تخمين هاي خطاي روش عناصر متناهي در حل يك مسئله كنترل بهينه مقيد به معادله برگرز

چكيده انگليسي :

Finite element error estimates for an optimal control problem governed by the Burgers equation Mehrnoosh Salehi m salehichegeni@math iut ac ir 2017 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir 2010 MSC 65N12 49J20 Keywords Finite element method Error estimates Optimal control problem Burgers equation Abstract In the present thesis we aim to study an optimal control problem constrained to a Burgers equation whilethe order of convergence for the optimal control variable in the L2 norm by using piecewise linear polynomials isinvestigated Solving optimization problems subject to some constraints involving distributed parameter systemsis one of the most challenging problems in the context of industrial medical and economical applications Almostall of these applications involve modeling based on some di erential equations and can be classi ed as topologyoptimization problems or optimal control problems in in nite dimensional spaces On the other hand since theBurgers equation is a one dimensional simple model for the nonlinear convection di usion phenomena optimalcontrol problems constrained to the Burgers equation are important models for the analysis and the developmentof numerical algorithms In this thesis rst of all we gather some preliminaries such as de nition of somesuitable function spaces and representing some useful de nitions and theorems Next we brie y comment onthe properties of the optimal control problem We describe the state equation and weak formulation of thehomogeneous Dirichlet problem for the Burgers equation It must be interesting that the Burgers equationhas a unique solution depending on the continuous right hand side function In order to express existence ofsolution for the optimal control problem a stronger assumption is considered and based on this assumptionone able to prove that considered optimal control problem has a solution Despite the strict convexity of theobjective functional and uniqueness of the solution of the state equation uniqueness of the optimal control cannotbe guaranteed since the feasible set is not necessarily convex hence we investigate the rst order necessary andsecond order su cient condition for local solutions of the optimal control problem Both conditions play importantroles in the derivation of error estimates On the other hand Burgers equation is approximated by using the nite element method and the corresponding error of convergence is dealt with A uniform mesh is constructedand the corresponding nite dimensional space is built Next we restate the nite element approximation ofthe optimal control problem with the aid of the piecewise linear functions for approximation of the state andthe control variables We reformulate the discrete optimal control problem associated to the problem and thena discrete admissible control space is rede ned The theory is nished by making a further error analysis bytaking into account a stronger assumption on the structure of the optimal control which is allowed to derive abetter interpolation error in the L1 norm which is crucial to make an improvement in the overall error estimate Finally for the sake of illustration of the theory some numerical tests are developed where the exact solutions ofthe optimal control problem are known One of the test problems has a continuous control whereas the other onehas a piecewise continuous control Discretized optimization problems are solved by using the projected BFGSmethod which implemented in MATLAB

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تخمين هاي خطاي روش عناصر متناهي در حل يك مسئله كنترل بهينه مقيد به معادله برگرز

روش عناصر متناهي , تخمين خطا , مسئله كنترل بهينه , معادله برگرز