Shape optimization for Dirichlet problems: Relaxed formulation and optimality conditions

1991 ◽  
Vol 23 (1) ◽  
pp. 17-49 ◽  
Author(s):  
Giuseppe Buttazzo ◽  
Gianni Dal Maso
Keyword(s):  
Shape Optimization ◽  
Optimality Conditions ◽  
Dirichlet Problems ◽  
Relaxed Formulation

scholarly journals Applications of implicit parametrizations

2021 ◽  
Vol 66 (1) ◽  
pp. 5-15
Author(s):  
Dan Tiba
Keyword(s):  
Optimal Control ◽  
Nonlinear Programming ◽  
Shape Optimization ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Equality Constraints ◽  
Control Problems ◽  
Global Type ◽  
New Algorithms

We review several applications of the implicit parametrization theorem in optimization. In nonlinear programming, we discuss both new forms, with less multipliers, of the known optimality conditions, and new algorithms of global type. For optimal control problems, we analyze the case of mixed equality constraints and indicate an algorithm, while in shape optimization problems the emphasis is on the new penalization approach.

NUMERICAL SHAPE OPTIMIZATION FOR RELAXED DIRICHLET PROBLEMS

1993 ◽  
Vol 03 (01) ◽  
pp. 19-34 ◽  
Author(s):  
S. FINZI VITA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Optimization Problem ◽  
Numerical Approximation ◽  
Elliptic Partial Differential Equation ◽  
Dirichlet Problems ◽  
Design Problems ◽  
Relaxed Formulation ◽  
Convergent Algorithm

We consider the numerical approximation of optimal design problems governed by an elliptic partial differential equation, in the relaxed formulation recently introduced by Buttazzo and Dal Maso. A discrete optimality condition is derived for the solution of the optimization problem in the finite element setting, by means of which a convergent algorithm is generated. We discuss the numerical results of its application on different examples.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

Sign in / Sign up

Export Citation Format

Share Document