A geometrical proof of the maximum principle for systems represented by difference-differential equations

1973 ◽  
Vol 11 (3) ◽  
pp. 245-248 ◽  
Author(s):  
M. A. Connor
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Geometrical Proof ◽  
The Maximum Principle

Phragmèn-Lindelöf and Comparison Theorems for Elliptic-Parabolic Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 864-871
Author(s):  
J. K. Oddson
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Comparison Theorems ◽  
Parabolic Differential Equations ◽  
The Maximum Principle ◽  
Comparison Functions

Theorems of Phragmèn-Lindelöf type and other related results for solutions of elliptic-parabolic equations have been given by numerous authors in recent years. Many of these results are based upon the maximum principle and the use of auxiliary comparison functions which are constructed as supersolutions of the equations under various conditions on the coefficients. In this paper we present an axiomatized treatment of these topics, replacing specific hypotheses on the nature of the coefficients of the equations by a single assumption concerning the maximum principle and another concerning the existence of positive supersolutions, in terms of which the theorems are stated.

scholarly journals The maximum principle for a type of hereditary semilinear differential equation

1994 ◽  
Vol 36 (2) ◽  
pp. 194-212
Author(s):  
Feiyue He
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
The Maximum Principle ◽  
Semilinear Differential Equation

AbstractAn optimal control problem governed by a class of delay semilinear differential equations is studied. The existence of an optimal control is proven, and the maximum principle and approximating schemes are found. As applications, three examples are discussed.

Application of Double Side Approach Method to the Solution of Fully Developed Laminar Flow in Duct Problems

2013 ◽  
Vol 29 (3) ◽  
pp. 471-479
Author(s):  
H.-W. Tang ◽  
Y.-T. Yang ◽  
C.-K. Chen
Keyword(s):  
Maximum Principle ◽  
Mathematical Programming ◽  
Laminar Flow ◽  
Differential Equations ◽  
Slip Boundary Condition ◽  
Optimization Method ◽  
The Maximum Principle ◽  
Weighted Residuals ◽  
Slip Boundary ◽  
Approach Method

AbstractThe double side approach method combines the method of weighted residuals (MWR) with mathematical programming to solve the differential equations. Once the differential equation is proved to satisfy the maximum principle, collocation method and mathematical programming are used to transfer the problem into a bilateral inequality. By utilizing Genetic Algorithms optimization method, the maximum and minimum solutions which satisfy the inequality can be found. Adopting this method, quite less computer memory and time are needed than those required for finite element method.In this paper, the incompressible-Newtonian, fully-developed, steady-state laminar flow in equilateral triangular, rectangular, elliptical and super-elliptical ducts is studied. Based on the maximum principle of differential equations, the monotonicity of the Laplace operator can be proved and the double side approach method can be applied. Different kinds of trial functions are constructed to meet the no-slip boundary condition, and it was demonstrated that the results are in great agreement with the analytical solutions or the formerly presented works. The efficiency, accuracy, and simplicity of the double side approach method are fully illustrated in the present study to indicate that the method is powerful for solving boundary value problems.

scholarly journals REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL

2020 ◽  
pp. 7-26
Author(s):  
Shahla Rasulzade ◽  
◽  
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Control Problems ◽  
The Maximum Principle ◽  
Distributed Parameters

One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

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