dimensional spaces. Expand
Discrete Approximations and Refined Euler--Lagrange Conditions forNonconvex Differential Inclusions
This paper deals with the Bolza problem $(P)$ for differential inclusions subject to general endpoint constraints. We pursue a twofold goal. First, we develop a finite difference method for studyingExpand
Stability and Euler Approximation of One-sided Lipschitz Differential Inclusions
Ordinary differential and functional-differential inclusions with compact right-hand sides are considered. Stability theorems of Filippov's type in the convex and nonconvex case are proved under aExpand
Discrete Approximations of Differential Inclusions in Infinite-Dimensional Spaces
In this paper we study discrete approximations of continuous-time evolution sy~tems governed by differential inclusions with nonconvex compact values in infinite-dimensional spaces. Our crucialExpand
APPROXIMATION OF LOWER SEMICONTINUOUS DIFFERENTIAL INCLUSIONS
We study the approximation of the solution set of lower semicontinuous differential inclusions having the form: by the solution set of the discretized one. When F(t, ·) is one sided Lipschitz inExpand
ON THE STRUCTURE OF THE SOLUTION SET FOR DIFFERENTIAL INCLUSIONS IN A BANACH SPACE
In this article a differential inclusion is considered, where the mapping takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect toExpand
On vector measures
The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space X to have the propertyExpand
Semicontinuous differential inclusions
Almost upper and almost lower semicontinuous differential inclusions in a Banach space with uniformly convex dual are considered. We suppose that the right-hand side is a sum of one-side LipschitzExpand
A Bogolyubov-Type Theorem with a Nonconvex Constraint in Banach Spaces
TLDR
An analogue of the classical Bogolyubov theorem, with a nonconvex constraint, is proved, based on a relaxation argument, as in the Filippov--Wazewski theorem, about a solution set of a Cauchy problem for a differential inclusion satisfying a Lipschitz condition. Expand
Existence of Solutions to Differential Inclusions
In what follows we shall deal with the existence and properties of solutions to differential inclusions of the form $$x'\left( t \right) \in F\left( {x\left( t \right)} \right)$$ (1) or Expand
...
1
2
3
...