This paper is concerned with, the proof of the existence and the uniqueness theorem for the solution of the state vector of a couple of nonlinear elliptic partial differential equations using the Minty-Browder theorem, where the continuous classical boundary control vector is given. Also the existence theorem of a continuous classical boundary optimal control vector governing by the couple of nonlinear elliptic partial differential equation with equality and inequality constraints is proved. The existence of the uniqueness solution of the couple of adjoint equations associated with the considered couple of the state equations with equality and inequality constraints is studied. The necessary conditions theorem and the sufficient conditions theorem for optimality of the couple of nonlinear elliptic equations with equality and inequality constraints are proved using the Kuhn-Tucker-Lagrange multipliers theorems

This paper is concerned with the proof of the existence and uniqueness theorem for the solution of the state vector of a couple linear elliptic partial differential equations using the Galerkin method, where the continuous classical boundary control vector is given. Also, the existence theorem of a continuous classical boundary optimal control vector governed by the couple of linear elliptic partial differential equation is proved. The existence and the uniqueness solution of the couple of adjoint equations associated with the considered couple of the state equations studied. The derivation of the Fréchet derivative of the Hamiltonian is developed. The necessary conditions theorem of optimality of this problem is proved.

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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.

The paper is concerned with the state and proof of the existence theorem of a unique solution (state vector) of couple nonlinear hyperbolic equations (CNLHEQS) via the Galerkin method (GM) with the Aubin theorem. When the continuous classical boundary control vector (CCBCV) is known, the theorem of existence a CCBOCV with equality and inequality state vector constraints (EIESVC) is stated and proved, the existence theorem of a unique solution of the adjoint couple equations (ADCEQS) associated with the state equations is studied. The Frcéhet derivative derivation of the "Hamiltonian" is obtained. Finally the necessary theorem (necessary conditions "NCs") and the sufficient theorem (sufficient conditions" SCs") for optimality of the state constrained problem are stated and proved.

In this paper, we consider a continuous classical optimal control for systems of nonlinear hyperbolic partial differential equations, with several equality and inequality state constraints. First, the considered continuous classical optimal control problem is discretized into a discrete classical optimal control problem by using the Galerkin finite element method in space and the implicit finite difference scheme in time. The classical continuous controls are approximated by picewise constants. Second the existence of a unique solution of the discrete state equations for fixed discrete classical control is studied. Third, we develop the existence theory for optimality of the discrete classical problem, and the discrete adjoint equations are developed corresponding to the discrete state equations. Finally the necessary conditions and a picewise minimum principle are developed for optimality of the discrete classical problem.

This paper deals with finding the approximation solution of a nonlinear parabolic boundary value problem (NLPBVP) by using the Galekin finite element method (GFEM) in space and Crank Nicolson (CN) scheme in time, the problem then reduce to solve a Galerkin nonlinear algebraic system(GNLAS). The predictor and the corrector technique (PCT) is applied here to solve the GNLAS, by transforms it to a Galerkin linear algebraic system (GLAS). This GLAS is solved once using the Cholesky method (CHM) as it appear in the matlab package and once again using the Cholesky reduction order technique (CHROT) which we employ it here to save a massive time. The results, for CHROT are given by tables and figures and show the efficiency of this method, from other sides we conclude that the both methods are given the same results, but the CHROT is very fast than the CHM.

In this paper the continuous classical boundary optimal control problem of a couple nonlinear 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 nonlinear parabolic partial differential equations for given (fixed) continuous classical boundary control vector. The theorem of the existence of a continuous classical optimal boundary control vector associated with the couple of nonlinear parabolic partial differential equations is proved. The existence of a unique vector solution of the adjoint equations is studied. The Fréchet derivative is derived; Finally The Kuhn-Tucker-Lagrange multipliers theorems is developed and then is used to prove the necessary conditions theorem and the sufficient conditions theorem of optimality of a couple of nonlinear parabolic equations with equality and inequality constraints.