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1

An existence result for nonlinear evolution equations of second order

100%
Kandilakis D.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1996
|
tom 16
|
nr 2
153-160
EN
In this paper we consider a second order differential equation involving the difference of two monotone operators. Using an auxiliary equation, a priori bounds and a compactness argument we show that the differential equation has a local solution. An example is also presented in detail.
2

Existence and relaxation results for nonlinear second order evolution inclusions

88%
Migórski S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
1995
|
tom 15
|
nr 2
129-148
EN
In this paper we study nonlinear evolution inclusions associated with second order equations defined on an evolution triple. We prove two existence theorems for the cases where the orientor field is convex valued and nonconvex valued, respectively. We show that when the orientor field is Lipschitzean, then the set of solutions of the nonconvex problem is dense in the set of solutions of the convexified problem.
3
Content available remote

Discontinuous quasilinear elliptic problems at resonance

63%
Kourogenis N. C., Papageorgiou N. S.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 2
213-223
EN
In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.
4
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Nonlinear multivalued boundary value problems

63%
Bader R., Papageorgiou N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 1
127-148
EN
In this paper, we study nonlinear second order differential inclusions with a multivalued maximal monotone term and nonlinear boundary conditions. We prove existence theorems for both the convex and nonconvex problems, when $domA ≠ ℝ^{N}$ and $domA = ℝ^{N}$, with A being the maximal monotone term. Our formulation incorporates as special cases the Dirichlet, Neumann and periodic problems. Our tools come from multivalued analysis and the theory of nonlinear monotone operators.
5
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Optimal control of nonlinear evolution equations

51%
Papageorgiou N., Yannakakis N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2001
|
tom 21
|
nr 1
5-50
EN
In this paper, first we consider parametric control systems driven by nonlinear evolution equations defined on an evolution triple of spaces. The parametres are time-varying probability measures (Young measures) defined on a compact metric space. The appropriate optimization problem is a minimax control problem, in which the system analyst minimizes the maximum cost (risk). Under general hypotheses on the data we establish the existence of optimal controls. Then we pass to nonparametric systems, which are governed by nonlinear evolution equations with nonmonotone operators. We prove two existence results for such evolution inclusions, which are of independent interest and extend significantly the results existing in the literature. Then we solve time-optimal and Meyer-type optimization problems. In Section 5, we derive necessary conditions for saddle point optimality in the minimax control problem. We conclude the paper with three examples of distributed parameter control systems.
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