Optimal control of two-dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone

2016 ◽  
Vol 37 (6) ◽  
pp. 1314-1328 ◽  
Author(s):  
Yuan Wang ◽  
Xiaochuan Luo ◽  
Yang Yu ◽  
Haijuan Cui
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuous Casting ◽  
Parabolic Partial Differential Equations ◽  
Cooling Zone ◽  
Two Dimensional ◽  
Secondary Cooling ◽  
Partial Differential ◽  
Secondary Cooling Zone

scholarly journals Optimal Control Method of Parabolic Partial Differential Equations and Its Application to Heat Transfer Model in Continuous Cast Secondary Cooling Zone

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Yuan Wang ◽  
Xiaochuan Luo ◽  
Sai Li
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Adjoint Problem ◽  
Parabolic Partial Differential Equations ◽  
Cooling Zone ◽  
Secondary Cooling ◽  
Continuous Cast ◽  
Partial Differential ◽  
Secondary Cooling Zone

Our work is devoted to a class of optimal control problems of parabolic partial differential equations. Because of the partial differential equations constraints, it is rather difficult to solve the optimization problem. The gradient of the cost function can be found by the adjoint problem approach. Based on the adjoint problem approach, the gradient of cost function is proved to be Lipschitz continuous. An improved conjugate method is applied to solve this optimization problem and this algorithm is proved to be convergent. This method is applied to set-point values in continuous cast secondary cooling zone. Based on the real data in a plant, the simulation experiments show that the method can ensure the steel billet quality. From these experiment results, it is concluded that the improved conjugate gradient algorithm is convergent and the method is effective in optimal control problem of partial differential equations.

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

2015 ◽  
Vol 32 (5) ◽  
pp. 1275-1306 ◽  
Author(s):  
R C Mittal ◽  
Amit Tripathi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Spline Functions ◽  
Parabolic Partial Differential Equations ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Burgers Equations

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value – First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

scholarly journals Existence theorems for optimal control of quasilinear parabolic partial differential equations

1979 ◽  
Vol 21 (1) ◽  
pp. 90-101 ◽  
Author(s):  
S. Nababan ◽  
E. S. Noussair
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Theorems ◽  
Parabolic Partial Differential Equations ◽  
Optimal Controls ◽  
Integral Functionals ◽  
Partial Differential ◽  
Lower Semi ◽  
Quasilinear Parabolic

AbstractThe question on existence of optimal controls for a system governed by quasilinear parabolic partial differential equations which is linear in the control variables is considered. It is shown that whenever the controls converge in the weak * topology of L∞, the corresponding solutions converge uniformly. Using this result and results on lower semi-continuity of integral functionals, existence theorems for optimal controls are proved.

scholarly journals The Classical Continuous Mixed Optimal Control of Couple Nonlinear Parabolic Partial Differential Equations with State Constraints

2021 ◽  
pp. 4859-4874
Author(s):  
Jamil A Ali Al-Hawasy ◽  
Ghufran M Kadhem ◽  
Ahmed Abdul Hasan Naeif
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
State Constraints ◽  
Nonlinear Partial Differential Equations ◽  
Parabolic Partial Differential Equations ◽  
Control Vector ◽  
Partial Differential ◽  
Vector Solution ◽  
Nonlinear Parabolic

In this work, the classical continuous mixed optimal control vector (CCMOPCV) problem of couple nonlinear partial differential equations of parabolic (CNLPPDEs) type with state constraints (STCO) is studied. The existence and uniqueness theorem (EXUNTh) of the state vector solution (SVES) of the CNLPPDEs for a given CCMCV is demonstrated via the method of Galerkin (MGA). The EXUNTh of the CCMOPCV ruled with the CNLPPDEs is proved. The Frechet derivative (FÉDE) is obtained. Finally, both the necessary and the sufficient theorem conditions for optimality (NOPC and SOPC) of the CCMOPCV with state constraints (STCOs) are proved through using the Kuhn-Tucker-Lagrange (KUTULA) multipliers theorem (KUTULATH).

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