scholarly journals Existence of optimal controls for semilinear elliptic equations without Cesari-type conditions

2003 ◽  
Vol 45 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Hongwei Lou
Keyword(s):  
Existence Theorem ◽  
Elliptic Equations ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Elliptic Partial Differential Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
State Control ◽  
Semilinear Elliptic ◽  
The Pontryagin Maximum Principle

AbstractOptimal control problems governed by semilinear elliptic partial differential equations are considered. No Cesari-type conditions are assumed. By proving an existence theorem and the Pontryagin maximum principle of optimal “state-control” pairs for the corresponding relaxed problems, we establish an existence theorem of optimal pairs for the original problem.

scholarly journals Optimal control of nonsmooth system governed by quasi-linear elliptic equations

1997 ◽  
Vol 20 (2) ◽  
pp. 339-346 ◽  
Author(s):  
Gong Liutang ◽  
Fei Pusheng
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Elliptic Partial Differential Equations ◽  
Necessary Condition ◽  
Control Problems ◽  
Nonsmooth Systems

In this paper, we discuss a class of optimal control problems of nonsmooth systems governed by quasi-linear elliptic partial differential equations, give the existence of the problem. Through the smoothness and the approximation of the original problem, we get the necessary condition, which can be considered as the Euler-Lagrange condition under quasi-linear case.

Existence of optimal controls for viscous flow problems

1992 ◽  
Vol 439 (1905) ◽  
pp. 81-102 ◽  
Keyword(s):  
Optimal Control ◽  
Flow Control ◽  
Viscous Flow ◽  
Existence Theorem ◽  
General Class ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Optimal Controls ◽  
Flow Problems ◽  
Existence Of Optimal Controls

A class of optimal control problems in viscous flow is studied. Main result is the existence theorem for optimal control. Three typical flow control problems are formulated within this general class.

scholarly journals A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems

2020 ◽  
Vol 77 (3) ◽  
pp. 831-869
Author(s):  
Veronika Karl ◽  
Ira Neitzel ◽  
Daniel Wachsmuth
Keyword(s):  
Optimal Control ◽  
Weak Convergence ◽  
Original Problem ◽  
Optimal Control Problems ◽  
State Constraint ◽  
Stationary Points ◽  
Control Problems ◽  
Constrained Optimal Control ◽  
Semilinear Elliptic ◽  
Augmented Lagrange Method

Abstract In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

scholarly journals Analysis of control problems of nonmontone semilinear elliptic equations

2020 ◽  
Vol 26 ◽  
pp. 80
Author(s):  
Eduardo Casas ◽  
Mariano Mateos ◽  
Arnd Rösch
Keyword(s):  
Elliptic Equations ◽  
Nonlinear Term ◽  
Optimal Control Problems ◽  
Semilinear Elliptic Equation ◽  
State Equation ◽  
Order Conditions ◽  
Control Problems ◽  
Local Optimality ◽  
Convection Term ◽  
Semilinear Elliptic

In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.

scholarly journals Sufficiency for Purely Essentially Bounded Optimal Controls

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 238
Author(s):  
Gerardo Sánchez Licea
Keyword(s):  
Nonlinear Dynamics ◽  
Optimal Control ◽  
Optimal Control Problems ◽  
Sufficient Conditions ◽  
Control Problems ◽  
Optimal Controls ◽  
Embedding Theorems ◽  
State Control ◽  
Discontinuous Solutions ◽  
Piecewise Continuous

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.

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