We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F(t,u,u′). For periodic systems ...we prove the existence of extremal trajectories, that is solutions of the system in which F(t,u,u′) is replaced by extF(t,u,u′) (= the extreme points of F(t,u,u′)). For Dirichlet systems we show that the extremal trajectories approximate the solutions of the “convex” problem in the C1(T,RN)-norm (strong relaxation).
Multivalued periodic Lienard systems Gasiński, Leszek; Papageorgiou, Nikolaos S.
Journal of mathematical analysis and applications,
09/2019, Volume:
477, Issue:
1
Journal Article
Peer reviewed
Open access
We consider a nonlinear multivalued periodic Lienard system driven by a general strictly monotone, nonhomogeneous homeomorphism which includes as a special case the vector p-Laplacian. The reaction ...has also a maximal monotone map which need not be defined on all of RN. We prove existence theorems for both the convex and nonconvex problems. We also show the existence of extremal trajectories, that is, solutions moving through the extremal points of the multivalued perturbation. We also show that every solution of the convexified system can be obtained as the C1(T;RN)-limit of a sequence of certain extremal trajectories. Finally as an illustration, we examine a nonlinear control system with a priori feedback.
We address the solution of Mixed Integer Linear Programming (MILP) models with strong relaxations that are derived from Dantzig–Wolfe decompositions and allow a pseudo-polynomial pricing algorithm. ...We exploit their network-flow characterization and provide a framework based on column generation, reduced-cost variable-fixing, and a highly asymmetric branching scheme that allows us to take advantage of the potential of the current MILP solvers. We apply our framework to a variety of cutting and packing problems from the literature. The efficiency of the framework is proved by extensive computational experiments, in which a significant number of open instances could be solved to proven optimality for the first time.
Periodic solutions for a class of evolution inclusions Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan D.
Computers & mathematics with applications (1987),
04/2018, Volume:
75, Issue:
8
Journal Article
Peer reviewed
Open access
We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the ...nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.
Nonlinear Multivalued Periodic Systems Gasiński, Leszek; Papageorgiou, Nikolaos S.
Journal of dynamical and control systems,
15/4, Volume:
25, Issue:
2
Journal Article
Peer reviewed
Open access
We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation
F
(
t
,
x
). We prove existence theorems for the ...“convex” problem (that is,
F
is convex valued and for the “nonconvex” problem (that is,
F
is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation
F
(
t
,
x
) is replaced by ext
F
(
t
,
x
), the extreme points of
F
(
t
,
x
)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.
We consider an anti-periodic evolution inclusion defined on an evolution triple of spaces, driven by an operator of monotone-type and with a multivalued reaction term
F
(
t
,
x
). We prove existence ...theorem for the “convex” problem (that is,
F
is convex-valued) and for the “nonconvex” problem (that is,
F
is nonconvex-valued) and we also show the existence of extremal trajectories (that is, when
F
is replaced by
ext
F
). Finally, we prove a “strong relaxation” theorem, showing that the extremal trajectories are dense in the set of solutions of the convex problems.
We have presented in this paper a new cluster Ansatz for the wave operator for open-shell and/or quasidegenerate states, which takes care of strong relaxation and correlation effects in a compact and ...efficient manner. This Ansatz allows contraction among the various cluster operators via spectator orbitals, accompanied by suitable combinatorial factors. Since both the orbital and the correlation relaxations are treated on the same footing, it allows us to develop a very useful direct method for energy differences for open shell states relative to a closed-shell ground state, where the total charge for the two states may differ. We have discussed a new spin-free coupled cluster (CC) based direct method and illustrated its performance by evaluating electron affinity of a neutral doublet radical. We have also indicated how the scope of the theory can be extended to compute the state energies of simple open shell configurations as well. In that case, the CC equations terminate after the quartic power of cluster operators – exactly as in the closed-shell situation, which is not the case for the current methods.
We consider nonlinear nonconvex evolution inclusions driven by time‐varying subdifferentials ∂ ϕ ( t , x ) without assuming that ϕ ( t .) is of compact type. We show the existence of extremal ...solutions and then we prove a strong relaxation theorem. Moreover, we show that under a Lipschitz condition on the orientor field, the solution set of the nonconvex problem is path‐connected in C ( T , H ). These results are applied to nonlinear feedback control systems to derive nonlinear infinite dimensional versions of the “bang‐bang principle.” The abstract results are illustrated by two examples of nonlinear parabolic problems and an example of a differential variational inequality.
PDF
