Necessary conditions on the extremum for convex differential inclusions with phase constraints

1985 ◽  
Vol 38 (3) ◽  
pp. 733-738
Author(s):  
V. N. Gurov
Keyword(s):  
Differential Inclusions ◽  
Necessary Conditions ◽  
Phase Constraints

scholarly journals Stratified Necessary Conditions for Differential Inclusions with State Constraints

2013 ◽  
Vol 51 (5) ◽  
pp. 3903-3917 ◽  
Author(s):  
Piernicola Bettiol ◽  
Andrea Boccia ◽  
Richard B. Vinter
Keyword(s):  
Differential Inclusions ◽  
State Constraints ◽  
Necessary Conditions

scholarly journals Overview on the Pointwise Constrained Liapunov Vectorial Convexity Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Clara Carlota ◽  
Sílvia Chá ◽  
António Ornelas
Keyword(s):  
Optimal Control ◽  
Open Problem ◽  
Differential Inclusions ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Real Life ◽  
Convexity Theorem ◽  
Realistic Case ◽  
Life Problems ◽  
Pointwise Constraints

In applications of the Calculus of Variations, Optimal Control and Differential Inclusions, very important real-life problems are nonconvex vectorial and subject to pointwise constraints. The classical Liapunov convexity theorem is a crucial tool allowing researchers to solve nonconvex vectorial problems involving single integrals. However, the possibility of extending such theorem so as to deal with pointwise constraints has remained an open problem for two decades, in the more realistic case using variable vectorial velocities. We have recently solved it, in the sense of proving necessary conditions and sufficient conditions for solvability of such problem. A quick overview of our results is presented here, the main point being that, somehow, convex constrained nonuniqueness a.e. implies nonconvex constrained existence.

Optimal Control in the Presence of Nondifferentiable State Constraints

1976 ◽  
Vol 98 (4) ◽  
pp. 432-439
Author(s):  
E. D. Eyman ◽  
D. P. Sudhakar
Keyword(s):  
Optimal Control ◽  
Integral Equation ◽  
Mathematical Programming ◽  
State Constraints ◽  
Necessary Conditions ◽  
Adjoint Equation ◽  
Phase Constraint ◽  
Maximal Condition ◽  
Phase Constraints ◽  
Differential Control

Necessary conditions are derived for optimality of differential control processes in the presence of nondifferentiable state (or phase) constraints. The techniques of general Mathematical Programming and the Dubovitskii-Milyutin Theorem are employed. The necessary conditions derived are in the form of an adjoint integral equation and a pointwise maximal condition. It is found that the gradient of the state (or phase) constraint can be replaced by the Gateaux differential of a certain form in the adjoint equation.

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