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).

scholarly journals The Continuous Classical Boundary Optimal Control of a Couple Linear Of Parabolic Partial Differential Equations

2018 ◽  
Vol 29 (1) ◽  
pp. 118
Author(s):  
Jamil Amir Al-hawasy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Control ◽  
Parabolic Type ◽  
Parabolic Partial Differential Equations ◽  
Control Vector ◽  
Partial Differential ◽  
Vector Solution ◽  
Classical Boundary ◽  
The Galerkin Method

In this paper the continuous classical boundary optimal problem of a couple linear partial differential equations of parabolic type is studied, The Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution of a couple linear parabolic partial differential equations for given (fixed) continuous classical boundary control vector. The proof of the existence theorem of a continuous classical optimal boundary control vector associated with the couple linear parabolic is given. The Frechet derivative is derived; finally we give a proof of the necessary conditions for optimality (boundary control) of the above problem.

An acceleration scheme for deep learning-based BSDE solver using weak expansions

2020 ◽  
Vol 07 (02) ◽  
pp. 2050012
Author(s):  
Riu Naito ◽  
Toshihiro Yamada
Keyword(s):  
Deep Learning ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Nonlinear Partial Differential Equations ◽  
High Dimensional ◽  
Parabolic Partial Differential Equations ◽  
Weak Approximation ◽  
Learning Method ◽  
Partial Differential

This paper gives an acceleration scheme for deep backward stochastic differential equation (BSDE) solver, a deep learning method for solving BSDEs introduced in Weinan et al. [Weinan, E, J Han and A Jentzen (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5(4), 349–380]. The solutions of nonlinear partial differential equations are quickly estimated using technique of weak approximation even if the dimension is high. In particular, the loss function and the relative error for the target solution become sufficiently small through a smaller number of iteration steps in the new deep BSDE solver.

scholarly journals Nonlinear parabolic problems in unbounded domains

1979 ◽  
Vol 82 (3-4) ◽  
pp. 201-209 ◽  
Author(s):  
Guy Mahler
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Unbounded Domains ◽  
Parabolic Problems ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Nonlinear Parabolic Problems ◽  
Existence Of Weak Solutions ◽  
Nonlinear Parabolic

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

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