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1
Content available Dynamic programming approach to structural optimization problem - numerical algorithm
Fulmański P., Nowakowski A., Pustelnik J.
Opuscula Mathematica
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2014
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Vol. 34, no. 4
699--724
EN
In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed.
2
Content available The maximum principle for viscosity solutions of elliptic differential functional equations
Karpowicz A.
Opuscula Mathematica
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2013
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Vol. 33, no. 1
99-105
EN
This paper is devoted to the study of the maximum principle for the elliptic equation with a deviated argument. We will consider viscosity solutions of this equation.
3
Content available remote Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints
Meyer C.
Control and Cybernetics
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2008
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Vol. 37, no 1
51-83
EN
We consider a linear-quadratic elliptic optimal control problem with pointwise state constraints. The problem is fully discretized using linear ansatz functions for state and control. Based on a Slater-type argument, we investigate the approximation behavior for mesh size tending to zero. The obtained convergence order for the L²-error of the control and for H 1-error of the state is 1 - ε in the two-dimensional case and 1/2 - ε in three dimensions, provided that the domain satisfies certain regularity assumptions. In a second step, a state-constrained problem with additional control constraints is considered. Here, the control is discretized by constant ansatz functions. It is shown that the convergence theory can be adapted to this case yielding the same order of convergence. The theoretical findings are confirmed by numerical examples.
4
Content available Stability of solutions of infinite systems of nonlinear differential-functional equations of parabolic type
Zabawa T.S.
Opuscula Mathematica
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2006
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Vol. 26, no. 1
173-183
EN
A parabolic initial boundary value problem and an associated elliptic Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations are considered. It is shown that the solutions of the parabolic problem is asymptotically stable and the limit of the solution of the parabolic problem as t → ∞ is the solution of the associated elliptic problem. The result is based on the monotone methods.
5
Content available remote Phragmen-Lindelof type theorems for some semilinear elliptic and parabolic equations
Celebi A., Kalantarov V., Tahamtani F.
Demonstratio Mathematica
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1998
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Vol. 31, nr 1
43-54
EN
Phragmen-Lindelof type theorems for some classes of fourth order semilinear elliptic and se
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