Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations

2013 ◽  
Vol 74 (12) ◽  
pp. 1969-2006 ◽  
Author(s):  
B. M. Miller ◽  
E. Ya. Rubinovich
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Space Time ◽  
Control Problems ◽  
Discontinuous Solutions ◽  
Singular Space ◽  
Time Transformations

scholarly journals Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xenia Kerkhoff ◽  
Sandra May
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Burgers Equation ◽  
Adjoint Equation ◽  
State Equation ◽  
Space Time ◽  
The State ◽  
Control Problems ◽  
Viscous Term ◽  
Distributed Optimal Control Problems

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

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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